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User Manual for IDEA 1.5 - optics.tugraz.at

1. 2 2 7 Internal Format of Colour Palette File 2 2 2 8 Internal Format of Mask File 2 2 9 Internal Format of File Pool File 2 2 10 Internal Format of Projection Angle File 22 11 External Graphic Formats 2 0042 4244 dee bbe PPE ES 2412 paying AS CIM data one 4 eva aga ade be bed G4 eS 2 3 Handling of floating point exceptions 2 4 Input Macros and Operators ss lt s 2 4 2 aea ee eee 2 5 The Graphical User Interface of IDEA 2 5 1 Data Selection in Active Window 20 2 The Status Bak eects 68s aaa a 2 0 8 Protocol Window lt lt lt sia diendo doe GEE ed de e a IDEA Menu Entries ae TUS Sora aa A ee ee E eee GS Sola News eae ok Ge aoe ASES A SZ Open 4 60 Bead ebehe awed Chee Eee BS A AA E OO00 0000 2000 CONTENTS 3 2 3 3 3 5 LA DIVE corsa ars di a a 26 3ko A A Reh eee 26 3 1 6 Convert Copy to Image so e casares seyta eap akt 26 3 1 7 Convert Copy to Picture o o e 26 3 18 Convert Copy to 2D Data o 27 3 1 9 Change Filetype sss s 22 6444 442406608 2 244d do 27 3 110 Protocoles s sox 2 s ele eR a ee BERS RE RR 27 S111 Preferences o o soe esc ee ee Pee eee a ee ae ees 27 A Stk ne ee ey a oe ae are ate a oD ETE 30 Bile Pool s 642 4 244 4044 4 daa e es 30 32 1 Add Piles og iris oe eee ER ee eh Owe GS 30 322 Adda File s o i sensi ib ye e
2. Figure 3 34 Two directional unwrapping scheme Starting from the vertical column including starting point x y all adjacent columns are unwrapped for each quadrant here the upper left To determine order in the next unidentified pixel p the already known step function values snp and Snv at inner pixels are taken into account This way an error check is provided AA A ANA Figure 3 35 Spiral scanning method From the central starting point x y data is unwrapped sequentially by spiraling around a growing region of data with known step function values To provide an error check the change of phase order from all available adjacent pixels are taken into account marked by the arrows If more than 50 agree the new step function value at unidentified pixel p is set accordingly Otherwise no decision can be made and p gets the reserved value for unreliable data The big arrow shows further scanning direction wrapped phase maps which could be handled by the 2D scan as this is the fastest branchcut option see Sec 3 10 10 1 3 10 2 Spiral Scan Method This method is more efficient than the two directional scan As for the 2D scan starting point s must be defined for unwrapping where phase data is reliable in a vicinity of 3 x 3 pixels For a phase map with several regions think of it as isles isolated from each other by areas of invalid data for each isle an
3. 64 3 6 6 Radial Data Convoluti0L o 66 3 0 7 Radial Data ART 4 264 445 444 misa A 66 3 6 8 Create Radial 2D Data o 66 dul EOMOSTApMY gt der a a A A 67 3 1 1 Get Single Projection lt a caas 085 05508544444 68 3 7 2 Build Tomographic Input File 68 3 7 3 Edit Tomographic Input File 70 3 7 4 Interpolate Projections 72 S00 COMVONHION sides dd ome ees 72 330 ARI a cas aaah GS 74 oat Calculate Projection Data ca wea dow ee ee eee ad 77 Sor OPE o s Saa ee ES Bee RWS eee ee ee Se RE RSS Ee 77 Seb Zao Padding 2 eae od AA 78 3 8 2 Gerchberg Fringe Extrapolation 78 38 3 Forward FET zos 85 6454 424 4AASEEE GEES 79 3 8 4 Filtered Back FFT to 2D Mod 2Pi Data 79 3 8 5 Filtered Back FFT to Image 79 3 8 6 Filtered Back FFT to 2D Real Data 80 38 7 Calculate 2D Mod 2Pi Dtos a ease aste DRESS 80 3 38 Calculate 2D Phase Data u sore ca ba dee ee ge ee es 80 3 0 9 Recalculate Image lt esaa a ee ek ee ee a ee 81 3 8 10 Recalculate 2D Data o 81 35 11 Show Real Part Only eos ods wea e hie d ds a a wD 81 3 8 12 Show Imaginary Part Only oo caase assiw o 81 3 8 13 Show Amplitide o ss ra ii 2668444046444 dete es 81 OLA Show Phase ca 4 oia ea EI a RE eS 81 39 Phase shift o s cia ma cd a t a tos bob SL aad 83 3 9 1 Three Frame Tec
4. Minimum Number of Valid Neighbours f I Always Allow Replacement of Invalids by Filter Result VV Allow Replacement of Invalids Only for Last Iteration OK Cancel Figure 3 20 Dialog for Trigonometric Filter The related dialog allows following entries e Filter Selection Select the kind of filter to be applied to the sine and cosine field Convolution with Unit Kernel Neighbourhood Mean The a linear spatial filter with unit kernel see Eq 3 1 is applied which is equivalent to calculate for each pixel the mean of the neighbourhood values Median Filter A median filter described in Sec 3 5 4 is applied Convolution with User Kernel Define any other kernel for linear spatial filtering in the dialog described at Sec 3 5 3 All filter parameters and options are then set in the following dialogs e Filter Width Width of the filter kernel e g horizontal extent of the neighbourhood taken into account for filtering e Filter Height Height of the filter kernel e g vertical extent of the neighbourhood taken into account for filtering e Iterations Number of iterations for the trigonometric filter process 3 5 FILTERING 57 e Invalid Data Treatment Rigourous Results only for all valid Neighbourhood A single invalid value within the neighbourhood of a pixels results in an invalid filter response Flexible Filterkernel adapts to valid Neighbourhood Only valid data within
5. 3 4 VIEW 47 Select Palette Eg eee Mapping Minimum C Grey C Jet2 C Spring C Red o e e c Order HSY C Summer Green E Prism C Spectrum C Autumn Blue e C Diff C Hot CWinter Pink idea C Cool C Bone C Read File N C Jet RedShade Copper Filename fc ideapalettes user pal OK Cancel Browse Figure 3 13 Dialog for Changing Palette 3 4 7 Invert Palette Inverts visualization by reversing order of colours within the palette The colour designated to maximum is then used for minimum and vice versa 3 4 8 Refresh Causes the operating system to redraw all IDEA windows which is necessary in rare cases when windows are not displayed properly after creation 3 4 9 Slide Show With the following submenu items you can configure and run a slide show created in File New Slide Show In principle this is a collection of pictures which can be displayed one after the other temporally separated by a definable timer interval With short interval this results in an animation The different pictures may have different Colour Palettes and size The final size of the Slide Show window is that of the biggest included picture Smaller elements are displayed in the upper left corner without refreshing window contents There is a size limitation here for the pictures to be shown which is somewhat smaller than the system display size Larger pictures are truncated to the size limits To manually lea
6. The common file ID PO is followed by the pathstyle parameter which can be either relative or absolute A relative pathstyle means that the filenames are given relative to the location of the File Pool In case of the absolute style the full paths are required The character for the folder separator can be either a backslash or a slash Both are accepted and converted to the appropriate style of the operating system in use 2 2 10 Internal Format of Projection Angle File This file contains raw ASCII data without header and ID The projection angles must be separated by a line separator or by an empty space 2 2 11 External Graphic Formats In addition to the self defined plain file formats we support three external formats Bitmap bmp X Pixmap xpm and the not so common IMAGING TECHNOLOGY ITEX format pic not to be confused with the better known PC Paint and Lotus PIC format with same extension The ITEX format has true eight bit binary pixel data giving directly the gray scales It differs from our own Image format img only by its longer header offset to image data 64 byte offset to width 4 bytes offset to height 6 bytes On the contrary the Bitmaps include a palette of 256 RGB encoded colours see Sec 2 1 1 For the format of Bitmap see 29 The X Pixmap has an unlimited colour palette and must be converted to an IDEA picture for further use A detailed description of the X Pixmap fo
7. Interpolation of Additional Projections For application of the convolution method see Sec 3 7 5 projections with equidis tant angles are required Following Sec 3 7 4 we interpolate 30 projections using To mography Interpolate Projections Expanding the number of projections from 8 to 30 should sufficiently eliminate the artifacts in this example Reconstruction with Convolution Method For reconstruction of the projections we apply the Filtered Backprojection with Con volution Tomography Convolution to the File Pool obtained from interpolation We use a Hanning Parameter of 0 54 and a Length Unit of 20 see Sec 3 7 5 For one 4 2 EXAMPLE 2 PHASE SHIFTING 117 idea File File Pool Edit View Filtering Abellnwersion Tomography 2D FFT Phase Shift Phase Mask Information Window Help a JOS eel MEA S S S al mie o Framom S 420 Framer S T OL p Mask Minimal Required Visibility Minimal Value for Imax Imin T Apply Mask File Mask Flie 63 Figure 4 4 Screenshot of IDEA demonstrating Phase Shift method From the three phase shifted interferograms Frame0 img Frame120 img and Frame240 img the phase modulo 27 is calculated using the 120 Three Frame Technique The corresponding images are selected using the Dialog Phase Shift The resulting phase modulo 27 is shown as PSI m2p The unwrapped phase Unwrap
8. 2D Fast Fourier Transformation 2D FFT Spatial Phase Shifting and methods for Speckle interferometry with phase shifting in reference state only Reconstruction algorithms for digital holograms Phase Unwrapping by Scanning Methods Branchcut or Fast Cosine Trans formation Abel Inversion by f Interpolation Fourier Synthesis or Backus Gilbert Technique Tomographical Reconstruction by Convolution or Algebraic Reconstruc tion Technique e Additional Features 1 3 REQUIREMENTS Multiple File Manipulation Batch Operation Animation Interactive Pseudo Colour Viewing 2D Data Field and Image Manipulation Spatial Filtering Interferogram Simulation 1 3 Requirements IDEA is a single file program using only the executable dea exe though pref erences can be saved and are then loaded from this file at startup No reg istry entries are made For Windows Systems the only requirement is the file CTL3D32 DLL in the System directory If the appropriate version of this file was not provided by your Windows 95 98 NT installation you can download it e g at http www chiropteraphilia com ctl3d index html This problem is common to many applications so searching the web brings up a lot of helpful links The minimal requirements for IDEA are e Pentium Processor or CPU with comparable power e 32MB of RAM e Graphic Card providing resolution 1024x768 at High Colour e Windows 95 98 NT 200
9. 3 27 as accurate as possible 3 7 5 Convolution This method approximates the Radon Inversion Eq 3 20 by applying the math ematical procedure of convolution eliminating direct calculation of the derivation within the integral which is critical for discrete measured data Since performing a convolution always goes together with a filtering effect the term for this method used in literature is Filtered Backprojection with Convolution A closer examination of Eq 3 20 reveals that this procedure can be interpreted as e a differentiation of the projections h p 0 to h p 0 e a subsequent Hilbert transformation of h p 0 and e finally a backprojection It can be shown 18 that both derivation and Hilbert Transformation can be approx imated by a simple convolution of projection data yielding A 00 T o0 re jm f af MpD aros 9 dp 62 0 00 with a fixed convolving function ners P RU cos 2nUu dU 3 23 This function contains the real monotonically non increasing function window func tion FA U which has to satisfy the following relations 0 lt F4 U lt 1 for U lt A 2 FA U 0 for U gt A 2 3 24 lim FA U 1 The implemented algorithm uses the recommended generalized Hamming Window 27U FA U a 1 a cos A 3 25 3 7 TOMOGRAPHY 73 with 05 lt a lt l 3 26 The implemented algorithm assumes equidistant projection angles according to Eq 3 27 so definit
10. Ref 90 cvPSIDatalmage img 5 File Pool Mode Rarden ersDetaimegezimg SS E Phase Stepped Reference Interferograms are at top ef alma Ime Phase Stepped Reference Interferograms are at bottom Ref 270 le PSIDatallmage3 img El fe Frelininary subtraction ct Object Interfercorenn Image Files Object Frame E PSIDatalObject img E Images taken from Filepao E Mask Mask Minimal Required Visibility 9 p Minimal Required Visibility 96 p Minimal Value for Imax Imin p Minimal Value for Imax lmin p IV Apply Mask File IV Apply Mask File Mask File fe PSiData Object msk Mask File le PSIDatalObjectmsk Figure 3 33 Dialog for the Speckle 4 1 method a Standard form of dialog b File Pool form of the dialog is introduced spatially e g from pixel to pixel within one image For a better under standing of the principle the experimental setup and procedure for the measurement of an object deformation shall be given as an example The diffuse reflecting object e g its speckle pattern is imaged with a lens to the CCD chip A nearly plane reference wave usually a laser beam emitted from a optical fibre at some distance to the camera actually forming a spherical wave front as the fibre is a quasi point light source is brought to interference with the irregular wave front from the object on the CCD The distance of the fibre to the center of the system s imaging aperture is then adjusted in a way that the
11. e Invalids Substitute input field Define a value which substitutes the invalid phase data e Number of Partners input field Define the number of residues n y with opposite sign which must be found before the scanning procedure is terminated see step 2 in the list above e Border Cost Factor input field To imply a handicap for connections to the border define a value here which will be multiplied to the cost calculated according to Eq 3 59 e Set Branchcuts to Invalid checkbox Check this to set all pixels below a branchcut to invalid in the unwrapped phase If not checked a final unwrapping procedure will add a multiple of 27 to these pixels to minimize deviations from neighbours at either side of the branchcuts Regard this as a mere visual enhancement of the result as there are no criterions to select the best result at branchcut pixels e Show Residues in Extra Image checkbox Check this to create an image where the residues are represented by pixels with colour e g value depending on the sign blue pixels values 1 mark negative residues white pixels values 252 mark positive residues e Create Mask to Show Branchcuts checkbox With this box checked a mask is created in the data field with unwrapped phase data which covers the branchcuts excluding the residues The same is done in the image showing the residues if created After finishing calculation information about overall squared cutlength is given in the
12. 3 4 3 Mirror Mirror Image or 2D Data Select axis in the submenu 3 4 4 Display Mode Changes scale of visualization of 2D Data All data z is virtually transformed to 2 by a scaling function before mapping see Sec 2 1 6 for visualization is done the data itself is not changed in any way therefore virtual transformation In a dialog you can choose between following mapping functions e Modified logarithmic scale In 1 z 4 e Quad Root scale 2 z e Linear scale No transformation used to re establish default state 3 4 5 Change Colour Palette Uses different Colour Palette see Sec 2 1 1 for visualization In the dialog shown in Fig 3 13 select one of the 22 predefined palettes or choose Read File to import Palette data for file format see Sec 2 2 7 Use Browse button to open standard file selector At the left side of the dialog the Mapping Minimum and Mapping Maximum values can be defined input of macros min and max is allowed see Sec 2 4 As described in Sec 2 1 6 the standard colours are then used to view only the selected data range Data out of range are marked with underflow or overflow colours 3 4 6 Extend Palette Remaps data for visualization using full range between minimum and maximum of all data This is equal to type min into text field for Mapping Minimum and max into field for Mapping Maximum in dialog of Fig 3 13
13. As for all other inverse problems those methods proved to be best suited which form a model function in the f domain from now on this term is used for radial domain and fit the parameters to the measured curve after applying forward transformation Eq 3 5 In comparison fitting and therefore smoothing in the h domain domain of integral data is susceptible to propagation errors especially in the center region Keep in mind that any fitting should be applied by the inversion algorithm itself or afterwards but never in the h domain Two such algorithms are implemented in IDEA the so called f Interpolation and the Fourier method which was developed at the institute for Experimental Physics Technical University Graz and for the first time introduced in 37 More detailed de scriptions are given in the specific menu explanation but for complete understanding of the techniques please refer to the mentioned publications With all these algorithms one has to gain some experience due to strong dependence of the result on the input parameter There is always concurrence between effective smoothing and small deviation between integral and measured data The better the smoothing the higher the overall deviation and vice versa In general for small deviation the inversion shows noise with high amplitude To help the user to develop the necessary feeling which of the rather different results may be the best one we implemented a routine
14. Depending on the purpose of the dialog the z values of the selected points the coordinates x y of the selected points or both selectable in this case with checkboxes in small dialog can be saved as ASCII data format x value space y value newline Which data is available for save is indicated by the label of the button e Load button Load either z values or coordinates x y from an ASCII file with the format described for the Save button e OK button Finishes the input and closes the dialog e Cancel button Closes the dialog discarding any input 3 3 16 Copy Selection Overtakes any selection rectangle line crosshairs polygon or multi point selection currently active in a different 2D Data field or image For rectangles lines and crosshairs different sizes of fields are allowed but be aware that only parts of se lections in range are overtaken For polygons and multi point selection all corners or points must be in range of the receiving window Apart from the multi point selection the selection of the receiving window are overwritten by the overtaken selection The multi point selection however is expanded by adding the points selected in the other window The procedure of copying a selection is similar to that in Sec 3 3 7 3 3 EDIT 45 Define Values at Coordinates gt 3 CU E p CS 2 8 8 Figure 3 12 Dialog to enter coordinates for
15. It is a compendium of pro grams that have been developed and used at the Graz University of Technology since the 1980 s covering phase stepping and Fourier domain evaluation as well as algo rithms for phase determination from Speckle interferograms and digital holograms The resulting modulo 27 data can be unwrapped by scanning methods branchcut or by a cosine transform technique To integral data Abel and tomographic algo rithms for the reconstruction of refractive index fields can be applied The software works with 8 bit Bitmaps ASCIl data and binary data in double format Interfero gram simulation image filtering as well as data field manipulation and pseudo colour visualization are included as tools Any two dimensional data can be manipulated in various ways like adding and multiplying constants averaging subtracting two fields or tilting the field Results can be visualized by using one of several 256 colour palettes Finally all the functions mentioned above can automatically be applied to a user defined collection of data files hence the program provides a powerful tool to deal with large number of acquired images or data The development of IDEA was funded by the Austrian Government within the frame work of an awarded grant covering activities on optical metrology in mechanical en gineering FWPF 457 1 2 Table of Features e Main Software Features Phase Shifting Algorithms Carre 3 4 5 and 7 Step Methods
16. n N with for 1 3 8 falr 1 1 cos nr 3 9 3 6 ABEL INVERSION 62 The Radius of whole integral distribution is denoted by R Inserting Eq 3 7 in Eq 3 5 leads to an integral which can be solved numerically The approximation of the measured integral distribution h r is done by determining the amplitudes An by applying the least square criterion The only problem here is the numerical computation of the transformation integrals including the cosine function To save time these integrals are pre calculated and stored as coefficients of 256 fitting splines unnoticeable by the user apart from the amount of memory allocated by IDEA Depending on the setting of File Preferences Abel Inversion Strict Symmetry Check ing you are warned if the integral 1D distribution does not fulfill all requirements for Abel Inversion The dialog appearing before calculation with Fourier Method is similar to that in Fig 3 23 explanation in Sec 3 6 2 but instead of the number of polynomials you have to define Minimum and Maximum Order of Model Function In Eq 3 7 these parameters are N and N and determines the range of spatial frequencies used to build f r The upper limit N should correspond to the bandlimit of measured data easy to determine if the phase data was evaluated by Fourier method as higher frequencies just reconstruct noise with astonishing high amplitude at regions near the center In most cases the maxi
17. the ASCII contents of this file After changing these settings it is possible to save the new preferences in a file with the same structure as idea cfg using the standard file selector Overwriting of idea cfg will redefine the startup settings Set Default Preferences Ignores any settings of preferences read from file and uses IDEA s internal default settings for maximal safety in data handling during work Set Preferences for Fast Works Ignores any settings of preferences read from file and uses IDEA s internal alternative settings for maximal data handling speed during work 1 0 3 2 FILE POOL 3 1 12 Exit Finish your work with IDEA and leave the program 3 2 File Pool Life would be easy if all problems could be solved by only one image acquisition and its evaluation In reality one has to bother with multi directional or temporally resolved measurements to mention just a few experimental requirements which leads to a large amount of image data Having spent hours applying the same evaluation step again and again on dozens of images or data fields we decided to avoid such a work with IDEA The solution we are able to present is the File Pool The idea was to allow the user to collect files in a pool and to apply most of the algorithms used for phase evaluation editing filtering and reconstructing to all collected files The File Pool itself includes just the filenames of the collected files not the data see Sec
18. 105 3 10 14 Mod 2Pi from 2D Phase Data 000 105 3 10 15 Remove Linear Phase Shift 00004 106 3 10 16 Remove Fitted Linear Phase Shift 106 MaSK d Gon ok Se dere ek ee ER ot ws Be AA 106 S111 Copy Mask lt xi 6 ies 4 4 aaa Obes i awa a 106 IIE AO Mis ic es ek SE oe eG we GOR ee 106 3 11 38 Save Mask lt lt eee REE a i 106 3 11 4 Remove Mask ee a 106 SUL Invern Mask ic a a aa oa ee da 106 3 11 6 Square Pen Enabled cocos ee ee ee wo 107 3 11 7 Circular Pem enabled o 2 oca 06 465 4 4444 e 4245 107 311 8 Mask Pen Width sie RR De OO 107 3 119 Mask Selected Points soste wie ada doe ey ee ee 107 SILO Mask Lines 2 aw ek AA ed de 107 S110 Mask Inside Ared 2 4448540056222 22 RRR eae De 107 3 11 12Mask Outside Area 2 i 107 3 1113 Mask Polygon lt 4 ra a A A AA 107 3 11 14Mask Minimal Values eens 108 311215 Mask Maximal Valles 2000 AAA 108 SLT 6 Mask Invalid Values cios aa da 108 ITLL Mask Interyal ra cc as e a oS 108 3 11 18 Mask Zero Frequency o 108 3 11 19 Mask Nyquist Frequencies e 108 3 11 20 Substitute Masked Values ee ee 108 3 11 21Symmetrze Mask ue eye rr oe oe owe oS 108 3 11 22 Mirror Mask Horizontally 108 3 11 23 Mirror Mask Vertically lt lt o 109 a sa r a owed et ed a ee ee be eS Ae A 109 3121 Wine Di
19. Data abl are checked for symmetry with axis at centerline and data outside of left border line is ignored Otherwise only data between left border line and center line is taken into account for calculation Symmetry is just assumed Open File Assuming Little Endian Byte Order With this menu item activated default on PCs the files to open are assumed to have little endian byte order eg files created at PCs Deactivate it if you want to open files with big endian byte order If you are not sure which byte order is used by 3 1 FILE 29 your machine look at the pre check entries in the first lines of the protocol window see 2 5 3 Save File with Little Endian Byte Order With this menu item activated default on PCs files are saved with little endian byte order Otherwise big endian byte order is used As long as this switch is in the same state as the one in the previous menu item you will not have any problems at least as long as you do not export data to other machines Obviously to have different states is very dangerous For exporting files with different byte order it is recommended to convert all files using a File Pool Substitute Invalid Data for Raw ASCIT Export The string representing invalid data in ASCII files created by exporting data in the raw ASCII format e g without header see Tab 2 3 is substituted by the string defined in the dialog box This has been included since invalid data is the mos
20. Format of Plain Image 2D and 1D Data For Images 2D Data and 1D Data we defined a most simple internal format The header of these classes consists of the ID and a information about the size of the saved data field or data vector For Images and 2D Data the size information is given by both width and height of the field Here width is the number of data per line and height is the number of lines The data is assumed to be stored line after line and this is also true for the visualization Lines are shown from top to bottom in the same order as they are read from file The Frequency File is excepted from this 2 2 FILE TYPES AND FORMATS 14 simple concept It includes complex spatial frequency data in the so called packed order See Sec 2 2 4 for more information For 1D Data the size information consists only of the number of data within the vector to read in In all cases data beyond the given size is ignored If less data is included than size parameters suggest an error message occurs and the file cannot be loaded The ID and size parameters are always in ASCII format even for binary files to avoid possible allocation faults if the file should be opened at machines with wrong byte order The common header line structure for 2D Data and Images is ID width height format string for ASCII files new line For 1D data it is even more simple ID number of elements format string new line Identification Code and size parame
21. Kernel requests these values and these values are valid As long as the center pixel of the Kernel can be placed at the outer pixels of the area without the periphery of the Kernel reaching out of the Image pixels outside of the area are taken into account All other available filtering methods are not linear The filter mask defines a neigh bourhood around a center pixel which is taken into account for the filter operation Further description is given directly in the following sections 3 5 1 Low Pass Low Pass Filter eliminate high frequency components in the fourier domain while leaving low frequencies untouched which can pass through the filter Edges and sharp details which are always characterized by high frequencies are suppressed by the resulting blurring of the image The following menu entries represent common low pass filters with different Filter Kernels The bigger the Kernel the slower processing but more blurring effect is obtained 3 5 1 1 3x3 Standard low pass filter with unity Kernel see Fig 3 15 Factor M D in Eq 3 1 is 1 9 With these values Eq 3 1 performs an averaging of all pixels covered by the unity Kernel 3 5 FILTERING 50 1 1 1L 1iyil 1 1 Figure 3 15 Filter Kernel for Low Pass 3 x 3 If a 2D Data field including invalid values shall be filtered a dialog appears where options for invalid data treatment have to be set which are e Invalid Data
22. a pixel might be a residue but if so the sign is undeterminable The only way to deal with this problem is to substitute the invalids with a valid phase data which then might be in the appropriate range to have no effect though the substituted value can deviate up to 7 from its neighbourhood or it will produce a pair of residues e g a branchcut which is no problem to deal with Additionally to a mask to determine the boundaries of valid phase data starting points for the final step of flood fill unwrapping have to be defined for each valid region as described in Sec 3 10 1 The input parameters to be defined in the dialog shown in Fig 3 10 10 1 are e Invalids Substitute input field Define a value which substitutes the invalid phase data e Scan Startpoint is Last in Multiple Point Selection checkbox Check this if the scan startpoint has been selected by the multiple point selection with the mouse The start point is assumed to be the last in the selection e Scan Start X input field Define the x coordinate column of the starting point for the residuum search path see step 1 in the list above if it has not been selected by the multiple point selection with the mouse 3 10 PHASE 101 Branchcut Nearest Neighbour x Invalids Substitute fo IV Scan Startpoint is Last in Multiple Points Selection Scan Start X Scan Start Y p IV Set Branchcuts to Invalid I Show Residues in Extra Image Create Mask t
23. a visualization It can also be designated as a Look Up Table In our case the information of one colour is given by its red green and blue portion RGB Code each of it with a depth of 8 Bits Each pixel value refers to one of these Codes by giving the 8 bit address offset within the palette IDEA provides several pre defined palettes which can be applied to any visualization In these palettes the 256th RGB entry is reserved for the individual mask colour see below whereas entry 255 is used for invalid values see Sec 2 3 Entries 252 and 253 are used to show under and overflow after remapping also explained below 2 1 2 Picture Following our definition a picture is the visualization of any 2D Data and Images The data itself is held in the background and is not affected by changing visualization parameters If you are interested not in the data itself but in its visualization you can convert the data to a stand alone picture and save it as a uncompressed Windows bitmap bmp with 256 colours This picture cannot be distinguished from the visual data representation The essential difference is that there is no data in the background anymore Therefore opening such a picture provides no usable data for processing unless it is converted to an Image see below 2 1 3 Image In the context of IDEA an Image is two dimensionally arranged data with a depth of 8 Bits Such an Image is by default represented by a gray scale Pic
24. and carrier fringes with a slant of 45 allow mean value calculation from results of a number of neighbourhood 3 10 PHASE 94 groups for circular speckle areas 7 in general elliptical or quasi rectangular speckles are used for preservation of spatial resolution in vertical direction The intensities are picked in groups of three starting at the most left column and advancing column by column The result calculated from each data triple is written then into a phase data field at the position of the second element where the algorithm assumes the phase shift i a 0 The phase data field has always the same size as the interferogram though the most left and right column have to filled with invalid data as no calculation is possible for those pixels A simple dialog asks for two inputs e Minimal Required Visibility Define a value for minimal required visibility in percent If this value cannot be reached a mask is set at the corresponding position in the phase data Nevertheless the phase result is not discarded e Minimal Value for Imax Imin Define a value for minimal required intensity modulation within the data triplet Tf the calculated modulation falls short of the defined value a mask is set with out discarding value for at the corresponding position in the phase data Be aware that there is concurrence to a defined Minimal Required Visibility see above A mask which is set in the interferogram when the
25. as T x y Lote y 1 V x y cos d x y i a x y 3 39 where x y are pixel coordinates Jj is the average intensity V is the interference fringe visibility is the wavefront phase and a is the known relative phase shift between object and reference beam introduced by a phase shifting device between or during recording of interferograms With each acquisition is incremented by one but depending on the algorithm used might start at a negative value it even might not be an integer The different values of i yield a system of equations which can be solved for the desired phase Depending on the phase shift and the number of recorded interferograms different relations can be found this way for which is calculated from the intensity values at same position x y in all images As there are three unknowns Io V and 9 at least three interferograms have to be recorded Most solutions for are taken from 11 where relations for visibility and information about susceptibility to errors of a and intensity values can be found as well Phase Shifting in Speckle Interferometry Especially for Electronic Speckle Pattern Interferometry ESPI 22 45 41 e g Dig ital Speckle Pattern Interferometry DSPI the introduction of the phase shifting method in the mid 1980 s was a huge step forward Before that the interest in this field of optical metrology had been declining as fringe tracking techniques had reached
26. bitmap representing the step function can be saved It can be removed from this window with Phase Remove Step Function see Sec 3 10 8 uncovering the wrapped phase data As the number of colours representing the phase orders is limited to 254 interfero grams featuring more fringes can not be treated by this algorithm However in this case the branchcut method is recommended as the flood fill algorithm used is not bound to a colour palette see Sec 3 10 10 1 3 10 3 One Step Unwrapping by Scan The menu entry subsumes the spiral scan for phase orders and the unwrapping pro cedure in a single step very handy for well behaved wrapped phase maps where no subscans are required and where the branchcut method would be an overkill However there is still the limitation to 254 phase orders e g fringes in the interferogram See Sec 3 10 2 for information about the scan method used The input of the starting point coordinates are the same as described there there is just the phase offset at the starting point s to be defined in the z column of a subsequently opened multipoint dialog see Sec 3 3 15 and Fig 3 12 3 10 4 Set Phase Jump Value for Scan Methods Define a value for the phase jump which is then used for both the 2D scan and the spiral scan method It is also used for any subscans 3 10 5 Sub Scan 2D Enabled Switch sub scan mode to two directional scanning method A checkmark shows which of both available modes 2D Scan
27. border remains unmasked Else the area must be defined in an input dialog by typing in coordinates of upper left and lower right corner 3 11 13 Mask Polygon If a polygon has been drawn in a Picture this commands executes flood filling the area starting from the x shaped start pixel stopping at any side of the polygon Otherwise the corners of the polygon have to be defined in the multiple points dialog described in Sec 3 3 15 where the last entry defines the start pixel for the flood fill 3 11 MASK 108 3 11 14 Mask Minimal Values After determining minimal value within an Image or 2D Data Field all pixels with this value are masked This is a more fast way to localize minimas than remapping with View Change Colour Palette 3 11 15 Mask Maximal Values After determining maximal value within an Image or 2D Data Field all pixels with this value are masked 3 11 16 Mask Invalid Values Masks all invalid values see Sec 2 3 within an 2D Data Field 3 11 17 Mask Interval Masks all pixels which value are within or outside of an interval In a dialog both mode within or outside of and limits of the interval must be defined see Sec 2 4 for macro inputs 3 11 18 Mask Zero Frequency Masks central pixel in an frequency field representing zero frequency Useful for eliminating this frequency before backtransformation to image 3 11 19 Mask Nyquist Frequencies Masks all Nyquist frequencies in an frequency field w
28. even higher maximum on either side is encountered Histogram Equivalization Another common technique to enhance contrast This technique is not linear as the two previous ones but is based on integration of histogram values to get a transfor mation function for gray levels It is well documented eg in 15 Please note that Images with already high dynamic range will change only marginally Square Intensity Creates a new unspecified 2D Data Field with all intensity values squared 3 3 EDIT 35 Invert Intensity Calculate a new Image Tiny from Torg by inverting pixel data using Liny u y 255 Lorg x y with x and y denoting the position of each pixel in the Images In terms of photography the negative of an Image is created this way Remap Intensity Expand or shrink the dynamic range of an Image range between minimum and maximum to a new range defined by user input of desired minimum and maximum For expanding no interpolation is done Mean Intensity All Images within a File Pool are averaged pixel by pixel to calculate a new one con sisting then of mean intensities In a new 2D Data Window the standard deviations sigma are shown All source images must have the same size Extract Interlace Field 1 Extracts all odd rows from an Image to form a new one For TV standard cameras in standard mode there is a specific time interval between exposures of the interlace fields Therefore when the shutter time
29. f Interpolation The higher the number the better is approximation of measured data but the lower is the smoothness of the inversion The highest valid input is the number of data points from border to center minus 1 e Show Integral of Reconstructed Data Activate this to show not only the Abel reconstruction f x a is Cartesian coordinate with left border of distribution at x 0 substituting r but also an additional graph of the analytically calculated integral data ha x using Eq 3 5 This data can be compared with measured data e Show Deviation from Input Data Activate this to show additional graph with deviation hm ha of analytically calculated integral data h x from measured data hm e Show Radial Symmetric 2D Data Activate this to show reconstruction f x two dimensionally The 1D distribution is simply rotated using the Bresenham algorithm whilst corners are filled with zeros 3 6 3 Abel Inversion Fourier Method In comparison to the Interpolation this method does not perform the Abel Inversion by gradually working from outer region to the center but computes the reconstruction in one step from the spatial point of view This avoids propagation of calculation or measurement errors from the periphery to the center The radial distribution f r is assumed to be a sum of N model functions fn with unknown amplitude An The implemented algorithm in IDEA uses cosine functions to form the radial distribution Nu
30. fast and reliable method for phase maps with medium noise The procedure is a hybrid between the minimum cost matching and the nearest neighbour method so it is referred to as local minimum cost matching It works as follows 1 Starting from the upper left corner of the phase map it is scanned for residues row by row 2 If a residuum has been found a spiral scan starts to search for a any other residues or border pixels until a specific number of ny residues of opposite sign has been found 3 A cost matrix for all residues and border pixels encountered is calculated from this selection of pixels as described in Sec 3 10 10 2 and from the resulting combinations of branchcuts only this with the shortest length is set Paired residues are marked and ignored for further processing 3 10 PHASE 104 Branchcut Local Minimum Cost Matching x Invalids Substitute fo Number of Partners F Border Costfactor P Y Set Branchcuts to Invalid J Show Residues in Extra Image T Create Mask to Show Branchcuts a A Figure 3 39 Dialog for branchcut unwrapping applying local minimum cost matching algorithm 4 After working through the whole phase map the procedure continues again with step 1 Refer to Sec 3 10 10 1 for definition of valid phase regions and corresponding starting points for the flood fill unwrapping The dialog for the local minimum cost matching algorithm features the following input elements
31. file containing angles Since the writers use the German version of Windows NT the labels in this File Selector Dialog are all German Subtracting Reference Phase and Additional Tilt The same procedure of phase evaluation must also be applied to the reference interfer ogram which was recorded to eliminate the influence of non ideal image properties For doing so the previously created masks can be used The final phase is subtracted from the result of the previously determined distribution by using Edit Subtract Image 2D Data as described in Sec 3 3 7 However due to the fact that the reference interferogram was recorded at slightly different wavelength a linear tilt remains To remove it we use Phase Remove Linear Phase Shift see Sec 3 3 5 yielding the final phase distribution shown as Arc pha in Fig 4 So far the phase shift within the plasma discharge was determined The zero padded area can be cut away and the Invalid values grey at the shadows of the electrodes are set to zero with Edit Edit 2D Data Substitute Invalid Values since otherwise they are not accepted by the tomographic algorithms cf Sec 2 3 After completion of all these evaluation steps for all directions we are now ready for the reconstruction of data 4 1 3 Tomographic Reconstruction The phase distributions calculated so far are line integrals along the path of light through the object In order to reconstruct the asymmetric object To
32. fint four Deviations00 bin shows overall deviation from each inversion obtained from f Interpolation to every result from Fourier Method As this deviation should be very small for reliable reconstructed data appropriate parameters can be found at dark violet regions of this field The 9 smallest values of this field are printed in the protocol window as shown in Fig 4 8 The 1D Data distribution displayed in window Data four Curvature dat shows the overall curvatures of Abel Inversions depending from the maximum order Equally 4 3 EXAMPLE 3 ABEL INVERSION 121 idea File FilePool Edit View Filtering AbelInversion Tomography 2D FFT Phase Shift Phase Mask Information Hooks Window Help ul ajaj wE 8 5 015 ele ml Calculation ready now evaluating results Sorted Sorted Deviations of results from both methods 0 000037 poly 2 0 000072 poly 10 491 68 0 000072 poly 8 492 3 0 000082 poly 11 118 64 0 000106 poly 10 4184 24 Abel Analysis x 0 000130 poly 248 24 i 0 000146 poly x 290 68 8 0 000150 poly 301 24 Hnterpolation Number of Cubic Splines ffos 9 0 000157 poly 6 ord 319 38 Created Multiline Graph Noname0 asc Fourier Method Maximum Order of Function fe ONE e lolo Data at x 25 0 603689 208x256 1 42 0 00176422 Figure 4 8 Screenshot of IDEA demonstrati
33. harddrive This way unwanted results of File Pool operations are easily removed without switching to other applica tions and redoing the file selections for deletion 3 2 7 Open Marked File s Entries marked by mouse selection for multiple selection use left mouse button in conjunction with SHIFT or CTRL key are opened and corresponding windows ap pear 3 2 8 Sort Alphabetically Use this to sort the filenames in a File Pool alphabetically The entries are accordingly reordered 3 2 9 Extract Every ith File Creates a new File Pool including every i file from the actual File Pool In the small dialog one has only to define the value 7 For example after an acquisition of 250 interferograms for temporally resolved measurement it s a good idea to start with the evaluation of a few representative images to see if everything worked out well So after creating a File Pool with all 250 images it is possible to evaluate only 10 images by choosing 7 25 in the dialog of this menu entry without selecting the according filenames by hand 3 2 10 Extract Marked Files Creates a new File Pool with all marked entries If nothing is marked the File Pool is duplicated 3 2 11 Extract Marked Images 2D Data It is not possible to perform file type specific operations on File Pools including files of different types To solve this problem we made it possible to create new pure File Pools by copying all entries of the selecte
34. ht h2 h3 close to a phase jump hl and h2 a 2 a by the low side h lt Y Nrs lt Nro T a T Naz lt B Nni Nhs h1 h2 h3 contains phase jump h3 Na gt Nm high value Nha lt V or Nri gt Nro T a ta 40 Nn2 lt B Nm Nhs hi h2 h3 contains phase jump hl Nai gt Nas low value Nhi 2 yV or Nrs gt Nn2 7 a 0 T In the original paper 8 Np lt yV is written I regard this as a typo 3 5 FILTERING 59 Repeat The number defined here determines how often the filtering process is repeated on the whole 2D Data field Width of Center Interval The width of the center interval for the histogram is equal to 2a In 8 this value is recommended to be set to 27 3 A lower value can be set for phase maps with quite high signal to noise ration where edges at phase jumps are quite clear Beta Factor This factor weights the sum of the populations N1 of interval 1 and Na of interval 2 See Tab 3 1 Gamma Factor The meaning of this factor can be seen also in Tab 3 1 It weights the number of valid pixels V and is mainly used to distinguish by comparison with Np and Nps if the filter window is near the jump or already containing it The higher this factor the more values in these intervals are needed to pick the mean just from there In 8 a value of 0 7 is recommended Minimum Valid Values If the number entered here is not exceeded by number of valid data within the filter window then no filt
35. if the here defined value is higher than the calculated correction value Minimum Iteration Error Break Off Condition After each iteration except the first one changes of corrections c in Eq 3 31 are calculated Convergence of the iteration procedure means that the sum of squared c corrections approaches zero Therefore a minimum value of this change can be used as a brake off criterion As soon as the calculated value falls short of the defined one the iteration is finished Size of Smoothing Matrix Defines the side length of the filter mask used for Selective Smoothing rou tine see Sec 3 5 8 A value of zero skips Selective Smoothing which is else performed after each completed iteration step Selective Smoothing Parameter The Selective Smoothing routine performs smoothing only in areas where edges actually covered by the filter mask are lower than a certain upper limit This limit can here be defined by the relation to the maximum value within the whole reconstructed field Apply Zero Correction Checkbox If all data within the reconstruction plane is known to be restricted to be all either positive or negative zero values in projections imply zeroes along the whole ray path through the plane Therefore instead of calculating Eq 3 31 and Eq 3 29 all intersected elements of the reconstruction x are set to zero Apply Limit Correction Checkbox If values of reconstructed data x are constrained by any physic
36. matter as long as the upper left corners coordinates 0 0 correspond to each other e Starting Point of Center Line 1 y1 Define x y coordinate of the starting point of the line which connects all re quired center pixels of projections center pixels are those pixels in a set of projections which correspond to a path through the origin center of the cross sections e Ending Point of Center Line x2 y2 Define x y coordinate of the ending point of the line which connects all required center pixels of projections Set here the same value as for the Starting Point to reconstruct only one cross section of the object e Coordinate of Starting Point Relative to Line Center dz dy Define the starting point of each projection relative to its center pixel located on 3 7 TOMOGRAPHY 70 center line For the upper projection center and starting pixel are connected and with required symmetry to center the final position is determined Be aware the dialog allows here more settings as make sense with tomography Only for dy 0 all projections correspond to the same reconstruction plane e Line separation Define the number of pixels which are located in vertical direction between two neighbouring projection centers This overrides settings for y2 For example if the vertical length of the center line y2 yl 1 30 and 20 is chosen for line separation only 2 projections can be drawn with vertical distance of 20 1 21 pixel Th
37. mouse cursor changes its shape to a little hand when a 1D Data window is entered The menu item keeps checked as long as you either click the hand on the window and copy it to the Multiline Window or reselect the menu item to end the adding mode compare procedure in Sec 3 3 7 Add 1D Data Files Add files from disk to Multiline Graph by using the file selector of the platform in use for which Multi File Selection is activated In Windows 95 98 NT use the SHIFT or CTRL key in conjunction with the left mouse button to choose a group of files in the filenames list of the selector Note The last selected filename appears always at the beginning of the text line showing the current selection located below the filenames list reversing the temporal order of your selection The data read from the selected files are copied to the Multiline Graph Add n 1D Data Files Add a specific number of files from disk to Multiline Graph by using an adapted file selector After defining the number n of files to open the file selector window appears with n text boxes Refer to Sec 3 2 2 and Fig 3 4 for further description Clear All Remove all contents of the Muliline Graph Remember this does not delete the files The contents of a Multiline Graph can be stored in Raw ASCII format The file will contain the curves as columns separated by spaces If there are shorter curves they are filled up with a value to define in a dialog 3 12 4 Histogram
38. own starting point can be selected This is done either by mouse interaction see Sec 3 3 14 or by the multiple points dialog see Sec 3 3 15 and Fig 3 12 The phase data modulo 27 is unwrapped sequentially by spiraling around a growing region of unwrapped data see Fig 3 35 The change of order As see Eq 3 57 to neighbours where step function is already known suggest orders for the pixel under investigation If more than 50 of all available changes unreliable data and masked pixels in wrapped phase do not count agree on the same order this value is accepted as new order of this pixel Else the value zero colour white is set for the new order marking it as unreliable Areas unreachable to scanning path are set to value 255 colour black marking undeterminable data Outside of the 3 x 3 starting area we assume only one available value for As to be too less to determine the new order Though more data is regarded as not reliable in this case tests showed that many error propagations can be prevented If there are several independent phase regions in the phase map the procedure is applied for each region separately with the other regions masked 3 10 PHASE 97 At any starting point the order value of 127 red colour is set allowing changes up and downwards After determination of the step function the corresponding Picture hides the modulo 27 data The window then includes both data fields but only the 256 colour
39. path of a mask file must be specified in a small dialog This menu entry was mainly designed to apply 2D FFT method to a File Pool of Images Since always the same filter mask is used the frequency domain of each interferogram should be nearly the same 3 8 8 Calculate 2D Phase Data Allows calculation of phase data from an Interferogram Image in one step by ap plying 2D FFT and unwrapping of resulting modulo 27 phase data successively This requires several inputs which have to be defined in the dialog shown in Fig 3 29 e FFT Filter Mask Define the path to a mask file determining frequency domain of interferogram for specifications see Sec 3 8 4 by typing into the textbox or click to button to the right to use the standard file selector e Unwrap Mask not required for DCT unwrap method Define the path to a mask file for modulo 27 data which is taken into account for unwrapping see Sec 3 10 This can be done by either typing into the textbox or by a mouse click on the button to the right and selection of file by standard file selector e Unwrap Method to select Select one of the unwrap methods provided by IDEA If DCT With Mask is cho sen the mask for the interferogram is just assigned to the resulting unwrapped data but not taken into account during calculation 3 8 2D FFT 81 e Phase Jump not required for DCT unwrap method For path dependent methods define phase jump which must be exceeded to be defined
40. pops up see Sec 3 2 It is followed by a dialog where projection angles 9 see Fig 3 24 corresponding to the 2D phase data in the input File pool same order can be defined see Fig 3 25 Projections TomoD ata00_tom Angle6 90 Angle Projection Name 0 projection00 pjn 15 projectionO1 pjn 30 projection02 pjn 45 projection03 pjn 60 projection04 pjn 75 projection05 pjn projection06 pjn projectionO pjn projection08 pjn projection09 pjn projection10 pjn projection11 pjn Ok Cancel Load anges Save anges Goto Figure 3 25 Grid for Defining Angles of Projections for Tomography This is done by typing in values into the grid or by loading an ASCII file containing a set of projection angles see Sec 2 2 10 The contents of the grid can always be saved in such a file Confirming angles pops up the next dialog see Fig 3 26 to define locations of projections in all phase distributions in the File Pool At the same time the marked or if no entry in the File Pool is marked the first 2D phase file is opened showing the default selections in the dialog Within this dialog the user not only can choose one set of projections for the reconstruction of one cross section of the object but also a number of projections sets for parallel and equidistant cross sections If any of the defined coordinates exceed the domain of the smallest file in the File Pool a warning message appears Therefore different sizes do not
41. positioned either by mouse action or by typing coordinates into text fields For integral 1D Data mentioned above you can move the blue border lines with mouse If you point the mouse cursor to such a line it changes its shape to a horizontal arrow By pressing and holding down the left mouse button the borderlines can be moved within the graph keeping symmetrical position relative to the center line For Abel 1D Data holding down the right mouse button unlocks the borderlines from the center and each of it can be shifted independently Likewise the red center line can be moved with the mouse The left mouse button shifts the center together with the border lines Again Abel 1D Data uses also 2 5 THE GRAPHICAL USER INTERFACE OF IDEA 1 2 3 4 Save Selected File 0 100 200 400 200 250 200x300 640x480 L 13 21 415 99 Figure 2 1 Basic structure of Status Bar 1 Menu Section 2 Coordinate Section 3 Draw Mode Section 4 Mouse Pointer Section the right mouse button which allows to move the centerline only There are some restrictions of movements e The centerline can only be moved between the border lines e Center and borderlines together can only be moved until one border line reaches the border of graph e If border lines are moved simultaneously left button but are asymmetrical to the center the opposite border jumps to symmetrical position when you click on a b
42. protocol window 3 10 11 Unwrap with DCT The problem of phase unwrapping can be met by solution of Poisson s equation with a specific form of the fast discrete cosine transform DCT 13 This approach is 3 10 PHASE 105 numerically stable computationally efficient and exactly solves Poisson s equation with proper boundary conditions But be aware Mask is not taken into account in the version which is implemented in IDEA Therefore it is not recommended to use this method for phase data with much irrelevant or erroneous phase data Additionally dimensions of the phase data field have to be powers of two Local pixel phase difference in a 4 neighbour sense should be identical in both the wrapped and unwrapped 2D Data The unwrapped solution is the one that minimizes M 2N 1 M 1N 2 5 5 Gig Pij A 5 5 Pi j Qij A 3 60 i 0 7 0 i 0 420 where the subscripts i j refer to discrete pixel locations in x y of 2D Data size M x N pixels The phase differences A j from the original wrapped phase data Y in the horizontal direction x and the vertical direction y are A W WVis15 Vij 1 0 M 2 j 0 N 1 Al W Wij41 Yig i 0 M 1 j 0 N 2 and 0 otherwise W denotes an operator that wraps all values of its argument into the range r 7r The normal equation leading to the least squares phase unwrapping solution can be written as 21 Gi41 3 264 53 di
43. subtraction are the last np files in the File Pool File Pool Includes Subtraction Fringe Interferograms Select this options if all included files are subtraction fringe interferograms The files must be ordered in subgroups where each subgroup consists of ny phase shifted subtraction fringe interferograms in a sequence according to increasing phase shift The result is then a File Pool where the sequence of wrapped phase entries corresponds to the sequence of subgroups Continue gt button Press this button to close this and open the next dialog which is in case of selection of an user kernel the User Kernel dialog Sec 3 5 3 in case of a File Pool operation the dialog for the saving options see Sec 3 2 and Fig 3 3 or in the regular case the file selection dialog for the POD method described just below Cancel button Press this button to cancel further input and close the dialog 3 9 PHASE SHIFT 90 Speckle Phase Of Difference Filter Options x Phase Shift Mode three Frame Technique 120 M Filter Options Convolution with Unit Kernel Neighbourhood Mean Convolution with User Kernel Filter Width Filter Height 5 x Filter Repeats fi iterations E Continue gt D Cancel Figure 3 31 First dialog for Speckle Phase of Difference The File Pool Mode section is only visible if applied to an active File Pool Window Image Files 420 fewsDataumaeimg E 120 sotaman O B Y
44. the height of the neighbourhood to take into account for the least squares fitting has to be defined as well as minimum number of valid neighbours A pixel is regarded to be valid if the minimum visibility and or the minimum value of Imin Imax 21m which are both to define near the bottom of the dialog are exceeded in the four phase shifted interferograms If the condition is not met the pixel location is masked in the phase field to mark the calculated value as unreliable In case the method shall be applied to files in a File Pool another new radio box la belled File Pool Mode is added below the neighbourhood section see Fig 3 33 b Here one has to define wether the four phase shifted interferograms are located at the bottom or at the top of the file name list of the File Pool 3 9 11 Spatial Phase Shifting 120 In contrast to the speckle methods above the spatial phase shifting does not require a temporal sequence of phase shifted interferograms for reference but the phase shift 3 9 PHASE SHIFT Phase Shifting for Speckle Interferometry x Neighbourhood Definitions for Fit Neighbourhood Width Neighbourhood Height fs y Phase Shifting for Speckle Interferometry x Minimum Valid Phases in Neighbourhood 5 AOA De temiors TOER Neighbourhood Width s Neighbourhood Height 5 a ls y ls y Minimum Valid Phases in Neighbourhood E Ref 0 C PSDataimage0 img E
45. the left to open standard file selector After starting the calculation for each iteration a forward and back transformation is performed After a defined number of iterations the result is presented with inverted mask of original interferogram 3 8 3 Forward FFT Applies Fast Fourier Transformation to Image or 2D Data The resulting data con taining complex amplitudes of all frequencies are presented by an Image which shows modulus of the complex amplitudes after mapping data in quad root mode see Sec 3 4 4 ignoring amplitude of zero frequency which is located in the center of the visualization Coordinates in the Status Bar see Sec 2 5 2 represent periods of horizontal and vertical direction in transformed image Negative values belong to negative frequencies and are located in the upper half plane since this corresponds exactly to coordinate system of fields where the y coordinate increases from top to bottom With this orientation the frequencies of a parallel fringe system are located along a line through zero frequency perpendicular to fringes Tip If only few details are visible switch to logarithmic display mode see Sec 3 4 4 and choose a different colour Palette Sec 3 4 5 Be aware that no data can be retrieved interactively from this window as it is just a visualization of a field containing data in the packed order see Sec 2 2 4 But real and imaginary parts can as well be accessed as the real amplitude by 2D F
46. the neighbourhood is taken into account The summation and division in Eq 3 1 is adapted to the number of valid data e Minimum Number of Valid Neighbours Only if more valid data is within the neighbourhood than the number defined here a filter result is calculated Otherwise the filter response is invalid e Allow Replacement of Invalids by Filter Results The standard filter procedure calculates filter results only for valid data How ever by checking this option the filtering process is also performed if an invalid data is at the center of the filter kernel The filter response then substitutes the invalid value 3 5 8 Selective Smoothing This filter is recommended in 18 for smoothing tomographical reconstructions while selectively preserving edges Filtering on an Image with gray level f i j is performed according to following yielding a filtered image fij Y Wirge girs fti j F i i 3 3 Y Wir jr Gin jm ms Here indices 7 7 denote mask coordinates with origin at center i j are coordinates in data field Expression w is a kind of weighting function taking into account the distance between element at location i 7 from center 0 0 j 3 tp 0 j 0 ea i j 2 2 else Expression gis depends on the minimum height t smooth limit of edges which shall be preserved 1 if i 0 7 0 glp 8 1 if FGF jt FEN lt E 0 else In the dialog one has to define the size of the mask and the smooth
47. the number of directions which have to be taken into account must be defined Of course the higher this number the higher is the precision of the reconstruction In addition to the text box for this input there are checkboxes to create additional data for feedback see Fig 3 23 and Sec 3 6 2 After pressing OK the dialog box for convolution appears see Sec 3 7 5 and Fig 3 27 With this data the tomographical reconstruction by convolution algo rithm is performed yielding an internal 2D Data field from which the horizontal center line is extracted as radially symmetric distribution If integral data is required by settings in the first Abel Dialog the integral data are calculated from the internal field by summing up all column data integrating in vertical direction Be aware that tomographical reconstructions are not recommended for radially sym metric data as Abel Inversion algorithms are less susceptible to error propagations 38 3 6 7 Radial Data ART Works like Sec 3 6 6 After confirming input in the Abel Dialog the more complex input dialog for ART see Sec 3 7 6 and Fig 3 28 pops up 3 6 8 Create Radial 2D Data Here data of any 1D distribution between left border line and center line is rotated around the position of center line yielding a radially symmetrical distribution with zeroes at regions out of range for visualization purposes mainly 3 7 TOMOGRAPHY 67 3 7 Tomography When using interf
48. their limits with the typical high amount of noise in speckle subtraction fringes With the classical phase shifting technique 10 no subtraction fringes are calculated but a phase is evaluated from a set of temporally phase shifted speckle pattern interfer ograms recorded both before and after a change of the object of interest One of the resulting phase maps represents the reference object state and consists of sta tistically distributed phase values changing usually from pixel to pixel The other phase map represents these speckle phases plus the change from the object which allows to calculate the object phase by subtraction of the reference phase map re mapping to the interval from r to 7 required This method is often referred to as Difference of Phase method Evaluation of subtraction fringes by the well known and established Fourier Transform technique cannot provide accurate results as the spectrum of a speckle pattern covers frequencies from zero up to an limit determined by the minimum speckle size which is not zero This stochastic frequencies adds to the carrier frequency of the fringes which flaws separation of background noise and fringe signal The Difference of Phase technique can be used in IDEA with the standard phase shifting algorithms by applying them to the phase shifted sets of speckle pattern interferograms for two states of the object Both results can then be subtracted by Edit Subtract Image 2D Data F
49. these neighbours and defined minimal jump value J a change in the step function or order respectively As 1 if Ad gt J As 1 if Am lt J 3 57 0 else is calculated for both pixels If the results are the same the new As is added ac cordingly in the step function but if they do not correspond s is set to invalid This value which is represented by the colour with offset zero in the palette see Sec 2 1 1 marks unreliable data By default the corresponding colour is white At positions where wrapped data is masked the palette colour with offset 255 black reserved for undeterminable data is set If there are several independent phase regions in the phase map the procedure is applied for each region separately with the other regions masked After determination of the step function the corresponding Picture hides the modulo 2n data The window then includes both data fields but only the 256 colour bitmap representing the step function can be saved It can be removed from this window with Phase Remove Step Function see Sec 3 10 8 uncovering the wrapped phase data As the number of colours representing the phase orders is limited to 254 interferograms featuring more fringes can not be treated by this algorithm However for such images the branchcut method is recommended as the flood fill algorithm used there is not bound to a colour palette Use the Nearest Neighbour method for 3 10 PHASE 96
50. to Sec 2 5 1 and 3 3 10 for how to draw a line The graph window for 1D integral data includes blue border line cursoers and a red center line curser which can all be moved by mouse action see Sec 2 5 subsection Select Borders and Center of 1D Data If 1D integral data shall be retrieved from a series of 2D Data Fields or Images put them all into a File Pool see Sec 3 2 Entering this menu pops up the dialog for File Pool saving options see Fig 3 3 followed by a dialog similar to that described in Sec 3 7 2 see also Fig 3 26 There the term projection should be regarded as integral data and reconstruction as the two dimensional Abel Inversion Note also the difference between the Data Fields in the source File Pools For tomography they refer to different viewing angles belonging to the same cross section with object 3 6 ABEL INVERSION A 1 x h gt 0 gt h yi Figure 3 22 Abel Inversion by f Interpolation The radial distribution f r is divided into rings P For inversion it is fitted within each ring by a polynomial of third degree minimizing deviation of the calculated optical path integral from measured data points As calculation proceeds from periphery to the center all contribution of outer rings intersected by the beam at y can be computed analytically only the contribution of ring P must be fitted to all measured points h y with Riy1 lt y lt Ri In ad
51. to the right show the factors of R G and B for the chosen option If Custom is selected the factors can be defined by user 3 Images are simply duplicated 3 1 7 Convert Copy to Picture This converts the visualization of Images and 2D Data to a 256 colour Bitmap using the current palette including mask under and overflow colours if existing Apply to a Picture to create a duplication 3 1 FILE 27 Greyscale Converter Xx Red 77 Green fist Figure 3 2 Dialog for Converting Picture To Image 3 1 8 Convert Copy to 2D Data Converts an Image to a Data Field by just expanding the gray values 8 bit to floats with double precision Data fields are duplicated 3 1 9 Change Filetype For 1D Data and 2D Data we allow the user to change to a different file type within these classes to eliminate any file type specific restrictions Be aware this is done by your own responsibility Don t be surprised when a 2D reconstruction from tomogra phy looks weird as a modulo 27 Field 3 1 10 Protocol Contents of Protocol Window can be saved and loaded only in this menu and not with standard Open and Save entries Load Protocol File Substitute contents of current Protocol Window with Protocol from file Save As Save current contents of Protocol Window to ASCII file User Edit Mode Disabled This menu entry is active by default In this case the Protocol is protected from user input Unch
52. 0 XP X Window System X11R6 e CTL3D32 DLL version 2 31 000 in Windows System Directory At lower resolutions than 1024x768 some help text in the status bar and the icon bar may appear truncated which does not further restrict functionality At colour depths smaller than 16 bit High Colour the 256 colour Bitmaps may not be displayed properly The RAM requirement results mainly from the 2D Fourier transform Lower amounts of RAM result in time consuming swapping Version 1 0 of the software was developed and tested mainly on Pentium 200 and 133 with 64MB RAM so this should be a good reference when data fields and images are smaller than 1024x1024 Version 1 5has been developed and tested with a Pentium III 750 equipped with 512MB RAM With the eliminated size restriction however the system running with Windows 2000 occasionally capitulated to phase evaluations of images with a size of 4096x4096 and larger Taking into account the amount of memory occupied for such operations this problem can be assumed to be attributed to limited system resources e g the management thereof and not to IDEA The required system power therefore depends on the typical image and data field size used with IDEA On basis of dimensions up to 2048x2048 the recommended system is something like e Pentium III 500 or CPU with comparable power e 256MB of RAM e 32MB Graphic Card resolution 1280x1024 True Colour 1 4 BACKGROUND OF DEVELOPMENT 1 4 Background
53. 1 5 Or j41 205 5 6i 5 1 Pijs 3 61 where pij AZ Af 13 Af Af 1 These equations relate wrapped and unwrapped phase differences in a discrete 2D grid form of the Poisson equation 2 82 gY aye y plz y with Neumann boundary conditions V n 0 which may be solved by the 2D DCT 36 The exact solution in the DCT domain is IR i j Pis 2 cos 34 cos 7 2 0 84 The unwrapped phase values can then be obtained by performing the inverse DCT of Eq 3 62 3 10 12 Interferogram from 2D mod 2Pi Data Calculates interferogram from modulo 27 data In the input dialog the modulation must be defined by values for minimal intensity Imin and maximal intensity Imaz Another free parameter is the constant phase offset By default this value is set to zero or to the modulo 27 phase at position of previously drawn crosshairs 3 10 13 Interferogram from 2D Phase Data Calculates full modulated interferogram data from phase distribution 3 10 14 Mod 2Pi from 2D Phase Data Calculates modulo 27 phase data from a continuous phase distribution The required constant phase offset can be defined indirectly by specifying coordinates where modulo 2r data should be zero By default coordinates of center pixel or those of previously drawn crosshairs are set 3 11 MASK 106 3 10 15 Remove Linear Phase Shift Subtracts a plane function from 2D phase data This is often required after phase e
54. 1D distribution B of same size from current active data A Follow instruc tion in Sec 3 3 7 Edit Values Manually This opens a window which lists all data of the distribution in a grid Click on data you want to edit and change entry in the text box on top of the list Mean Value Calculates mean values and standard deviation of equally located data elements in 1D Distributions in a File Pool which all have to be of same size 3 3 EDIT 40 c Figure 3 8 Left Right Side only Symmetrizing a distribution a by mirroring data located between center line and border line to the other side center line is axis Using left side yields b whereas c is the result of mirroring the right side 3 3 EDIT 41 OO a b Figure 3 9 Averaging left and right side of a source distribution a Values of data points within left border and center line and of corresponding data point at same distance from center are substituted by their mean value The result b is a symmetrical data distribution The hump at the left side of a is now also visible at the right side though weakened by the averaging 5 a b Figure 3 10 Averaging left and right side of a source distribution a with asymmetrical position of right border line Averaging is not possible for data points with longer distance from center line than right border line Therefore data out of range
55. 2 2 9 You are not restricted to only one filetype within a File Pool Feel free to comprise all files related to the same project For file type specific operations it is possible to extract all files of appropriate type from the main File Pool into a new one This way you can easily overlook your work and save much time The operation performed on a File Pool is repeated file after file and the results are saved after completion of each step Structure of filenames and the format of saved data can be defined by the user All created filenames are collected in a new File Pool To create a empty File Pool use File New File Pool see Sec 3 1 1 By double clicking a file name in the File Pool Window you can open it in a separate window Whenever an operation is applied to a File Pool a dialog titled Saving Options for File Pool pops up before the calculations start see Fig 3 3 In this dialog you can define a Saving mode in the upper left hand corner In general you have the choice between overwriting the files in the File Pool with the results Overwrite existing files or to save the results with new filenames Save as new file New files require new filenames A part of this is overtaken from the filenames in the source File Pool name A second part has to be defined by the user in the left text box at the bottom of the dialog At the upper right hand corner you can choose between setting the user defined part in f
56. 3 2 13 Change File Counters Define a number which is added to all filenames in the File Pool The files are immediately written to the output folder Files without counter are just copied 3 2 14 Convert Files Converts formats of all files in a File Pool The parameters must be defined in the dialog Saving Options for File Pool which is described in the introduction to this chapter Sec 3 2 and shown in Fig 3 3 This can be used to copy files from CD to a local folder without changing the filetype if option Same Format is used as Saving Format 3 2 15 Update Test the existence of all files collected in a File Pool Files not existing any more are removed 3 2 16 Set Output Folder In general when an operation is applied to files collected in a File Pool the resulting new files are written into the same folder as the corresponding source file By explicitly setting a output folder all created files can be directed into this folder important for example for files on CD 3 3 Edit This menu entry covers all basic operations on data apart from filtering routines which are comprised in an extra menu Data type specific operations are organized in submenus whereas operations valid for several types can be found directly in this menu 3 3 1 Copy Copy contents of active window to clipboard Windows 95 98 NT only In the present version this works only for 1D Data graph or grid window and pictures 3 3 2 Paste Paste c
57. 3 7 TOMOGRAPHY 72 3 7 4 Interpolate Projections For back projection with Convolution Method see Sec 3 7 5 equidistant projection angles according to Eq 3 27 are required As this cannot always be provided by experimental setup it is necessary to create a set of appropriate projections from the measured data This is done by linear interpolation of the measured projections respective to the viewing angles 9 see Fig 3 24 The only input parameter to define is the number M of projections which shall be calculated from the projections within the active Tomographic Input Window After calculation a new Tomographic Input File is created containing all new projection data Another application is to increase the number of projections by interpolation to visu ally eliminate so called artifacts which occur in resulting data field outside of proper reconstructed zone with radius R see Eq 3 21 Be aware that this elimination is just for cosmetic reasons as physical information within R cannot be enhanced by this procedure Obviously a linear interpolation cannot be very accurate but several tests proved this method to be sufficient for most applications Nevertheless precision of recon struction may be reduced especially if missing projections were calculated by inter polation Therefore it is recommended to design the optical setup for tomographical data acquisition to provide projections angles according to Eq
58. 6th entry of the palette offset 255 is reserved for the mask colour 2 1 6 Mapping When an Image is opened the data range from 0 to 255 refers to palette entries 0 to 251 So for a Data Field the range between minimum value and maximum value is divided into 252 intervals each represented by one colour Mapping is a term we use for redefining the data range for which the 252 palette entries are used Values which are beyond the defined data range get the colour for overflow offset 253 those below are marked with the colour for underflow offset 252 The modulo 27 data is handled a little bit different For visualization values higher than 7 are always regarded as overflow values lower than 7 as underflow 2 2 File Types and Formats The various implemented algorithms require input data of different types or origin For instance the phase unwrapping methods are restricted to modulo 27 phase data The results of calculations must also be distinguished To keep to the previous exam ple the result of phase unwrapping can only be phase data whereas tomographically reconstructed data may be of any type e g intensity To prevent any confusion we decided to do a strict differentiation of all data types This allowed us to organize availability of menu entries in IDEA Only those menu entries are available which can be applied to the data represented by the currently active window The others are grayed out We hope you agree th
59. A histogram of an image is the function Nk Pd 3 63 with nz is the number of pixels with gray level J and N is the overall number of pixels of the Image Loosely speaking the Histogram function gives an estimate of the probability of occurrence of gray levels The histogram window included the graph p 1 and two vertical cursor lines To each cursor line belongs a text box below the graph showing I and p J corresponding to current position of the cursor line Y 3 13 WINDOW 111 3 12 5 Data at Selected Points If a polygon has been drawn or multiple points have been selected in a Picture the values at the corresponding coordinates are shown in the z column multiple points dialog described in Sec 3 3 15 Otherwise also the coordinates of the pixels have to be defined in this dialog before the z values are displayed 3 12 6 Sum Shows sum of all elements in any data field If there are masked values in a 2D Data field or Image the additional information of the sum of all masked data is delivered If a rectangle was previously drawn and Rectangle Draw mode is still active the sum of all elements in this area again in and excluding masked pixels is shown additionally 3 12 7 Sum of Rows For 2D Data fields the elements of each row are summed up and the results are plotted in an Integral Abel Data Window 3 12 8 Sum of Columns For 2D Data fields the elements of each column are summed up and the results are p
60. Add n File s 3 2 2 Add n File s Add a specific number of files from disk to File Pool by using an adapted file selector After defining the number n of files to open the file selector window appears with n text boxes Type in full paths or use the Browse button at the left side of each box to select the file by the standard file selector Compared to multiple file selection in Sec 3 2 1 you can easily handle files in different folders In addition to that we added a special feature for files with counters at the end of the filename eg image23 bmp Put the cursor in a text box containing the full path of such a file and press ENTER key The subsequent text boxes are then filled with the same path but with counter incremented from box to box Previous contents are overwritten 3 2 3 Add Files in Folder Select a folder or directory for X Window Systems to add all files located there to a File Pool Don t change the character in the filename box as it is used as a dummy 3 2 FILE POOL 32 3 2 4 Remove Marked File s Entries marked by mouse selection for multiple selection use left mouse button in conjunction with SHIFT or CTRL key are removed from the File Pool 3 2 5 Clear All Remove all contents of the File Pool Remember this does not delete the files from disk as in Sec 3 2 6 since a File Pool only contains paths 3 2 6 Delete Marked Files From Disk All files collected in a File Pool are deleted from the
61. FT Show Real Part Imaginary Part Amplitude 3 8 4 Filtered Back FFT to 2D Mod 2Pi Data Applies backtransformation with Fast Fourier algorithm to active frequency window with filter mask To get modulo 27 phase data corresponding to intensity distribu tion of the forward transformed interferogram this mask has to cover all frequencies outside of the first order of the interferogram s frequency domain in only one of the quadrants using the second order leads to unwrapped phase multiplied by two Be fore backtransformation is computed all masked complex amplitudes are set to zero and only unmasked data contributes to modulo 27 phase data use View Zoom to mask single frequencies If complex amplitudes remain unmasked in symmetrical positions in respect to zero frequency backtransformation results in real data and ap plication of Eq 3 36 makes no sense In this case an error message appears Note the sign of the phase data depends on the quadrant where unmasked frequencies are located 3 8 5 Filtered Back FFT to Image Applies back transformation with Fast Fourier algorithm to active frequency window with filter mask With this mask unwanted spatial frequencies can be marked which are set to zero before backtransformation is applied use View Zoom to mask single frequencies However since this procedure should yield real data it is required that the mask is symmetrical in respect to zero frequency The reason for that is that
62. For the Enlarge Mode all pixels are reproduced in hori zontal direction by the selected number For shrinking only every ft pixel in the lines is kept to form the resized Data Field No averaging is applied e Vertical Factor The same as for Horizontal Factor in this time of course on vertical direction Add 2D Data Add two 2D Data fields A and B of same size by the procedure described in Sec 3 3 7 which yields a new window Multiply by 2D Data Multiply data of two 2D Data fields A and B of same size by the procedure described in Sec 3 3 7 which yields a new window Divide by 2D Data Divide data of two 2D Data fields A and B of same size by the procedure described in Sec 3 3 7 which yields a new window Mean Value All data fields within a File Pool are averaged element by element to calculate a new one consisting of mean values The standard deviation sigma is also shown in a new 2D Data Field All source fields must have the same size 3 3 6 1D Data Add Constant Value Define a value which is then added to each data element of the 1D Data distribution Use negative sign for subtraction Here macros min and max are allowed see Sec 2 4 3 3 EDIT 38 Multiply by Constant Value Define a value with which each data element of the 1D Data distribution is then multiplied Here macros min and max are allowed see Sec 2 4 Substitute Invalid Values Any invalid values or respec
63. INTERFACE OF IDEA 19 Table 2 4 Input Macros In dialogs of IDEA macros can be defined in text boxes instead of values The input is scanned non case sensitive for these macros Macro Interpretation nan Not a number see Sec 2 3 invalid Same as nan inf positive Infinity see Sec 2 3 inf negative Infinity see Sec 2 3 min Minimum of related Image 2D or 1D Data max Maximum of related Image 2D or 1D Data pi Constant 7 pi Constant 7 2 5 The Graphical User Interface of IDEA If Windows 95 98 NT is used all opened or created data fields are visualized in child windows managed by IDEA s main window You have always access to the Menu Bar on the top of the main window where all entries are grayed out which are not allowed for active child window data X Window does not allow this hierarchy Each child window has to have its own Menu bar with allowed entries The Status Bar at the bottom of the Main Window or Child Window in case you use the X Window Version is designed to give the user as much information as possible about the active window and the current interaction In addition we added an automatically updating protocol window to help you keeping track of your evaluation steps Both elements are described in this chapter The basic kind of interaction with data windows is the selection of specific data We provide the possibility to select areas investigate
64. Interferometrical Phase Data pha dat ph PH 2D Data Interferometrical Phase Data modulo 27 m2p dat m2 M2 2D Data Frequency Data Complex Result from frq dat fr FR 2D Data 2D Fast Fourier Transform Data reconstructed by Tomographical Al tor dat tr TR 2D Data gorithm Projection 1D integral data used together pjn dat pj PJ 1D Data with other projections to perform tomo graphical reconstruction 1D integral data to be Abel inverted abl dat ab AB 1D Data Abel reconstruction 1D data recon abr dat ar AR 1D Data structed by Abel Inversion Tomographic Input File containing all tom to Tom Input data necessary for tomographical recon struction File Pool Collection of filenames to per fpl PO File Pool form collective processions Image in raw ASCII format without header bim aim Image export only 1D Data in raw data format without bld ald 1D Data header export only 2D Data in raw data format without b2d a2d 2D Data header export only General 1D Data bin 1D 1D Data General 2D Data bin 2D 2D Data File containing data for user defined filter Alt FL Filter File kernels Mask File contains info about location of msk ms Mask File masked data Colour Palette File list of RGB values pal input only Multiline Window Collection of 1D Data asc distributions export only Angles Projection Angles ASC Preferences cfg 2 5 THE GRAPHICAL USER
65. Preliminary Subtraction of Object Interterogram Object Frame e PSIDataObjectims E Mask Minimal Required Visibilility 96 o Minimal Value for Imax Imin 5 IV Apply Mask File Mask File Marea msk OK N Cancel Figure 3 32 Second dialog for Speckle Phase of Difference for file selection The second dialog shown in Fig 3 9 8 is quite similar to the file selection dialog of the standard phase shifting algorithms see Sec 3 9 1 and Fig 3 41 for explanation of the dialog elements For non file Pool operations there is just the addition of a checkbox with the label Preliminary Subtraction of Object Interferogram Checking this activates the input field for a filename which is by default greyed out By entering a file name for a object interferogram or pressing the small button at the right for file browsing the interferogram which is subtracted from the set of phase shifted interferograms above is determined If the checkbox is left blank the files selected in the upper input fields have to be subtraction fringe interferograms 3 9 9 Speckle 4 Frame for Speckle Subtraction Fringes This method taken from 31 allows phase extraction from speckle subtraction fringes almost without low pass filtering It is restricted to phase shifting corresponding to the 4 Frame method with a 90 Using the denotation of section Sec 3 9 3 the phase shifting formula for the Four Frame method can also be expressed in te
66. Treatment Rigourous Results only for all valid Neighbourhood A single invalid value within the neighbourhood of a pixels results in an invalid filter response Flexible Filterkernel adapts to valid Neighbourhood Only valid data within the neighbourhood is taken into account The summation and division in Eq 3 1 is adapted to the number of valid data e Minimum Number of Valid Neighbours Only if more valid data is within the neighbourhood than the number defined here a filter result is calculated Otherwise the filter response is invalid e Allow Replacement of Invalids by Filter Results The standard filter procedure calculates filter results only for valid data How ever by checking this option the filtering process is also performed if an invalid data is at the center of the filter kernel The filter response then substitutes the invalid value The dialog appears for all convolution filters if all kernel elements are 1 For all other filter kernels the rigourous mode applies Invalid Data Treatment x Invalid Data Treatment Rigorous Results only for all valid Neighbourhood Flexible Filter Kernel adapts to valid Neighbourhood Minimum Number of Valid Neighbours ps Allow Replacement of invalids by Filter Result ok X Cancel Figure 3 16 Dialog for Invalid Data Treatment during Median and Convolution Filtering of 2D Data fields 3 5 1 2 5x5 Blur Applies low pass filter more effect
67. USER MANUAL FOR IDEA 1 5 Software for Interferometrical Data Evaluation Martin Hipp Peter Reiterer Institut fiir Experimental Physik Technische Universitat Graz Supported by START PROGRAMM Y57 FWF July 2003 Contents 1 Introduction 3 1 1 1 2 1 3 1 4 1 5 What is IDEA 23 sa 364 daa dae dd Tablevot Features ra 4 44 04 cia Aa Wee ea a A Requirements ociosos a ra EEE GSS See eS Background of Development o 1 4 1 About the Manual What s new im Version 1 5 o e sc c cc ee cda be eka DA Lol New features sc aes be Re ak ee EDS Lo2 Amendments s 6 Lr arpata atela aed Sa e h y a AP KES 15 3 Known Meses dorar SA a ak a a ee ee a Conventions and Definitions 2 1 Definitions of IDEA Specific Terms 2AM Palettes e ori 44 4 bee eee bbe ea ww a ee oS A o o eaaa eee ea hee aos BERS e es ad BEER 203 MALE dd A ag A de eh we BOE EE ead A 214 2D Date or Data Field ccoo oak Lk kh ee ee Dale Mask oo iaa aaah Pa eae Ss ZO Mapping sa voce eee a kd SG ee oe ee AA Se 2 2 File Typesiand Formats i e 4444600864844 644 545 40444 221 File Type Cl sica o e BER a ORE RE ORES SD 2 2 2 File Format Conventions 04 2 2 3 Internal Format of Plain Image 2D and 1D Data 2 2 4 Internal Format of Frequency File 2 2 5 Internal Format of Tomographic Input File 2 2 6 Internal Format of Filter File
68. Update If you have large File Pools or a slow machine deactivate this switch Otherwise contents of File Pools are constantly updated Files not existing any more are then automatically removed from the list works only for Windows95 98 NT version Zoom Window Automatically Update Activating this menu item causes zoom windows see 3 4 1 to retrieve possibly changed master window information on activation Windows 95 98 NT only Image 2D Data Enable Operations in Selected Area To restrict edit and filter operations to an area selected by drawn rectangle activate this menu item Resize and Rescale Commands excluded You will be asked every time if you want to perform the operation on the whole data set or only on the selected area After some time this may become annoying therefore the option is deactivated by default Mask Enable Operations in Selected Area To restrict mask operations to selected area rectangle drawn in window activate this menu item You will be asked every time if you want to perform the operation on the whole data set or only on the selected area After some time this may become annoying therefore the option is deactivated by default Be aware mirroring and symmetrizing cannot be restricted to selected area Abel Inversion Strict Symmetry Checking To be on the save side for Abel calculations this menu item should be activated default In this case before inversion is performed all integral 1D
69. a and F Francini Phase shifting speckle interferometry a noise reduction filter for phase unwrapping OptEng 36 1997 no 9 2466 72 53 54 55 T E Carlsson and An Wei Phase evaluation of speckle patterns during continuous deformation by use of phase shifting speckle interferometry Appl Opt 39 2000 2628 37 91 92 K Creath Phase shifting speckle interferometry Applied Optics 24 1985 no 18 3053 3058 83 91 Temporal phase measurement methods Interferogram Analysis D W Robinson and G T Reid eds Institute of Physics Publishing 1993 pp 94 140 83 W D Fellner Computer Grafik BI Wissenschaftsverlag Mannheim 1988 19 D C Ghiglia and L A Romero Robust two dimensional weighted and un weighted phase unwrapping that uses fast transforms and iterative methods J Opt Soc Am A 11 1994 no 1 107 117 104 R M Goldstein H A Zebker and C L Werner Satellite radar interferometry Two dimensional phase unwrapping Radio Science 23 1987 no 4 713 20 98 R C Gonzales and R E Woods Digital image processing Addison Wesley Reading 1992 34 49 58 123 BIBLIOGRAPHY 124 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Jie Gu Y Y Hung and Fang Chen Iteration algorithm for computer aided speckle interferometry Appl Opt 33 1994 5308 17 88 B Gutmann and H Weber Phase unwra
70. a 44440665545 4 4448 43 3 3 15 Draw Selection by Coordinates 43 3 3 6 Copy selection lo a raos dak aa bee ee ee ee ee ed 44 VIEW 2 bb aa Oe Sak Se Oo ee bee oe ee A 46 3 4 1 Zoom Selected Area 2 ee ee 46 A II E eR RE BEAR SOO eG 46 SAS MOE adds aaa ee ane be a Pak oes 46 34A Display Mod yi 2c hoo eee Swe ake amp ee ed PE 46 3 45 Change Colour Palette lt 2 0 0 4 6 bee eee ee 46 34 6 Extend Palette a a 6 acca ea ed e e EEE eS 46 S47 Invert Palette ca c a 4 4s rana BAS RARE aa s 47 348 Refresh os s sos eS ee ee ee eae heen nea ee oS 47 349 Slide SHOW A OS Soe BE eae ee eae ee 47 EPICA 22 cae di dan e bbe ea aa A 48 SD Low Pass a8 aca Sek a DOO 49 CONTENTS 3 0 2 High Passi S S63 bey peto ri DE PS 51 3 0 3 User Kernel ya a RRR RD RE EER a SS 51 soc Medid cocoa aaa dias ea ed 53 3 5 5 Selective Median 2 2 cee ee es 53 30 6 Adaptive Median 244440444444 4 244 eae B45 53 3 0 Trigonometrie Filter 22444424248 64688 244444 55 3 5 8 Selective Smoothing 57 3 0 9 Local Enhancement os eae eee ee eee ees 57 3 0 Abel INVersion gt saa 4 dake eR ea eae SSS a AA 58 3 6 1 Get Intesral Data ce so e eater ea OF 4G oo 444444 59 3 6 2 Abel Inversion f Interpolation 60 3 6 3 Abel Inversion Fourier Method 61 3 6 4 Abel Inversion Backus Gilbert Method 62 3 6 5 Abel Inversion Problem Analysis
71. added utilizing nearest neighbour and minimum cost matching algorithms to minimize the overall branchcut lengths To test the quality of modulo 27 Data the number of residues inconsistences where scanning unwrapping algorithms are likely to fail can be determined in Info Number of Residues There is now an additional method to remove a linear phase shift e g phase plane from unwrapped phase data The plane to subtract can be determined by planar regression of masked data which reduces error influences from noise For ASCII export one can choose now between tabulator and space as delimiter in File Preferences The ASCII output for invalid values so far NAN can now be substituted by a user definable string in File Preferences The treatment of invalid data in the filtering process can now be defined for 2D Data fields Two filter routines for modulo 27 phase data have been added both preserving the sawtooth edges In Edit 2D Data Substitute Invalid Values the possibilities to replace invalid data by neighbourhood mean or median have been added It is now possible to extract both interlaced fields from an Image Image data can be squared When applying Edit 1D Data Rescale invalid values are now substituted by the spline routine This way be setting the scale factor to 1 one can interpolate missing data To remove linear tilt from 1D Data one can choose now to fit a line through data outside of the bo
72. al integral phase distribution for a burning candle After transfor mation to required symmetry the resulting distribution Edited Data abl can be Abel inverted The Fourier method yields Reconstruction abr which can be analytically forward Abel transformed to the integral distribution Integral abl which then can be compared to original data Edited Data abl by showing the difference of both graphs in Deviation abl To obtain a two dimensional distribution the result Reconstruction abr can be rotated and displayed as a 2D Data field Rotate bin In the experiment phase stepping was applied at i reference condition air and ii object condition burning candle For four phase shifted object and reference speckle interferograms the phase modulo 27 was calculated using the Carre tech nique Phase Shift Carre Technique The two resulting modulo 27 fields were sub tracted using Edit Subtract Image 2D Data After applying a 5 x 5 selective me dian filter to the obtained phase modulo 27 Filter Selective Median the data were unwrapped using the 2D Scan method cf Sec 3 10 1 Due to noise suc cessive subscans cf Sec 3 10 5 were necessary until the data could be unwrapped using Phase Unwrap with Step Function In the resulting two dimensional phase distribution a horizontal line was selected by activating line draw mode Edit Draw Line and the data along this line were extracted with Abel Invers
73. al means this can also be taken into account Checking this box activates the following two textboxes where upper and lower limit of valid data range can be defined Whenever Eq 3 29 yields values of x out of this data range they are set back automatically to the nearest limit value Initialize with Precalculated Data Obviously the choice of initialization of reconstruction data x has a large effect on the outcome of the iterative procedure especially if the number of iterations is limited In 18 it is recommended to set all data to the same value near to estimated average density of the field or to use the output of a reconstruction with Convolution Here saved data can be used to initialize x Type in the path in the textbox or click the Browse button to use the standard file selector for searching This way a too early interrupted and saved reconstruction can be continued with some extra iterations Note that contrary to Convolution the viewing angles included in the Tomographic Input are only restricted to be smaller than 180 and must not be equally distributed During calculations a window with a progress bar gives additional information about time used so far estimated time to wait last calculated sum of squared corrections and quite handy the change of this value compared of the iteration before Negative values show that corrections have decreased indicating convergence Pressing the 3 8 2D FFT 77 Cancel butto
74. al system is high enough during time of phase shifting and the recording of seven interferograms this method should be used as it reduces any influences of detection errors significantly With phase shifts of a 60 and i 3 2 1 0 1 2 3 between acquisitions the phase is calculated from data for nearly two full modulations a 7 V3 Ip Is Is 16 4 7 I 3 44 arctan 3 Iz 2I I5 lh I I Iy Though spatial dependencies x y are omitted here they are still existent The indices of intensities J correspond to number of interferogram in order of acquisition The required input parameters are set in a dialog similar to this in Fig 3 30 but with seven text boxes and browse buttons For detailed explanations see Sec 3 9 1 3 9 6 Carre Technique As all other methods the Carre Technique requires a constant phase shift but here the value of a must not be known As the system of equations is over determined it 3 9 PHASE SHIFT 87 is still possible to obtain a solution for the phase VII La 22 19 B Is h 1a La Is 11 14 arctan 3 45 Though spatial dependencies x y are omitted here they are still existent The in dices of intensities J correspond to number of interferogram in order of acquisition This algorithm is per definitionem immune to linear miscalibration of the phase shift ing device and also immune to spatially non
75. ar to this in Fig 3 30 For detailed explanations see Sec 3 9 1 3 9 3 Four Frame Technique This technique uses four frames to get an over determined system of equations with a 90 i 0 1 2 3 It is very popular since the solution of this system is less susceptible to detection errors than this for three frame methods la x y La x y x y arctan Non 3 42 The indices of intensities J correspond to number of interferogram in order of acqui sition The required input parameters are set in a dialog similar to this in Fig 3 30 but with four text boxes and browse buttons For detailed explanations see Sec 3 9 1 3 9 4 4 1 Frame Technique With five frames recorded with phase shifts of a 90 i 2 1 0 1 2 the accu racy can be enhanced even further Nevertheless the rather high number of required acquisitions makes this technique unattractive for slow or instationary experimental systems The solution of the over determined system of equations yields 2 13 I4 arct A 3 43 arctan E RL 3 43 where spatial dependencies x y are not written but are still existent and indices of intensities J correspond to number of interferogram in order of acquisition The required input parameters are set in a dialog similar to this in Fig 3 30 but with five text boxes and browse buttons For detailed explanations see Sec 3 9 1 3 9 5 6 1 Frame Technique If stability of the optic
76. as phase step see Sec 3 10 1 e Unwrap Start x y not required for DCT unwrap method For path dependent methods define coordinates x y of reliable phase where scanning for phase steps starts e Zero Phase z y Define coordinates where phase should be zero after unwrapping This is done by subtracting original value after unwrapping at this position from all phase data 3 8 9 Recalculate Image Performs Forward and Backtransformation to an Image using a symmetric filter mask see Sec 3 8 5 to eliminate spatial frequencies before inverse transformation Equiv alent to Sec 3 8 5 this menu entry was established mainly for File Pools allowing several images to be filtered with one filter mask in a single step 3 8 10 Recalculate 2D Data Equivalent to Sec 3 8 9 but instead of Images 2D Data Fields are created omitting the re mapping required for Images see 3 8 5 3 8 11 Show Real Part Only Extracts real parts from all complex amplitudes of a frequency fields forming a new 2D Data Field Note Coordinates are not longer shown in the mode for frequency data where origin is at center pixel but in standard mode with origin at upper left corner 3 8 12 Show Imaginary Part Only Extracts imaginary parts from all complex amplitudes of a frequency fields forming a new 2D Data Field Note Coordinates are not longer shown in the mode for frequency data where origin is at center pixel but in standard mode with origin at uppe
77. at the ease to obtain a general view makes up for the somewhat pedantic differentiations For those who feel uncomfortable by these restrictions IDEA allows to change the internal data type see Sec 3 1 6 to Sec 3 1 9 on one s own responsibility 2 2 1 File Type Classes The various file types or data types respectively can be separated into six classes You can find this classification also in the standard file selector dialog where in the 2 2 FILE TYPES AND FORMATS 13 box entitled File Types you can select from different types distinguished by their extensions In the case of IDEA you do actually not select a file type but a class of file types These classes usually comprise files with different extensions Here they are listed up again together with other unique file types which cannot be loaded or saved with the standard file selector e Image Two dimensional 8 bit data binary or ASCIT regarded as gray scale values See Sec 2 1 3 for further description e 2D Data Any two dimensional data in double precision format binary or ASCII See Sec 2 1 4 e 1D Data Any one dimensional data in double precision format binary or ASCII e Tomographic Input File containing all input data for tomographic recon struction For detailed file structure see Tab 2 2 e File Pool Collection of file names for collective processing e Graphic Format Picture Colour picture in standardized format For our interpret
78. ation of the term picture within the context of IDEA see Sec 2 1 2 Filter File 2D Filter kernel data Can only be saved and loaded in Filtering User Kernel e Mask File Contains location of masked pixels Saving is only possible in menu Mask Save Mask loading only in Mask Add Mask File e Colour Palette File RGB Values for customized palette ASCII Loading is possible only in Edit Change Palette e Angles Contains angles of projections for tomographic reconstruction Load ing and Saving is only possible in Tomography Edit Projection Angles 2 2 2 File Format Conventions By default IDEA saves data in the byte order used by the detected machine type result of detection shown at top of Protocol window As most computers PCs use the little endian byte order with the least significant byte of a word first Avoid porting data saved with such machines to big endian systems which use reverse byte order and vice versa It is clear that all data would be completely wrong interpreted To allow data exchange between different machine types we provide a conversion between little and big endian order Refer to Sec 3 1 11 for details The identification of most supported file formats is made by the first two bytes of the opened file We call these two bytes Identification Code ID It overrides the file extension and is the same for data in binary and ASCII format In Tab 2 3 the ID is given for all file types 2 2 3 Internal
79. border pixels This is undesirable with computation time of order N x N and not only complicates coding the algorithm for a sparse cost matrices as suggested in 6 but also diminishes the advantages of this approach Therefore ways have been found to take into account a reduced number of border pixels as described in the explanation of the dialog elements below Of course reduc ing the border might prevent to find the true global minimum though this is quite unlikely For border pixels there is a special rules for cost evaluation connections of border pixels to each other and to virtual residues to have cost zero In general requirements of memory resources and computation time are very high for this algorithm Computers having at least 256 MB RAM and clock speeds of more than 1 GHz work at full capacity with only a few hundred pairs of residues if the neighbour border reduction mode is enabled Nevertheless results are always convincing It is convenient to first check the number of residues and borderpixels as described in Sec 3 12 12 to estimate the computation time will be bearable Refer to Sec 3 10 10 1 for definition of valid phase regions and corresponding starting points for the flood fill unwrapping The dialog for the minimum cost matching algorithm features the following input elements e Invalids Substitute input field Define a value which substitutes the invalid phase data e Border Cost Factor input field To i
80. c 3 10 1 The resulting step function Order bmp in Fig 4 shows the different orders of the modulo 27 phase field which are the areas between the jumps in the data Each area is represented by a different colour Using Phase Unwrap with Step Function the final 2D Phase distribution can be calculated yielding Window Carrier pha in Fig 4 In this Window there is still the information of the carrier fringes included 4 1 EXAMPLE 1 TOMOGRAPHIC RECONSTRUCTION 115 idea File File Pool Edit View Filtering AbeHnversion Tomography 2DFFT Phase Shift Phase Mask Information Window Help al fie Taela o 08 ele melee 510 01 Please Specify a File to Open a E PhaseniData 2 pha PhaseniData 3 pha PhaseniData 4 pha PhaseniData 4 pha PhaseniData 3 pha PhaseniData 2 pha PhaseniData 1 pha Dateiname fangles asc Dateityp fio Raw Ascii Data asc Abbrechen T Mit Schreibschutz ffnen Projection Name Phasen Data 1 pha Phasen Data 2 pha 2 Phasen Data 3 pha Phase Phasen Data 1 pha Phasen Data 4 pha E S l yA Phasen Data 4 pha Phasen Data 3 pha Phasen Data 2 pha Phasen Data 1 pha Figure 4 2 Screenshot of IDEA showing how to set projection angles for Tomography In the dialog Projection Angles the Load button has been pressed popping up the standard Windows File Selector dialog in order to select an already created
81. called Problem Analysis which performs inversion with a series of input parameters compares the different results and shows several interesting trends in graphs Fully aware of this problem we added a new method based on the Backus Gilbert algorithm 2 With this method the algorithm tries to find a way to get a smooth curve with as small deviation of the integral from measured data as possible taking the whole distribution into account The weighting of both criteria can be given with a single input value called tradeoff parameter Compared with the other methods the basic shape of the result keeps the same but the matrix calculation is rather time consuming In addition to this true Abel Inversion methods you have the possibility to use the tomographical algorithms ART and Convolution However according to 38 their effectiveness cannot be compared to the straight forward methods but they have impressive smoothing power and may serve at least for comparison of results All methods require symmetry of measured data h r and zero value at the borders of the radial distribution This often requires manipulation of data with Edit 1D Data where some menu entries are dedicated to symmetrizing just for that purpose But as for smoothing it is recommended to manipulate original data as less as possible 3 6 1 Get Integral Data Extract previously selected line data from 2D Data Field or Image to perform Abel Inversion afterwards Refer
82. can be interpreted the same way At the upper right corner for 1 up to 141 the maximum number of polynomials all results are shown row by row in the 2D Data Window In the Pro tocol Window the 9 smallest values of Data fint four Deviations00 bin see Fig 4 7 are printed Bibliography 10 11 12 13 14 15 H A Aebischer and S Waldner A simple and effective method for filtering speckle interferometric phase fringe patterns Opt Comm 162 1999 205 56 G E Backus and F Gilbert Uniqueness in the inversion of inaccurate gross earth data Phil Trans R Soc London Ser A 266 1970 123 192 59 D J Bone Fourier fringe analysis the two dimensional phase unwrapping prob lem Appl Opt 30 1991 no 25 3627 32 98 100 M Born and E Wolf Principles of optics sixth ed Pergamon Press Oxford 1980 T Bothe J Burke and H Helmers Spatial phase shifting in electronic speckle pattern interferometry minimization of phase reconstruction errors Appl Opt 36 1997 5310 6 93 J R Buckland J M Huntley and S R E Turner Unwrapping noisy phase maps by use of a minimum cost matching algorithm Appl Opt 34 1995 no 23 5100 8 99 101 102 J Burke H Helmers C Kunze and V Wilkens Speckle intensity and phase gradients influence on fringe quality in spatial phase shifting ESPI systems Opt Comm 152 1998 144 52 94 A Capanni L Pezzati D Bertani M Cetic
83. consumption is possible if the iteration number is set to 3 or 4 The input dialogs for this procedure are exactly the same as these for the Phase of Difference method see Sec 3 9 8 for detailed explanation 3 9 10 Speckle 4 1 Frame This method proposed in 9 takes advantage of the otherwise disturbing stochastic nature of a speckle pattern As with the other two implemented speckle methods the phase shifting takes place at stable reference conditions Here four frames In x y n 1 2 3 4 with a required phase shifting angle of 90 have to be acquired From the 3 9 PHASE SHIFT 92 phase shifting the mean intensity Jp the visibility V or modulation intensity Im respectively and the initial speckle phase 4 can be obtained With this information the cosine of the overall phase of an pixel au combined of the speckle phase and the phase change Ag that is present at an altered object state can be obtained from another speckle interferogram fop Ioj Lo COS ball Im 3 54 The ambiguity of the arccos function prevents us from directly determining au and subsequently the phase shift Due to the identity cos cos and the fact that the function is usually defined to give the result in the interval 0 71 there is a sign ambiguity that has to be removed This problem is solved by regarding not only a single pixel but to take into account a pixel neighbourhood with a certain extension Assuming the pha
84. ction center is always 2 2 FILE TYPES AND FORMATS 16 0 0 0 new line 1 1 1 new line 255 255 255 2 2 8 Internal Format of Mask File Completely in binary format this file type includes information about the location of masked pixels or data respectively of a master picture To save disk space we decided to set one mask bit for every pixel of the master picture If the pixel is masked the bit is set to 1 else to 0 The bits are packed together in groups of eight to form bytes which is the data format used to save a mask For saving the master picture is scanned row by row for mask colour setting the bits in the corresponding bytes At the end of the file some lower significant bits in the last byte are not set if the number of pixels in the master picture cannot be divided by eight This surplus on bits keeps its initial values of zero which has no effect since they are never referred to The masked data is internally treated as a vector so the size parameter has to rep resent the length in bytes The size check made before adding a mask to a picture is therefore limited For instance the mask of a 256 x 512 Image fits well on an Image with size of 1024 x 128 No error message would appear in this case 2 2 9 Internal Format of File Pool File The File Pool is in principle a collection of file names All data is in ASCII Format and in the following order PO pathstyle new line paths separated by line feeds
85. d a a ho A a Roe oll EE e 109 3 12 2 Line Graph is maraca A A SE a a a 109 3 123 Edit Multiline Graph sx as ae a ee ee a 109 3 124 Histogram s diia a ee a ae we we we a es A 110 3 12 5 Data at Selected Points 0000 eens 111 SIMAO UM 4 a a a a eas A GAS RR RE A ee A 111 SDF QUO ROWS 4 ec ob a e a ee A 111 3 12 8 Sum Of Columns ee ee aR RRR Yh wee RE ww 111 3 12 9 Extreme Valles xo y adios E E AAA 111 3 12 10 Average Value msm ade a ee E 111 3 12 11 Number of Masked Pixels 02 00008 111 3 12 12 Number of Residues o s as e ma o 002 eee eee ee 111 WildOW oe cnn RR A a REA EAR a REGGE RS OO 111 CONTENTS 4 Example 113 4 1 Example 1 Tomographic Reconstruction 113 ALI Backsround c s s 2460045 cee E ba ee ee a A 113 4 1 2 Phase Evaluation with 2D FFT 113 4 13 Tomographic Reconstruction o 115 4 2 Example 2 Phase Shifting lt lt lt lt lt 117 4 2 1 Experimental Background 117 4 2 2 Phase Shift Evaluation lt lt 117 42 3 Phase Unwrappime Viera a AA 118 4 3 Example 3 Abel Inversion e o 118 A3 Background ss goig ge A Ad a oe 118 ASO INDeIMVETSION 2 bata a ae ANA 119 Chapter 1 Introduction 1 1 What is IDEA IDEA Interferometrical Data Evaluation Algorithms is a software developed for evaluation of phase information from interferograms
86. d as unspecified 1D Data and can be copied to Clipboard by Edit Copy It can also be zoomed see Sec 3 4 1 3 12 3 Edit Multiline Graph With the following submenu items you can manage contents of a Multiline Graph after creating an empty Window by File New Multiline Graph The Multiline Graph is used to plot up to 10 1D Data distributions in a single graph which is scaled to maximum x and y value of all data In Fig 3 12 3 an example for a Multiline Graph is shown In the lower part the names of the different distribution filenames or window titles are shown in the colour of the corresponding curve in the graph In the vicinity of a name the mouse cursor changes its shape to a hand Then a right mouseclick shows a popup dialog with the following entries UU 3 12 INFORMATION 110 e Extract Selecting this entry by mouse click creates a 1D Data Window including the selected curve e Delete Removes selected curve from Window e Colour Opens a the standard dialog for colour selection There a new colour can be defined for the curve The procedure of adding and deleting data from this Multiline Window is similar to that used for Slide Shows File Pools and Tomographic Input Files Nevertheless the submenu entries are explained here in detail once more Add Single 1D Data To add a data distribution corresponding to any 1D Data Widow at the IDEA desktop select this menu item to enter adding mode In this mode the
87. d by drawing a rectangle in B The same is true for destination x y Select the point with crosshairs in A All according data will be up to date in the dialog Insert Image Picture 2D D ata Ea Source Valid Coordinates 0 511 0 511 a Size of Source Area w oo h fe00 Destination Valid Coordinates 0 511 0 511 x fioo y f 00 Insert Only MASKED Values of Source Area N C Insert Only UNMASKED Values of Source Area C Copy Only the MASK but NOT the Values C Copy Only the INVERTED MASK but NOT the Values OK Cancel Figure 3 11 Dialog for Insert Image Picture 2D Field e Source x y Coordinates of upper left corner of area in B which is to be inserted in A e Size of Source Area w h Width and height of the area to insert e Destination x y Coordinates in A where upper left corner of selected area in B shall be inserted Never mind if area does fit completely into A e Insert Mode of Source Area Select if you wish to insert all masked or unmasked data It is also possible to overtake just the mask with or without preceding inversion 3 3 9 Rescale Image Picture Rescales Images or Pictures to new width and height This is done by calculating the corresponding address of each destination pixel in the source image reverse mapping 28 In general the result is a non integer value If interpolation is applied intensity and distance of the four pixels that surround the calculated position
88. d type from the superior File Pool If some entries are marked only those are taken into account This allows a convenient file organization especially for files spread around different folders Collect all files of a project in a main File Pool and create all possible pure subordinates Perform evaluations on these File Pools At the end of the session merge all relevant results into the main File Pool This keeps your project files together and allows direct access all the time 3 3 EDIT 33 3 2 12 Subtract Every ith File In the upcoming dialog one has to define the Subtraction Group Size i Starting from the top all filenames are then divided into subgroups each of it containing i filenames Remaining names are ignored The selection in the dialog checkbox sets either the leading file or the last file represented in a subgroup as the argument of the subtraction This file is then subtracted from the other 1 files in the group A new File Pool is then created containing the filenames of the files created as results of all subtractions This In FilePool Subtraction has been designed for series recordings of sets of phase stepped Speckle interferograms each followed by one interferogram with altered object state This File Pool operations allows to create a series of subtraction fringes interferograms also referred to as secondary Speckle interferograms which can be further processed by phase shifting procedures
89. d value e Mask File available only if checkbox is active If desired define the path of Mask File Either type in the path or click on the Browse button to the right to open standard file selector The specified mask data is automatically assigned to the resulting modulo 27 data As masks and invalid values for P x y are set by several means a short overview is given here e A mask is set when 1 previously created mask data in general for the interferograms is taken into account or 2 the minimal visibility or modulation of intensity values of all phase shifted interferograms falls short of user defined value e An invalid value is set when 1 previously created mask data in general for the interferograms is taken into account in this case also a mask is set covering this invalid values 2 the arctan of the fraction cannot be calculated as both denominator and numerator are zero Invalid values are shown in colour specified in the actual colour palette see Sec 2 1 1 3 9 2 Three Frame Technique 120 This technique uses also three frames but here with a 120 phase shift between recording of interferograms and i 1 0 1 The phase shift z y in each pixel can be calculated in this case by x y arctan va aoe 3 41 3 9 PHASE SHIFT 86 where indices of intensities J correspond to number of interferogram in order of ac quisition The required input parameters are set in a dialog simil
90. dard deviation computed in a neighbourhood around i j M is the global mean of f The local gain factor A can be modified by a constant factor k In the dialog the size of the matrix respectively the neighbourhood must be defined as well as the Gain Factor k A i j k O lt k lt l 3 6 Abel Inversion When radially symmetrical objects shall be investigated by interferometry the ob served result will be the phase shift integrated along the optical path of the light beams ray bending is neglected To get the desired radial distribution of whatever causes the phase shift one has to apply Abel Inversion algorithms to the integral data which leads to phase shift in one line element pixel as a function of radius r The relation between integral data h y and radial distribution f r see Fig 3 21 is given by the forward Abel Transform dl r h y 2 far 3 5 The result h y of this transformation is measured as integral phase shift but we want to retrieve the radial distribution f r from the integral The analytical solution of this problem is the Abel Inversion _ 1 f dhy dy 3 f r a 7 ea 3 6 3 6 ABEL INVERSION 59 For measured data points this inversion can of course not be applied directly hence different numerical methods have been developed to solve this inverse problem Some commonly used methods are thoroughly evaluated and compared in 38 with special attention paid to the propagation of errors
91. de is given whereas section 4 shows mouse position and data value located there In Fig 2 2 the contents of the Status Bar for 2D Data and Images is shown depending on Draw Mode For 1D Data refer to Fig 2 3 2 5 3 Protocol Window After startup a child window titled Protocol is automatically created At this time it contains some information about pre checks From then on all your important actions are protocolled in this window as short text lines Most important every creation of data is reported including information about the automatically created window title If write protection is disabled the user is allowed to add personal notes or delete contents It is not possible to close this window but it can be made invisible 2 5 THE GRAPHICAL USER INTERFACE OF IDEA Rectangle Line Draw Mode Draw Mode Mask Pen Shape Coordinates of Coordinates of end point L Line S Square starting point corner opposite corner i C Circle R Rectangle xy y C Crosshairs Mask Pen Size Save Selected File 0 100 200 400 200 250 200x300 640x480 L S13 21 415 99 Size of Image Data Field width x height Mouse pointer coordinates x y Value in Image or Field at location of mouse pointer Active menu entry Coordinates of center pixel x y width x height of rectangle including line as diagonal Crosshairs Draw Mode Coordinates of previous sel
92. ded as an supplemen tation to the Phase of Difference method However the fact that the phase values can be calculated without direct application of the despised low pass filtering makes this method interesting Phase maps obtained with this algorithm look similar to those calculated with the Difference of Phase method 10 see also introduction to this section though the noise is still visibly higher The phase accuracy that can be achieved is claimed to be 27 30 rad after substitution of phase results for pixels with Im less than 5 grey levels at 8 bit resolution by neighbourhood mean followed by a final filtering process 3 x 3 median This error is smaller by factor two than this of the Phase Of Difference method without enhancement by iteration The improve ment can be visually confirmed at the discontinuities in wrapped phase data e g at the border line where the phase changes from z to 7 For data evaluated by the Phase of Difference method there is a typical ripple in this border line However it should be mentioned such a straight border appearance as shown in 31 can not be achieved with the implementation in IDEA not even for simulated data The computation time required for this method with the iteration number set to one is comparable to that of the iterative Phase of Difference method as for comparable accuracy some additional data treatment is required Otherwise an improvement at cost of significantly increased time
93. defined Noise Density noise is added for this pixel Entering of 1 causes noise in every pixel defining 0 5 means that to 50 statistically of all pixels noise is added Relative Absolute Noise Amplitude Defines range of noise Amplitude The value of noise which is added to the pixels of the ideal interferogram those which passed density test see above is generated by a random number generator It is statistically distributed between noise amplitude and noise amplitude The sum representing new pixel value may be out of range 0 255 so overflows are set to 255 underflows to 0 The amplitude can be given absolute or relative see below Define 0 for amplitude if you do not want to have noise added m 3 1 FILE 26 e Use Absolute Noise Level to check If the box is active the value for Noise Amplitude is interpreted as absolute value Else the absolute noise amplitude for each pixel is calculated from pixel valuexrelative Noise Amplitude In this case minimums of a interferogram have less noise than the maximums e Vertical Horizontal Circular to select Determines appearance of fringes 3 1 2 Open Read file from disk using the standard file selector of Windows 95 NT or X Window Select desired file type see Sec 2 2 1 to be viewed in file list or for X Window from the submenu IDEA accepts only files with correct ID see Tab 2 3 Consider byte order settings in File Preferences Open File Assumi
94. dition to this least square criterion smoothing must be provided by forcing neighbouring polynomials to overlap For Abel Inversion they are results from different measurements yielding different inversions This dialog not only allows simultaneous extraction of integral data from multiple source files but also the definition of several vertically separated integral data distri butions within all source files corresponding each to a cross section with the radially symmetric object Extraction of data creates a File Pool including all corresponding integral data distributions To distinguish between those of different source 2D Data Fields and different vertical positions the user defined part of the filenames see Fig 3 3 is internally extended by the name of the source file and the coordinates of the distribution s intersection with the center line see also Fig 3 26 If integral data shall be retrieved at different vertical positions from only a single 2D Data Field use a File Pool containing only this file 3 6 2 Abel Inversion f Interpolation This numerical method interpolates the resulting distribution f r see Eq 3 6 with polynomials of third degree The number M of polynomials can be defined by the user see Fig 3 23 The distribution to be Abel inverted is separated into M zones P each represented by a own set of coefficients for the corresponding polynomial in the f domain To provide sufficient smoothness the poly
95. drawing a polygon or selecting multiple points or editing data at selected points z data The appearances here is for entering z data the coordinates have been selected before by drawing a polygon or multiple point selection with the mouse or by previously entering data into another version of this dialog where the z column is missing and the x y fields are editable 3 4 VIEW 46 3 4 View This menu contains functions to change appearance of data visualization It does not effect data itself apart from reversible reordering procedures 3 4 1 Zoom Selected Area Enlarges Selected Area with zoom factor to be selected in submenu The enlarged area is shown in a new slave window which cannot be saved and exists as long as the original master window Coordinates in status bar relate to the master window All operations done in slave window e g drawing a mask are simultaneously performed in the master window When changes are made in original window they are overtaken from the zoom window on activation if File Preferences Zoom Window Automatically Update is activated Windows 95 98 NT only in X Window Systems it is necessary to use View Refresh Zooming is also possible for data in a Line Data Window see Sec 3 12 1 but actually the distribution is just enlarged by reproducing every pixel according to the zoom factor 3 4 2 Rotate Rotate Image or 2D Data in 90 steps Select direction in the submenu
96. e 2D Data or Image window see Sec 2 5 1 3 3 13 Draw Polygon Activate this to globally switch draw mode to draw a polygon in active 2D Data or Image window see Sec 2 5 1 3 3 14 Select Multiple Points Activate this to globally switch draw mode to multiple point selections where a number of points can be selected by mouseclick within an Image or 2D Data window see Sec 2 5 1 3 3 15 Draw Selection by Coordinates Depending on the currently active draw mode a dialog pops up where the corre sponding number of coordinates can be entered for the subsequent drawing in the active window For polygons or multiple point selection the dialog shown in Fig 3 12 appears The dialog elements are note that this dialog has several appearances de pending on the menu entry calling it some items might not be available since here all of them are listed e x y z input fields Here the x column number y row number and for other functions of IDEA z values data at pixel x y are shown or can be entered The Windows 32 version shows non editable data in grey 3 3 EDIT 44 e Next gt button If more than 10 points have been selected or entered this button switches to the next sheet with another 10 rows of input fields e Previous gt button Switches back to the previous sheet of entries e Clear Entries Removes all editable entries from the input fields e g sets z values to zero e Save button
97. e applications of phase stepping unwrapping Abel inversion 2D FFT and tomo graphic reconstruction with convolution method are demonstrated All shown figures are actual screenshots from IDEA Using this as a tutorial with frequent references to Chapter 3 should provide a fast method to get familiar with the software The required files can be downloaded from our web page http optics tu graz ac at 4 1 Example 1 Tomographic Reconstruction from Holographic Interferograms 4 1 1 Background This example demonstrates the evaluation of the experiment presented in Ref 51 Multidirectional heterodyne holography is applied to a glow discharge near the reso nance line of sodium which evaporates from the cathode into the plasma To obtain the density of sodium atoms in horizontal planes between the electrodes it is nec essary to determine the phase shift respectively the refractive index in each pixel area of this plane This requires determination of integral phase shifts in different di rections and the application of tomographic reconstruction algorithms to these data which is performed by using 2D FFT on interferograms with carrier fringe system For each direction this fringe system is recorded with and without discharge 4 1 2 Phase Evaluation with 2D FFT The IDEA screenshot in Fig 4 demonstrates the evaluation of an interferogram Origi nal img that corresponds to one direction As shadows one can see the tipped radially s
98. e eae Pe ee ee ee 31 3 2 3 Add Files in Folder 00220000 31 3 2 4 Remove Marked File s 32 320 Clear Al aro Ar a SS 32 3 2 6 Delete Marked Files From Disk 32 3 2 7 Open Marked File s 32 32 0 Sort Alphabetical ii a bee eee dae ee 32 3 2 9 Extract Every ith File 2 2 0828 se eb 10000 32 3 2 10 Extract Marked Files o o 32 3 2 11 Extract Marked Images 2D Data o o 32 3 2 12 Subtract Every WA File sewa ssa a eb ee ee nds 33 3 2 13 Change File Counters cocoa 33 32 1d Convert Miles e x drei a aro RR A a a 33 Aal Update osa e Ey ee eS 33 3 216 Set Output Folder i ee ecsaeaaadewn eee ba dew A 33 Edi ge te ace ereere epee eee ane oa 33 Dol o taboo oO heat wees RAGE e EE EE Ea A 33 So Paste coin rr AA 33 3 33 Cp e on 5 Se a ida oe a SS 34 Soe IMAGE a kk a ae ee ea bo eG ee es 34 3300 2DDitas 20 Phd oe dADEADLA Ge OEE Ea Dee eS 35 3 30 ID Dd 6 666 046 Cee Oe eS SS aoe be he ra 37 3 3 7 Subtract Image 2D Field cocer eb ee eee es 41 3 3 8 Insert Image Picture 2D Field 42 3 3 9 Rescale Image Picture 220 048 42 S10 Draw Lines oo g aun eee OO ED eo eek be ee eee ee 43 3 93 11 Draw Rectangle y lt lt Tacora na Ta ba eR Aa wee 43 3 3 12 Draw Crosshairs e s 6 44 22 5 8628 bea Sean s 43 3 3 l3 Draw Polygon s cnn O eds eee a AA 43 3 3 14 Select Multiple Points o i c
99. e end of the central line is corrected to this value The equally spaced projections correspond to reconstruction planes with same distance e OK Cancel Show Buttons After changing settings in the dialog the display in the corresponding picture of phase data can be actualized to current values by pressing the Show button Finally the settings are confirmed by pressing OK button or operation is inter rupted by Cancel button Select Line s xi Starting Point of Center Line x1 so y1 50 Ending Point of Center Line x2 20 Y2 foo Coordinate of Starting Point 7 Line Relative to Line Center Separation dx 200 dy Po Line Separation jo OK Cancel i a b Figure 3 26 Dialog for Building Tomographic Input File a and explanation of the input pa rameters b Confirmation of dialog inputs starts the extraction of all projections from the 2D phase fields included in the Input File Pool The result is a File Pool with as much Tomographic Input Files see Sec 2 2 5 as parallel projection were selected in the sample window Each Input File includes the set of projections at the same vertical position in the 2D phase fields defining a reconstruction plane of the object at corresponding height 3 7 3 Edit Tomographic Input File Changes contained projection data and angles of Tomographic Input Files All ma nipulations are similar to those available for Slide Shows see Sec 3 4 9 and File Pools see Sec 3 2 so user
100. e gives some feedback about the overall curvature c N C Y Fi 3 16 i 1 which is plotted in dependence of the number of polynomials n used for inversion N is number of measured data points Typically this curve is of parabolic form 6 Curvature of inversions with Fourier Method four Curvature This curve gives some feedback about the overall curvature c N c of eM 1 Nu 3 17 i 1 which is plotted in dependence of the maximum order of model function N N is number of measured data points 2D Data Windows 1 All Abel Inversions with f Interpolations fint AllAbelInversions 2D Data window comprising all results of f Interpolations Vertical coordinate corresponds to number of used polynomials n whereas x coordinate is correlated with radius of distribution Therefore as n 0 is not allowed the first line is set to invalid values From the visualization one can see dominant noise rising out of smooth data with increasing n 2 All Abel Inversions with Fourier Method four AllAbelInversions The same as in previous item for Fourier Method Here the vertical coordinate corresponds to maximum order of model function N As N 0 is not allowed the first line is set to invalid values 3 Deviations of all inversions with f Interpolations from Fourier inversions fint four Deviations A further criterion for good parameters is when both inver sion methods lead to the same result Therefore each result of f Interp
101. e menu by High Pass Low Pass and User Kernel include coefficients and are referred to as Filter Kernels An example of an 3 x 3 Kernel is shown in Fig 3 14 The filtering is performed by placing the Kernel at the upper left corner of the image and to sum products between the kernel coefficients and the intensities values of the pixels currently covered by the Kernel Following the notation in Fig 3 14 the response of the linear Kernel is M R paz C222 C929 3 1 Here the z denote the gray levels of pixels at locations corresponding to location of Kernel elements The divisor D and the multiplier M is used to scale the sum to the valid gray level range The result R is written to an equally sized image at the location of the Kernel s center pixel After that the Kernel is shifted to the right and the whole procedure is repeated until the Kernel reaches the right side of the Image then the next row is processed The filtering effect depends on the coefficients in the Kernel and increases with its size Tt is clear that Eq 3 1 cannot be applied to pixels at the outer rim of the images which is unreachable to the center pixel of the Kernel This rim is just copied from the original to the filtered image For filtering restricted to an area File Preferences Image 2D Data Enable Operations in Selected Area is checked this rim effect does not occur In this case pixels outside the selected area are taken into account if
102. eck to activate edit mode Window Visible If active default the Protocol Window is present on IDEA s desktop Else it is hidden but still existent Note The Protocol Window cannot be closed Clear All Delete all entries of the Protocol Window 3 1 11 Preferences Here some preferences for IDEA can be set saved or loaded After startup of IDEA the current directory and afterwards the directory of idea ere is searched for a con figuration file If no cfg could be found the options for maximum safety in data handling are activated by default 3 1 FILE 28 Set Working Folder By default any window title includes the path of the file relative to the folder from which IDEA has been started For the elements in a File Pool the path is given relative to the location of the File Pool file For both cases the master folder can be redefined to a common working folder with this menu entry All window titles are actualized afterwards Image 2D Data 1D Data Create New Window If this menu item is active for every edit and filter operation on data of chosen type a new window is created default If you like experimenting but not piling windows on your desktop deactivate the switch Creation of new Images is then only done if two different data sets are used to create a new one eg Edit Subtract Image 2D Data or Edit Insert Image Picture 2D Data apart from any Mask operations File Pool Automatically
103. ected ixel x PREI Coordinates of pixel selected before previous pixel x y Coordinates of last selected pixel x y 10 25 8 220 50 96 10 250 253 Values of Image Field at coordintes x y Step Function Crosshair Coordinates of selected pixel Draw Mode L Line R Rectangle C Crosshairs Modulo 2 Pi data at location of mouse pointer 50 100 5 640x480 C 2D 21 89 3 14 Value of Step Function at selected pixel Slide Show Number of pictures in Slide Show Size of Step Function picture width x height Mouse pointer coordinates uy Sub Scan Mode 2D 2D Scan SP Spiral Scan 1 20 Image00 bmp 640x480 10ms 21 89 Number of active Filename of active Slide Slide Size of active Slide width x height Mouse pointer coordinates uy Current Timer Interval Figure 2 2 Contents of Status Bar for Images and 2D Data Contents of Coordinate Section see Fig 2 1 depend on active draw mode Identical contents are omitted For pure Pictures there are no mask entries in Draw Mode Section For the Step Function data type which is saved as a Picture the Status Bar is quite different from standard configuration and therefore shown separately 2 5 THE GRAPHICAL USER INTERFACE OF IDEA 23 Integral Data and Line Graph x coordinate of p center line x coo
104. ectrum and the spectrums of the transformed signals present 3 5 FILTERING 56 greater overlapping than for unfiltered data and due to the nonlinearities in in the filter process distortions might be generated These are most significant where the cosine of the signal is close to zero That s why one should always be critical with results and should not be too glad by the visually stunning smoothness of the filtered data However this filer approach shows very good performance when compared to other techniques In 1 an iterative implementation of this algorithms is suggested This is done by restarting the procedure described above at step 1 using further on the filter result gf obtained at step 5 of the previous iteration For 20 to 30 iterations the following effect is claimed to occur dense fringes are perfectly filtered after a few iterations and are not further affected by later iterations Sparse fringes however continue to be filtered more strongly from iteration to iteration This is the favourable behaviour of an automatic adaptive filter which here of course comes at the expense of quite long calculation times Trigonometric Filter x Convolution with Unit Kernel Neighbourhood Mean Median Fitter Fitter Selection Convolution with User Kernel ee dea iterations fi Invalid Data Treatment Rigorous Results only for all valid Neighbourhood Flexible Filter Kernel adapts to valid Neighbourhood
105. efining the number n of files to open this file selector window appears with n text boxes Refer to Sec 3 2 2 and Fig 3 4 for further description Remove Marked Projections Entries marked by mouse selection for multiple selection use left mouse button in conjunction with SHIFT or CTRL key are removed from the Tomographic Input File Clear All Removes all projection data from the Tomographic Input File Open Marked Projections Projections marked by mouse selection for multiple selection use left mouse button in conjunction with SHIFT or CTRL key are opened and corresponding graph windows appear Sort Alphabetically Use this to sort the projections in the Tomographic input file alphabetically The en tries are accordingly reordered and projection angles are reset according to Eq 3 27 Edit Projection Angles Opens a grid with projection angles in the same order as projection data are contained in Tomographic Input File Values can be typed in but also loaded from an raw ASCII File see Sec 2 2 10 Closing the Tomographical Input window is possible only after closing the grid Edit Marked Projections Opens a grid for each marked projection for marking multiple entries in the Tomo graphic Input File use left mouse button in conjunction with SHIFT or CTRL key where data can be changed manually see Sec 3 12 1 and Fig 3 12 1 Note Closing the Tomographical Input window is possible only after closing all grids
106. en pixel in the phase map the closed residuum is searched by a square spiral path as described in Sec 3 10 2 2 The found residuum is then the center of a new spiral path to search for the closest partner either a residuum of opposite sign or a border pixel 3 When found the connection by a branchcut is established and both residues are marked to be ignored for the further evaluation 4 The procedures restarts at step 1 until all residues have been connected to a partner One should not expect too much from this method Only if the density of residues is limited the result will be acceptable In some cases the spiral scan method for unwrapping is more powerful than this application However for the scanning meth ods there is a limit to 254 phase orders in IDEA Therefore the nearest neighbour branchcut has been implemented as an alternative with comparable processing speed for interferograms of rather good quality As for all branchcut methods it is essential to provide the information of the bound aries of valid data This is done by setting a mask in the wrapped phase map outside of the valid regions Note that invalid data is no criterion for localizing the bound aries as these have to be substituted by a valid value in order to set the branchcuts The reason for that is that also in valid regions pixels with invalid phase data might occur for example due to insufficient intensity modulation in a speckle interferogram Such
107. eometry is peculiar or measured data has errors imposed the ART usually well outperforms the convolution method To apply ART to projection data within a Tomographic Input File in Win dows 95 98 NT its window must be active on desktop several values must be defined in a dialog shown in Fig 3 28 e Size of Reconstructed 2D Data By defining a number of pixels smaller than original length of projections re construction can be performed with same center but accordingly reduced extend to both sides This is quite handy if domain of object was overestimated when projection data were obtained and computation time should not be wasted e Maximal Number of Iterations Primary Break Off Condition whenever this number of iterations is finished calculation will be stopped e Relaxation Parameter RP Defines a value for A in Eq 3 31 The upper limit for this value is 1 since higher values might lead to divergence 3 7 TOMOGRAPHY 76 RP Adjustment after Iteration Number Automatic adjustment according to Eq 3 32 is as soon applied as the current number of iteration exceeds the here defined value Before that the settings in upper textbox is used for A Minimum Correction Break Off Condition For each iteration all values for c in Eq 3 31 are squared and summed up According to Eq 3 29 this value represents the overall change of the recon struction made in the just finished iteration Further iterations are skipped
108. er Transform is slightly changed by the imposed step function these changes are lower than those caused by resizing algorithms 3 8 2 Gerchberg Fringe Extrapolation The 2D FFT algorithm assumes the Image to be extended periodically in all direc tions With other words the result of transormation belongs to a plane tiled with interferograms all adjacent and identical Therefore discontinuities at edges and lim ited domain of interferogram within acquired Image are an important source of error This is known as boundary problem In 43 an interesting solution is presented using a simple iterative algorithm proposed by Gerchberg In IDEA it can be applied the following way 1 Forward transform the interferogram suffering from boundary problem 2 Find the domain of the fringe system by creating masks see Sec 2 5 1 around it and performing back transformation to Image Remember that for that purpose a mask symmetrical to central pixel is required If the fringe system is well reproduced while noise and other discontinuities are suppressed save the filter mask 3 In the original interferogram mask all pixels outside of fringe domain 3 8 2D FFT 79 4 With the original interferogram still active select this menu item A dialog pops up where the number of iterations and the file containing the filter file created in 2 must be specified This can be done by typing the path into the textbox or use the Browse Button to
109. er of analytical projections Therefore it cannot be applied directly to measured data and different numerical methods have been developed to solve this inverse problem In IDEA two of these methods are included which were actually developed for medical X ray computer tomography CT The filtered back projection based on the mathe matical procedure of a convolution 18 is rather fast but restricted to constant angle between viewing directions and concerning precision inferior to the algebraic recon struction technique ART 18 The latter is an iterative technique which roughly speaking fits the asymmetrical distribution to the projections Its disadvantage of high calculation time is not only compensated by higher precision but also by ar bitrary viewing directions and its capability of considering physical criteria for the object under investigation For many experimental setups it is not possible to provide viewing directions sepa rated by constant angles and equally distributed within m which is required to perform back projection with convolution method To create a approximated set of projec tions fulfilling this requirement IDEA provides a routine Tomography Interpolate Projections which applies linear interpolation to measured projections This can also be used to eliminate disturbing artifacts in the reconstructed data field Obviously quality of reconstruction increases with number of viewing directions As suggested i
110. er result is processed for the center pixel of the window but it is set to invalid NaN Required Deviation After calculating the filter result for a pixel the deviation to the original value is calculated Only if this deviation is higher than the value to be defined here the filter result is accepted Otherwise the original value is set in the filtered 2D Data field If the filter parameters are set in a way that none of the five conditions in Tab 3 1 are met at some pixels these values are inserted unchanged into the new 2D Data field 3 5 7 Trigonometric Filter This is a widely used filter designed for modulo 27 phase data Its principle is as simple as effective With i j denoting the wrapped phases at locations i j in the field to be filtered the procedure is as follows 1 Calculate the field sin p 1 7 2 Apply a low pass filter to this field resulting in a data field s i j Calculate the field cos i 7 gt Ww Apply the same low pass filter to get c i j 5 The filtered phase data modulo 27 can then be calculated by p arctan 2 3 2 By transforming the phase data to sine and cosine fields the problem of the typical sawtooth edges in conjunction with filtering is elegantly evaded The comparable smooth data distributions can safely be low pass filtered in these fields before the arctan function transforms the data back to modulo 27 data Drawbacks are that the noise frequency sp
111. ernel from the list Note that the ability to define rectangular filter kernels allows even better adaption of the convolution window to the minimum fringe pitch for the IPOD method Filter Repeats Define the number of successive filter operations to be done before the wrapped phase is calculated It is not recommended to set this value higher than 1 for the IPOD method Filter Iterations Define the number of iterations for the IPOD method As mentioned above 3 to 4 iterations should be sufficient File Pool Mode This dialog section is visible only in case the menu entry had been selected with an active File Pool window There are three possibilities concerning type and order of the images included in the File Pool Phase Stepped Reference Interferograms are at Top In case the File Pool contains only uncorrelated speckle interferograms there have to be np phase shifted speckle interferograms included in the File Pool which have been recorded at stable reference conditions By selecting this option the procedure picks the np phase shifted reference interferograms from the top of the file list in the File Pool All other files belong each to a different object state and are subtracted from the phase shifted interferograms according to step 3 of the POD procedure listed above Phase Stepped Reference Interferograms are at Bottom By selecting this options the four phase shifted interferograms to which all others are correlated by
112. erometry for the quantitative optical investigations of asymmetric transparent objects only integral phase information can be obtained very similar to X ray imaging where only integral absorption data are derived These applications require tomographical reconstruction algorithms for further evaluation which are able to obtain the inhomogeneous spatial distribution of the desired data from integral data Obviously reconstruction of an asymmetrical object needs multi directional measurement of integral data Tomographical procedures can be applied to various optical diagnostic techniques for instance spatially resolved measurements of emission coefficients absorption X ray tomography laser induced fluorescence and of course interferometrical phase measurements 49 51 35 The analytical relation between integral data distributions h p 0 which we call pro jections and the local distribution f r for explanation of the parameters see Fig 3 24 is the so called Radon Transformation 00 h p 0 1 ds f ve s2 arctan 5 0 3 19 o0 P This relation assumes straight ray paths within the object neglecting any ray bending effects which could occur at high gradients of refractive index It was named after the Austrian mathematician Johann Radon who was able to invert Eq 3 19 to i f T 1 Oh p 0 f r 22 ao dp ee a ar a o 3 20 The application of the Radon Inversion Eq 3 20 would require an infinite numb
113. es is displayed 2D FFT frq After creation of the filtermask Masked frq the backtransformation can be applied resulting in a field of phase modulo 27 data Back FFT m2p For unwrapping a Step Function is necessary Order bmp The obtained phase field Carrier pha has already been clipped to original size but still contains the information of the carrier fringes After subtracting the reference phase field of the carrier fringes and after removing the linear tilt the true phase field showing a glow discharge is obtained Arc pha solute values of the resulting complex data field To obtain modulo 27 data only the frequency bandwidth of the fringes in the fre quency data has to backtransformed We mask everything which should be set to zero for backtransformation using Mask Mask Outside Area The result is shown as Masked frq in Fig 4 As this mask must be used again for the reference interfero gram it should be saved Frequency msk As described in Sec 3 8 4 we calculate modulo 27 phase data using 2D FFT Filtered Back FFT to 2D Mod 2Pi The resulting field is corrupted at locations where no fringes appeared in the interferogram To hide these areas we copy the mask of Padded img using Mask Copy Mask The result is shown as Back FFT m2p in Fig 4 Unwrapping To obtain the final distribution of phase shift the modulo 27 data are unwrapped using Phase 2DScan Method see Se
114. est method is to connect to to the nearest partner which is obvious due the fact that they appear naturally in pairs However if the density of residues is high this might not be the best solution Since discontinuities of the unwrapped phase across branchcuts are allowed it is obvious to define the min imum overall branchcut length as the criterion for optimized unwrapped phase data 1 Proposed optimizing algorithms are for example simulated annealing 40 stable marriages algorithm 39 and minimum cost matching algorithm 6 The latter has 1It has also been proven that under certain conditions even better results can be achieved if the gradient of the phase field is also taken into account 17 3 10 PHASE 100 been implemented in IDEA as it is the only one which guarantees to yield the global optimum With the branchcuts defined and acting as barriers the unwrapping is completely independent of the path taken Therefore to carry out the last remaining task of the unwrapping scanning methods comparing more than two neighbours to each other are not required A simple and fast flood fill algorithm borrowed from graphics computing is sufficient 3 3 10 10 1 Nearest Neighbour Branchcuts The branchcuts are set to the nearest neighbour in binary form only pairs are al lowed which is the most simple approach to the problem without any optimization for minimizing the overall cutlength 1 Starting from a giv
115. f through the pictures move the mouse cursor into the slide show and click on the left button to display the next picture or the right button to show previous picture With active Slide Show the contents of the Status Bar change The Coordinate Section see Fig 2 1 shows index of current picture in Slide Show the number of pictures in slide show and the path of the currently displayed picture In the Draw Mode Section dimension of current picture and timer interval are shown Whereas timer driven Slide Shows can run in background without disturbing operation on other windows the idle Status Bar is still updated by the Slide Show window The initial idea behind the Slide Show was to visualize differences of visibility and location of fringes This can be quite handy if one wants to evaluate the quality of the chosen frequency mask Compare the backtransformed image with the original using a slide show The location of fringes should be the same in the two slides at least in relevant areas If so the frequency mask was well chosen Later we expanded such Flipping Windows to Slide Shows allowing more pictures to be displayed sequen tially For instance you can use this to animate interferograms recorded for phase shifting making the fringes run across the image or show results of tomography corresponding to different heights of the reconstructed slices u L 3 5 FILTERING 48 Add Picture To add a Picture correspondin
116. for a scale added to a pseudo colour visualization of any data Resizing is possible using Edit Rescale Image Picture 3L FILE 25 Simulated Interferogram This creates a simulated interferogram with optional noise added Several parameters must be specified in the dialog shown in Fig 3 1 interferogram x Number of Fringes 20 Width of Interferogram 512 Height of Interterogram 512 Minimum Intensity Imin 0 254 Maximum Intensity Imax Imin 255 255 Phi Shift Noise Density 0 1 Relative Absolute Noise Amplitude i IV Use Absolute Noise Level Fringe Shape e Vertical Horizontal Circular Figure 3 1 Dialog for Creating New Simulated Interferogram Number of fringes For circular fringes the number refers to number of fringes in the diagonal of the interferogram to be created Width of Interferogram Height of Interferogram Minimum Intensity Imin 0 254 Defines minimum intensity of ideal cosine distribution without noise Maximum Intensity Imax Imin 255 Defines maximum intensity of ideal cosine distribution without noise Must be greater than Imin Phase Shift Phase shift in degrees By default phase is O at center for circular fringes at left side for vertical fringes and at upper side for horizontal fringes Noise Density 0 1 A random number generator creates a noise density value between 0 e and 1 for every pixel If this value is below the
117. forward transformation of real data yields complex values at symmetrical positions which are conjugated to each other therefore the visualization showing absolute values is also symmetrical Obviously both of these complex amplitudes are required to obtain real data again This way Image filtering in the frequency domain can be applied Unfortunately elimination of spatial frequencies may also change dynamic range of an image after 3 8 2D FFT 80 Fast Fourier Transform Ea FFT Filter Mask C phase image1 _frqmsk Unwrap Mask le phaselmagel_m2p msk Unwrap Method Phase Jump fi i N Unwrap Start Discrete Cosine Transform Without Mask x pss Y ss Discrete Cosine Transform With Mask Zero Phase x fp y p O a Figure 3 29 2D FFT Dialog for Calculate 2D Phase Data backtransformation and even worse extreme values may be out of range especially if zero frequency is omitted Therefore data is always re mapped between 0 and 255 after backtransformation which could pretend enhanced contrast To provide a feedback about this the extreme values are written into the protocol window before re mapping is applied 3 8 6 Filtered Back FFT to 2D Real Data The same as for Sec 3 8 5 but here a 2D Data Field is created instead of an Image but without remapping 3 8 7 Calculate 2D Mod 2Pi Data Allows calculation of modulo 27 phase data from an Image in only one step Of course as an filter mask is required the
118. g c r 5 Calculate modulo 27 phase by Im represents imaginary part and Re the real part Im r arctan Re 3 36 This procedure assumes a completely parallel carrier fringe system which is hard to obtain In general it is recommended to use a slightly different application of the 2D FFT The transfer of vo to origin is skipped yielding r r U r after calculations of Eq 3 36 with U r denoting carrier fringe phase If Y corresponds to a single carrier frequency vo as assumed in previous explanations then can be obtained by subtracting the linear tilt corresponding to vp In IDEA this can be made by selecting Phase Remove Linear Phase Shift where three points in must be defined to calculate the tilt This should lead to same result as Eq 3 36 But if the carrier fringe system is not so perfect it can be recorded and evaluated separately yielding U r Principally a powerful method the 2D FFT is restricted to monotonic phase distri butions and therefore fails with circular fringe systems 3 8 1 Zero Padding The two dimensional Fast Fourier Transformation requires input fields of size 2 2 a and b are natural numbers To make data fields of different size also accessible to 2D FFT IDEA puts the original Image or 2D Data Field into the center of a field with next valid size and fills unoccupied pixels with zeroes Additionally a mask is set there This is a rather common method and although the Fouri
119. g to any 2D Data or Image on the IDEA desktop select this menu item to enter adding mode In this mode the mouse cursor changes its shape to a little hand when a Picture is entered The menu item keeps checked as long as you either click the hand on the picture and add it to your Slide Show or reselect the menu item to end the adding mode compare procedure in Sec 3 3 7 Add File s Add files from disk to Slide Show by using the file selector of the platform in use for which Multi File Selection is activated In Windows 95 NT use the SHIFT or CTRL key in conjunction with the left mouse button to choose a group of files in the filenames list of the selector Note The last selected filename appears always at the beginning of the text line showing the current selection located below the filenames list reversing the temporal order of your selection Special Feature If a File Pool already contains all data you want to view activate the File Pool Window and use this menu entry to transform all files in the File Pool to visualizations in a Slide Show which is automatically created Add n File s Add a specific number of files from disk to Slide Show by using an adapted file selector After defining the number n of files to open the file selector window appears with n text boxes Refer to Sec 3 2 2 and Fig 3 4 for further description Remove Picture Removes all contents of Multiline Graph Clear All Remove all contents of the Sl
120. gins causing the curve moving up again However sometimes more than one minimum can be observed or even worse there is no distinct minimum at all In this case one of the other curves should provide better feedback The x axis of this graph is the higher number of used polynomials For example the value at x 6 is the overall difference of inversions with 6 and 5 polynomials 2 Deviations for Fourier Method four Deviations See previous item Deviation for f Interpolation but regard number of poly nomials as maximum order of model function Nu The lower number of poly nomials is always 1 The x axis of this graph is the higher maximum order of model function For example the value at x 6 is the overall difference sum of squared deviations of inversions with order 6 and 5 of model functions 3 Deviations from measured data and inversions with f Interpolation fint Chi Depending of the used number of polynomials n the sum of squared deviations from measured data is plotted in this graph Typically this curve is hyperbolic and ideal parameter value should be chosen in the vicinity of the curve s knee 3 6 ABEL INVERSION 65 4 Deviations from measured data and inversions with Fourier Method four Chi Depending of the maximum order of model function the sum of squared devi ations from measured data is plotted in this graph 5 Curvature of inversions with Interpolation fint Curvature This curv
121. gle invalids but gets into trouble if there are invalid areas larger than the filter dimension Then a single filter pass is not sufficient to eliminate the invalid pixels The procedure must be repeated by setting the number of iterations in the dialog greater than 1 However do not expect the substitutions to be always smooth as valid data is spread from different directions starting from eventually different values into the invalid area 3 3 EDIT 36 Substitute Invalid Values xj Substitute by Neighbourhood Maian C Substitute by Fixed Values Filter Parameters Filter Vvidth Iterations 3 y n Filter Height Minimum Number of Valid Neighbours a y 5 mured Valle Substitution Substitution for intinity ma Substitution for infimity frin Substitution for Not a Number p OK Cancel Figure 3 5 Dialog for Substituting Invalid Values For the substitution with fixed values one can define separate values for Infinity Infinity and Not a Number in the text boxes near to bottom of the dialog Here macros min and max are allowed see Sec 2 4 Shift Minimum Maximum Average to 0 Shift the current distribution of data to set new zero level Modulus Get the absolute values of 2D Data Remove Linear Tilt Subtract a user defined plane from 2D Data The definition of the plane is done by selecting three point coordinates If three x y coordi
122. hich are located int outermost rows and columns In many applications these frequencies are automatically set to zero after forward transformation 36 but not in IDEA Therefore if such behaviour is desired this can be achieved with this command before any backtransformation is performed 3 11 20 Substitute Masked Values All currently masked pixels are set to a value which must be defined in small dialog see Sec 2 4 for macro inputs 3 11 21 Symmetrize Mask E Mask Priority Regard the whole field consisting of pairs of pixels symmetrical in relation to central pixel Then this command sets mask at pixels whose partner is currently masked This way masked areas are symmetrized E Image Priority Regard the whole field consisting of pairs of pixels symmetrical in relation to cen tral pixel Then this command removes mask from pixels whose parter is currently unmasked This way holes in mask are symmetrized 3 11 22 Mirror Mask Horizontally H Mask Priority Mirror mask with horizontal axis through central pixel to either side This means mask from upper half is mirrored to lower half but also vice versa 3 12 INFORMATION 109 Image Priority Mirror unmasked condition of pixels with horizontal axis through central pixel to either side This means holes of mask in upper half are mirrored to lower half but also vice versa 3 11 23 Mirror Mask Vertically Mask Priority Mirror mask with vertical axis through central p
123. hnique 90 aaau 84 3 9 2 Three Frame Technique 120 85 3 9 3 Four Frame Technique 86 3 9 4 4 1 Frame Technique 86 9 9 50 6 1 Frame Technique 0 lt caacaea ced 2444 eee es 86 3 9 6 Carre Technique 2 22454008 204e eee ee bee Pa dees 86 3 9 7 6 Frame with Nonlinearity Correction 87 3 9 8 Speckle Phase of Difference 020 0000 87 3 9 9 Speckle 4 Frame for Speckle Subtraction Fringes 90 3 9 10 Speckle 4 1 Frame o 91 3 9 11 Spatial Phase Shifting 120 92 3 9 12 Spatial Phase Shifting 90 o o o ooo 94 ILMO PHASE oso ora a CRAG EOE eee Coe be eee ews 94 3 101 2D Scam Method sica Sek eke a RR DOO i 95 CONTENTS 3 13 3 10 2 Spiral Scam Method lt s v s weste ada a eS 96 3 10 3 One Step Unwrapping by Scan o o 97 3 10 4 Set Phase Jump Value for Scan Methods 97 3 10 5 Sub Scan 2D Enabled 2 646 a a cts we 97 3 10 6 Sub Scan Spiral Enabled 97 310 7 Add Step Functions sia 44 98 3 10 8 Remove Step Function e 98 3 10 9 Unwrap with Step Function o 98 3 10 10 Unwrap with Branchcut Method 98 3 10 11 Unwrap with DCT cena eae oa 104 3 10 12 Interferogram from 2D mod 2Pi Data 105 3 10 13 Interferogram from 2D Phase Data
124. ide Show Remember this does not delete the files Start Slide Show Forward Start timer driven Slide Show displaying pictures in the same order as they were added Stop Slide Show Backward Start timer driven Slide Show displaying pictures in the reverse order as they were added Timer Interval Define temporal interval between display of two consecutive pictures in milliseconds 3 5 Filtering In some cases acquired images show unwanted noise e g due to Speckle Effect To remove noise is one example for application of image enhancement techniques which are not limited to low pass filters required here but include also high pass edge enhancement and median filters to mention only a few All of this filtering methods are available in this menu where we concentrate on spatial filtering algorithms filtering in frequency domain can be done in menu 2D FFT using spatial masks for image processing This technique is well known and 3 5 FILTERING often used due to easy implementation and fast processing It should be found in any comprehensive literature dealing with image enhancement e g 15 In IDEA this algorithms can also be applied to 2D Data though the following description only mentions images C1 Ca C3 C4 C5 C6 C7 Cg C9 Figure 3 14 Example for Kernel of Linear Spatial Filtering Usually masks are square matrices with odd sidelength Linear filters represented in th
125. ield or by using the corresponding button The phase mapping is done automatically for modulo 27 data fields Within a File Pool one can include alternating phase shifted sets of initial sate and final state interfer ograms The resulting phase maps can be automatically subtracted from each other by using File Pool Subtract Every t File where i 2 3 9 PHASE SHIFT 84 The drawback of the Phase of Difference method is obviously the large number of interferograms to be recorded at least 6 and the requirement of object stability during the phase shifting process That is why the method often fails in the presence of external disturbances like vibration rigid body motion or temperature and flow fluctuations turbulence for phase objects Therefore alternative methods based on temporal phase shifting have been developed which require acquisition of a single speckle pattern interferogram at the final object state Together with the information of a phase shifted set recorded at a stable reference state the phase information can be obtained from this interferogram given that no decorrelation effects took place between measurements All methods include a spatial filtering in some way which reduces spatial resolution in the phase result Three of such methods are implemented in IDEA but only the Phase of Difference method can be regarded to be well known see Sec 3 9 8 Sec 3 9 9 and Sec 3 9 10 With the availability of CCD camera
126. ifferent approaches to set connections with a minimized overall length four pixels is referred to as residue either positive p or negative p Residues appear if a fringe within an interferogram is disrupted for example by noise The break is likely to occur also in the corresponding phase discontinuity curve where the starting point and the end point turn out to be residues with opposite sign just like in Fig 3 10 10 Investigating all 2 x 2 squares within a wrapped phase map reveals any residues so this is the initial step for the branchcut unwrapping procedure As shown with the example in Fig 3 10 10 a path enclosing an equal number of p and p will be a permitted unwrapping route since s 0 To ensure that only permitted paths are taken connections of either pairs of p and p or more generally groups of residues with an equal number of and signs can be set With these connections so called branchcuts acting as barriers for unwrapping no phase inconsistencies are encountered by a scanning unwrapping procedure One has also to take into account isolated residues located near the boundary where the partner is out of view This monopole have to be connected to the border To set the branchcuts is the second step of the procedure Definitely this is the most critical part and many publications are concerned with the problem how to optimize the result by finding the right combi nations of connections The simpl
127. inclination angle of the plane wave to the CCD results in a linear phase change of 27 within the extend of three camera pixels this corresponds to a dense carrier fringe pattern when compared to the technique of spatial heterodyning with phase determination by Fast Fourier transform With the mean speckle size adjusted to the same extension by an aperture and an inclination axis parallel to the CCD columns the speckle phase can be calculated by the three frame algorithm with the intensities corresponding to a phase shift of 27 3 0 and 27 3 taken from three neighbouring pixels in a row The evaluation is done with interferograms before and after object deformation and subtraction of resulting phase fields give information about phase change due to local surface displacement As indicated above the phase evaluation of the interferograms can alternatively be done with the Fourier Transform method as described in Sec 3 8 which is of course much slower On the other hand the inclination of the reference wave front does not have to be adjusted to a specific amount This more general approach is usually referred to as Digital Fourier Holography The phase shift algorithm used here is the same as for the Three Frame Technique under assumption of a spatial phase shift in horizontal direction only It has to be mentioned that this is the most simple approach More sophisticated methods require measurements of object and reference wave separately 5
128. ing phase shifting algorithm 4 If maximum number of iterations is not reached a calculate a set of synthetic phase shifted interferograms Ij j 1 2 nr np is the number of frames recorded for one phase shifting cycle from the wrapped phase T z y 1 cos lp x y 7 a 3 48 b increase iteration counter by one c repeat steps 3 and 4 otherwise leave iteration loop and take last calculated as the final result In 16 a proof of convergence and the ability of suppressing noise that fluctuates at least twice as fast as the information is given for Four Frame phase shifting There is also an estimation of the phase error caused by and increasing with fringe frequency changes The error also increases with the number of iterations but only by factor two after five iterations In summary it is recommended to select the dimension of the convolution window to be between one quarter and one third of the smallest fringe pitch and to iterate three or four times The phase error should then be less than 5 e g 27 20 rad Without iteration the phase error is at least 27 15 As proven in the cited paper the iterative method is more accurate than the repeated averaging method which simply repeats the low pass filtering process several times before the smoothed subtraction fringe interferograms are fed to the phase reconstruction algorithm With only slightly longer computation time for large dimensions of the conv
129. invalid data the dialog for invalid data treatment described in Sec 3 5 1 1 appears in case all of the defined Filter Kernel elements are 1 Otherwise the rigourous filter mode is applied 3 5 4 Median This non linear filter is used for noise reduction rather than blurring The value of each pixel is replaced by the median of the values in a neighborhood of that pixel instead of the average To do that all pixel values covered by the mask are sorted and the median is determined by taking the central value of the sorted set of values which is then written to location of mask center in filtered image For example using a mask size 3 x 3 leads to 9 sorted values equal values have to be grouped from which the 5th is the median In case a 2D Data field containing invalid values is to be filtered there appears another dialog as described in Sec 3 5 1 1 3 5 5 Selective Median Though standard median filtering is used to spare edges its effect on high edges like in modulo 27 data is not negligible To avoid data corruption due to filtering we implemented an adapted median filter which detects edges under filter mask and performs filtering in this case only if height of the edge is below an user defined limit 42 Edge detection is done after sorting when median of the higher half and of the lower half both including central value of the data set are determined For example for a mask of size 3 x 3 these are the third and seventh elemen
130. ion Get Integral Data The final integral data are shown as Candle abl in Fig 4 6 4 3 2 Abel Inversion Abel Inversion requires symmetric data cf Sec 3 6 Hence the asymmetric dis tribution Candle abl in Fig 4 6 is modified using Edit 1D Data Remove Linear Tilt and Edit 1D Data Average Left and Right to obtain the symmetric distribution Edited Data abl This symmetric 1D data are then Abel inverted using the Fourier Method described in Sec 3 6 3 with 6 model functions The result is shown as window Reconstruction abr in Fig 4 6 To get some feedback about the quality of inverted distribution the analytical form 4 3 EXAMPLE 3 ABEL INVERSION 120 el 6 aloja Cae A SH 2D Data Data four AllAbellnversions00 bin TE El Datafour Curvature dat Fel ES Line Graph 5 1 Data four Chi dat ME x Olne Graph 5 1 Datatour Deviations dat Cl x 78 0 000181829 520x205 C 51 95 18 0 00986731 Figure 4 7 Screenshot of IDEA demonstrating Problem Analysis for Fourier Method This window shows Data distributions belonging to the Fourier Method calculated by Abel Inversion Problem Analysis 1D Data distributions were extracted from a Multiline Window and zoomed by factor 5 Data four Curvature dat shows the overall curvatures of Abel Inversions depending from the maximum order Equally organized Data fo
131. ion of the used number M of projections is sufficient Of course in many applications this necessity for equidistant projections will collide with the actual experimental setups Then non equidistant projection data have to be trans formed into a new appropriate set of equidistant data see Sec 3 7 4 The Convolution implemented in IDEA uses contents of a tomographic input file For Windows 95 98 NT its window must be active on desktop to be able to select this menu entry and input values defined in a dialog shown in Fig 3 27 e Size of Reconstructed 2D Data By defining a number of pixels smaller than that of the original length of pro jections reconstruction can be performed with same center but accordingly reduced extend to both sides This is quite handy if domain of object was over estimated when projection data were obtained and no calculation time should be wasted e Parameter for Hanning Window This is the value for a in Eq 3 26 It controls the filter effect of the win dow function Eq 3 25 increasing with smaller values of a In principle if data collection is very reliable and the bandlimit of projections is very close to half the Nyquist frequency then it is appropriate to choose a 1 yielding FA U 1 bandwidth window No spatial frequencies are suppressed Oth erwise if there is considerable noise in the data or if aliasing errors occur then lower values for a are more likely to produce good results si
132. is substituted by corresponding values at left side Area in which averaging can be performed is shown gray in a As indicated here discontinuities might appear at borders of this region 3 3 7 Subtract Image 2D Field Subtract other Image or 2D Data B of same size from current active data A For Images negative results are substituted by their modulus To perform subtraction follow instruction below 1 Click on window of data A to activate it 2 Enter this menu The menu entry gets a check mark showing that the system waits for selection of field B 3 Point the mouse to window of data B You will notice a change of the cursor s shape to a hand This signals proper size and data type to perform operation In all other windows the cursor gets a shape similar to a No Parking sign 4 Click on window with data B if you change your mind and want to cancel the operation enter this menu again the checkmark will disappear and the system will return from waiting to normal state 5 A new window showing the result of the operation is created 3 3 EDIT 42 3 3 8 Insert Image Picture 2D Field Copy selection of data from other window B into current active window A Both must include same data type The procedure is similar to that in Sec 3 3 7 apart from an additional dialog appearing after clicking on B see Fig 3 11 The parameters needed to define are listed below Source x y w and h may be preliminary selecte
133. is sufficiently short an image of a fast moving object will consist of two recorded states each of it in one interlace field The one consisting of the odd lines usually denoted as field 1 can be extracted here Extract Interlace Field 2 Extracts all even rows from an Image to form a new one See previous menu item 3 3 5 2D Data Add Constant Value Define a positive or negative value which is then added to each data element of the 2D Data Here macros min and max are allowed see Sec 2 4 Multiply by Constant Value Define a value with which each data element of the 2D Data is then multiplied Here macros min and max are allowed see Sec 2 4 Substitute Invalid Values To get rid of invalid values see Sec 2 3 which for example prevent Fast Fourier Transform to work properly three simple algorithms are implemented in IDEA see Fig 3 3 5 Substitution by neighbourhood mean or median or by fixed values In the first two cases the Data field is scanned for invalids Infinity Infinity and Not a Number are all treated the same way starting from the top left then the mean value or median of the valid data within the neighbourhood with dimensions Filter Width and Filter Height is determined The invalid is substituted by this value only if more than a definable number of valid neighbours have been found This simple procedure works well for data fields peppered with sin
134. is the same as for the adapted file selector described in Sec 3 2 2 The order of this files corresponds to the indices of intensities in equations for the phase modulo 2r e Minimal Required Visibility Define a value for minimal required visibility in percent At positions x y where this value cannot be reached a mask is set Nevertheless the result for is not discarded The defined value is ignored if the value in the lower text box is more restrictive e Minimal Value for Imax Imin Define a value for minimal required modulation of pixels at corresponding po sitions x y in the interferograms At any position x y where the calculated modulation falls short of the defined value a mask is set without discarding value for P x y Be aware that there is concurrence to a defined Minimal Required Visibility see above as only the more restrictive value is taken into account 3 9 PHASE SHIFT 85 Phase Shift X Image Files 45 CAPSMNmagel bmp 3 435 IC PSiimage2 bmp 225 C PSlimage3 bmp Mask Minimal Required Visibility 96 p Minimal alue for Imax Imin E Mask File fePstimaget msk OK Cancel Figure 3 30 Input Dialog for Phase Shifting here for Three Frame Technique e Apply Mask File checkbox Check this box if you want a mask e g created from domain of interferogram to be taken into account Masked pixels are simply ignored and phase is set to invali
135. ive than standard filter described in previous menu Like there all 5 x 5 Kernel elements are 1 but here the factor M D in Eq 3 1 has to be 1 25 to scale result R to valid data range For filtering 2D Data fields containing invalid values refer to Sec 3 5 1 1 3 5 1 3 7x7 Applies very effective low pass filter Like for the standard 3 x 3 filter kernel see Fig 3 15 the 7 x 7 all Kernel elements are 1 but here the factor M D in Eq 3 1 has to be 1 49 to scale result R to valid data range For filtering 2D Data fields containing invalid values refer to Sec 3 5 1 1 3 5 FILTERING ol 3 5 1 4 Gauss 3x3 This low pass filter uses a Kernel representing a Gaussian distribution see Fig 3 17 weighting influence of pixels by their distance to the center The factor M D in Eq 3 1 is 1 16 1 2 1 2 4 2 1 2 1 Figure 3 17 Filter Kernel for Gaussian Low Pass 3 x 3 Since some Kernel elements are not equal to one only the rigourous filter mode can not be applied to 2D Data fields containing invalid data see Sec 3 5 1 1 3 5 1 5 Pillbox 5x5 This low pass filter uses a Kernel representing a Gaussian distribution see Fig 3 18 The factor M D in Eq 3 1 is 1 33 Rir rin n RIN N NDN R RI IN PR DN r RIN N NDN R Rir ea i n n Figure 3 18 Filter Kernel for Pillbox Low Pass 5 x 5 Since some Kernel elements are not equal to one only the rigouro
136. ive window Since mask data is saved bit wise in an one dimensional data array see Sec 2 2 8 the size check made before adding a mask to a picture is limited For instance the mask of a 256 x 512 Image fits well on an Image with size of 1024 x 128 No error message appears in this case 3 11 3 Save Mask Saves mask of current active window to file using standard file selector The format of a mask file is described in Sec 2 2 8 3 11 4 Remove Mask Removes and discards mask from current active window without warning so be sure that useful data was saved before 3 11 5 Invert Mask For each pixel of the active Picture the binary mask information is inverted so previously masked data is unmasked and vice versa This can be used as an eraser Invert the original mask and set a mask where it was unwanted before Inverting again will remove the mask in this area 3 11 MASK 107 3 11 6 Square Pen Enabled Changes shape of mask pen to a square This means a single click with right mouse button in a Picture not in Step Function Window draws a square in mask colour The size of the square is determined by current setting of Mask Pen Width see Sec 3 11 8 Both shape and size of mask pen are shown in Draw Mode Section of Status Bar see Fig 2 1 3 11 7 Circular Pen enabled Changes shape of mask pen to a circle This means a single click with right mouse button in a Picture not in Step Function Window draws a circle in
137. ixel to either side This means mask from left half is mirrored to right side but also vice versa Image Priority Mirror unmasked condition of pixels with vertical axis through central pixel to either side This means holes of mask in left half are mirrored to right side but also vice versa 3 12 Information The menu includes all commands related to retrieving and showing data 3 12 1 Line Data Retrieves all data from pixels forming a previously drawn line and shows it in a grid window see Fig 3 12 1 In the text box located at the uppermost part of the window the value of the currently selected grid entry can be edited At the left side of this box the number of the grid entry with prefix y is shown In the left column the rows are enumerated The right side shows values of pixels in order from starting point to endpoint of drawn line The Goto button opens a dialog where a row number can be entered After confirmation the grid id scrolled until this row shows up at the top With slider and arrow buttons on the right side of the window the grid can be scrolled manually Grid entries are selected by a left mouseclick on the grid 3 12 2 Line Graph Retrieves all data from pixels forming a previously drawn line and shows it in a line graph window the simplest of all 1D Data windows This includes a red cursor line which can be moved to show value at current x coordinate in Status Bar see Fig 2 3 The plotted data can be save
138. izes the FPU to raise an exception for division zero by zero which terminates the running program although the result of this operation is a specific encoded symbol NaN not a number To override such settings IDEA re programs the FPU at startup to be less rigid In fact all exceptions are deactivated Results of mathematically non defined values are represented by the already mentioned symbol NaN results of infinity e g 1 0 are referred to as Inf or Inf With the FPU settings used for IDEA other software in use is not concerned it is possible to perform any operation on data including NaNs and Infs The results are well defined and of course again non values When data including such non values is saved in ASCII format the symbols are writ ten as NAN NAN INF or INF if sign is printed depends on compiler used for actual version of IDEA In binary data they are encoded in TEEE standard binary format Exchanging such data with other software could lead to serious problems Therefore we implemented routines to substitute the symbols by valid user defined values Edit 2D Data Substitute Invalid Values Not all algorithms especially those for reconstruction are able to tread invalid values which may lead to corrupted results if they are applied to Data Fields containing NANs or INFS Such data need to be cleaned For 2D Data Fields this can be achieved for example by substituting invalid values and
139. just the contrast of the active Image Enhancing contrast is in principle done by defining a range of gray levels in which areas of interest appear and stretching it to full range 0 255 Here the gray scale range can be chosen from a histogram window see Sec 3 12 4 to have a feedback about probability of occurrence of the gray levels within an Image Bad contrast corresponds to low flanks in the histogram due to under representation of dark or bright areas Move the red border lines see Sec 2 5 1 how to do this in the histogram window to cut away these flanks The grey scale range between the borders is then remapped to full data range Note that it is not possible to reduce contrast by this interactive technique Contrast Threshold Enhance the contrast by defining a threshold value t 0 lt t lt 1 The histogram is scanned from the left and right side until a value exceeds t The interval between the corresponding gray levels is remapped to full data range 0 255 As in Edit Image Brightness adjustment is done by a slider Here the value at the left side of the slider means relative contrast enhancement A value of 0 corresponds to the original Image 100 means full contrast If the histogram has more than one local maximum you will notice jumps of contrast during movement of the slider instead of continual change This is due to the scans from both sides where outer local maximums freeze the current data range until an inner
140. l l h o co Figure 3 19 User Kernel Dialog Filter Mask Field on the right side of the dialog to show current selected Kernel and its elements Factor Factor to multiply sum in Eq 3 1 with there this factor is 1 D Usually the reciprocal value of the sum of all Kernel elements is used to prevent from exceeding the data range of the original data For Images this factor can be used to adjust the gain Repeat How often the filtering with the selected Kernel is applied to the data field Use Absolute Value of Sum Checkbox For Kernels with one or more negative elements the result of Eq 3 1 may be negative If this box is activated the modulus of the negative value is taken as result Set Value Button and text box For editable Kernels set all elements to value in the text box at the right side of the button Load Kernel Button Pressing this button opens the standard file selector where a user defined Kernel can be selected to load A Kernel file has very simple structure see Sec 2 2 6 and can be created with any text editor Save Kernel Button Save the Kernel shown at the left side of the dialog Filter Mask see Fig 3 19 to file OK Button Press to apply selected Kernel to the active Image or 2D Data 3 5 FILTERING 53 e Cancel Button Cancel operation and close the dialog window immediately If a 2D Data field shall be filtered which includes
141. lag M nchen 1994 16 26 T J Loredo and E I Epstein Analyzing gamma ray burst spectral data Astro physical Journal 336 1989 896 919 63 A J Moore J R Tyrer and F M Santoyo Phase extraction from electronic speckle pattern interferometry addition fringes Appl Opt 33 1994 7312 20 90 91 S Nakadate and H Saito Fringe scanning speckle pattern interferometry Appl Opt 24 1985 2172 80 87 W Osten Digitale Verarbeitung und Auswertung von Interferenzbildern Akademie Verlag Berlin 1991 77 R L Parker Understanding inverse theory Ann Rev Earth Planet Sci 5 1977 35 64 63 H Philipp T Neger H Jager and J Woisetschlager Optical tomography of phase objects by holographic interferometry Measurement 10 1992 no 4 170 182 67 74 BIBLIOGRAPHY 125 36 W H Press B P Flannery S A Teukolsky and W T Vetterling Numerical 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Recipes in C second ed Cambridge University Press Cambridge 1995 14 62 63 105 108 G Pretzler A new method for numerical Abel inversion Z Naturforsch 46 a 1991 639 641 59 G Pretzler H Jager T Neger H Philipp and J Woisetschlager Comparison of different methods of Abel inversion using computer simulated and experimental side on data Z Naturforsch 47 a 1992 955 970 59 60 66 J A Quiroga A G
142. led up with single projections from desktop or from previously saved files see Sec 3 7 3 Note Projections must consist of an odd number of data as a center pixel is required for calculations the center pixel belongs to the path through the origin of the reconstructed plane The graph window for projections includes blue border line cursors an one red center line curser These can be moved by mouse though this is restricted by the required symmetry see Sec 2 5 subsection Select Borders and Center of 1D Data 3 7 2 Build Tomographic Input File For many viewing directions it can get rather tiresome to retrieve all projection data by hand Under ideal conditions when pixel scale and position of center pixels of the projections are the same for all directions projections can be retrieved in one step from all 2D phase data collected in a File Pool see Sec 3 2 The best way to fulfill these conditions is to accordingly design the experimental setup nevertheless if loss of accuracy is acceptable it is possible to manipulate data afterwards e g by resizing interferograms or the phase distributions with other application the latter method is 3 7 TOMOGRAPHY 69 much more recommended due to lower spatial frequencies in phase data After selection of a File Pool containing the 2D phase data from all directions e g a number of phase files from a set of previously evaluated interferograms the dialog for File Pool saving options
143. limit t men tioned above Edges higher than this limit are less concerned by the smoothing The checkbox Relative Smooth Limit must be activated if limit is given relative to dynamic range of the whole data field maximum minus minimum else the value is interpreted as absolute value To be more rigorous with edge preserving acti vate Strict Filtermode This sets Fli i of Eq 3 3 immediately to f i j if any fli i j 9 f i j lt t edge preserving condition 3 5 9 Local Enhancement From an Image f a new Image g is created with local enhanced contrast by remapping gray levels in the neighbourhood of pixels In detail the following 3 6 ABEL INVERSION 58 A y Figure 3 21 Abel Transformation and Inversion In interferometry a radially symmetrical distribution f r cannot be measured directly but only through the optical path integrals h y For example if a light beam crosses the distribution along the path s the measured phase shift h y left side is the integral of f r An r ds where An r is the difference of refractive index in all elements P hit by s The distribution f r is assumed to be zero outside of Radius R To get the radial distribution f r from measured data h y Abel Inversion algorithms must be applied is used 15 where M a i j In this formulation i j are the coordinates within Image g and f m i j and a i j are the gray level mean and stan
144. line vertical must be defined as well as length of the projections and vertical distance in pixels The highlighted 2D distribution Data2 pha is opened to show the intersections with the reconstruction plane or position of projections respectively The resulting File Pool TomoInput fpl contains one Tomographic Input File for each selected plane The Multiline Window Projections asc shows 4 projections of such a Tomographic Input File Af ter interpolation of projections to remove artifacts and applying the convolution method a File Pool containg reconstructed 2D Data is created not shown One such a reconstruction is shown as Arc Reconstruction tor culated in the same way as described in Sec 4 1 2 from a larger database and saved as Data 1 pha Data 2 pha To enable retrieval from all these fields in one step with the help of a File Pool the anode s tip was brought to the same position in all Images by window clipping Additionally for Tomographical reconstruction all phase distributions must have the same pixel scale pixel mmm Applying the procedure Tomograpy Build Tomographic Input File described in Sec 3 7 2 we build Tomographic Input Files from the File Pool Phase fpl This one and the resulting File Pool TomoInput fpl are displayed in Fig 4 3 In the Mul tiline Window Projections asc four projections are shown simultaneously for one of the Tomographic Input Files
145. lotted in an Integral Abel Data Window 3 12 9 Extreme Values In Images or 2D Data Fields maxima and minima are determined of all data and last selected area if Rectangle Draw Mode is still active treating masked data separately 3 12 10 Average Value In Images or 2D Data Fields the mean value is determined of all data and data within last selected area if Rectangle Draw Mode is still active treating masked data separately 3 12 11 Number of Masked Pixels Shows the number of all masked pixels 3 12 12 Number of Residues This evaluates the number of residues for modulo 27 phase data fields This is in tended to be used preliminary to applying branchcut algorithms to estimate compu tation time especially for the minimum cost matching algorithm see Sec 3 10 10 2 which is not recommended for a large number of residues In detail information is given about number of positive and negative residues and of border pixels for the whole data field as well as for the area selected by an eventually drawn rectangle 3 13 Window This standard main menu entry exists only for Windows95 98 NT and is created and managed by the operating system as usual 3 13 WINDOW 112 Figure 3 40 Grid Window showing Line Data Figure 3 41 Muliline Window containing three curves Chapter 4 Example This chapter is intended to present the main features of IDEA In three examples th
146. ly calculated integral data h x using Eq 3 5 This data can be compared with measured data e Show Deviation from Input Data Check this to show additional graph with deviation Am ha of analytically calculated integral data ha x from measured data hm e Show Radial Symmetric 2D Data Check this to show reconstruction f x two dimensionally The 1D distribution is simply rotated using the Bresenham algorithm whilst corners are filled with Zeros 3 6 ABEL INVERSION 64 3 6 5 Abel Inversion Problem Analysis After few experiments with f Interpolation and the Fourier Method everybody will inevitably notice the weak point of these techniques For different input parameters the results may look completely different For example increasing the number of polynomials for f Interpolation just by one can result in a different basic shape with a minimum in the center instead of a maximum The Fourier Method is even more sensitive To obtain the true reconstruction it is necessary to suppress as much of noise influ ence without smoothing away relevant information from the ideal distribution If no a priori knowledge about the result is available much experience is necessary to find the convenient parameters To help the user to develop this experience we decided to provide a tool which automatically probes a given distribution of integral data by applying Abel Inversions with a series of input parameters analyzing the results wi
147. m measured data are small enough for N gt 5 e Regarding only data between first two minima of Data four Deviations dat as reliable gives us the already fulfilled restriction 3 lt Ny lt 10 e The comparison of all results from f Interpolation to those of the Fourier Method recommends N 8 or N 9 e Taking a closer look to inversions for 5 lt N lt 9 reveals that a depression in the reconstruction appears for Nu gt 7 which is unlikely for the investigated object e Finally as the reconstruction for N 7 is slightly tipped we choose N 6 to be the best input parameter Typically for inversion problems an a priori knowledge about the object is VERY useful Without that Nu 8 would have been the best choice for this example A draw back here is that the measured integral data distribution required manipulations to become true radially symmetric Especially removing a linear tilt is very critical In such cases evaluation of different distributions extracted at different heights if object is homogeneous in y directions or evaluation from consecutive measurements e g with rotated object is recommended The screenshot in Fig 4 8 shows windows with the 1D Data distributions belonging to the f Interpolation method which were also extracted from the original Multiline Window and zoomed by factor 2 The 1D Data distribution are the same as those of the previously described screenshot for the Fourier Method and
148. many cases actions are restricted to the Active Window a term we use for this window which is currently in the foreground either after creation after a mouse click on the window or after selection in the main menu entry Window not for X Window System It is marked by a different colour of the window s title bar 3 1 File This main menu entry deals with all basic kinds of file and file type handling creation of empty data structures and with preferences of IDEA 3 1 1 New File Pool Tomographic Input File Slide Show Multiline Graph Here an empty window dedicated to the corresponding data type can be created which can be filled up with file data or data already available on the IDEA desktop 1D Integral Data Abel Reconstruction Since it is possible to paste 1D Data from the clipboard Windows 95 NT only empty 1D Data windows graphs can be created to be filled up with data from the clipboard by Edit Paste Of course values can also be typed in directly By default the new 1D Data are initialized with zeroes As input parameter you have to define the number of data you want to include in your graph Scale Bar A Scale Bar is a picture with size of 128 x 256 From the lowest line of the picture up to the top the colours of the actual palette from 0 to 255 are used including mask over and underflow colours if they exist By default the grey scale picture palette is used This bar may serve as colour reference
149. mask colour The radius is determined by current setting of Mask Pen Width see Sec 3 11 8 Both shape and size of mask pen are shown in Draw Mode Section of Status Bar see Fig 2 1 3 11 8 Mask Pen Width Here the width for the mask pen can be defined in an small dialog which is then taken for side length or radius of mask pen depending on its shape see Sec 3 11 6 and Sec 3 11 7 Only odd values are accepted as center pixels are required for calculation 3 11 9 Mask Selected Points If a polygon has been drawn or multiple points have been selected in a Picture this commands draws a mask with set shape and pen width at the coordinates these points or corners respectively Otherwise the coordinates of the pixels have to be defined in the multiple points dialog described in Sec 3 3 15 3 11 10 Mask Line If a line is drawn in active window this command sets masks with actual pen width and pen size around each pixel of this line Therefore if pen size is larger than 1 the mask will stand out from the end pixels of the line 3 11 11 Mask Inside Area If a rectangle has been drawn in a Picture this command sets the mask within the selected area including drawn border Else the area must be defined in an input dialog by typing in coordinates of upper left and lower right corner 3 11 12 Mask Outside Area If a rectangle has been drawn in a Picture this commands sets the mask everywhere outside of the selected area drawn
150. mask fill is set which is the last entry in the related coordinate list Take this into account when drawing polygons by values see Sec 3 3 15 The start pixel can be reset independently from the drawn polygon by pressing the left mouse button when the Control key Ctrl is held The new position of the start pixel is then set at the mouse cursor position A mask filled into the polygon does not cover the polygon border lines Select Multiple Points In this draw mode global switch see Sec 3 3 14 a left mouse click selects a pixel by marking it with a x The whole selection can be reset by pressing the left mouse button when the Control key Ctrl is held The first point of the new selection is then set at the position of the mouse pointer Note that the left mouse click function overrides the draw mask function Therefore when a mask should be drawn in the Image or 2D Data field one has first to change the draw mode There is another distinctness of this draw mode When copying a multi point selection into a window see Sec 3 3 16 then the actual point selection is not overwritten but expanded by the points in the other window Select Borders and Center of 1D Data For Tomography and Abel Inversion borders and position of center must be defined within a graph showing integral data The borders are represented by blue vertical lines through the graph initially located at the left and right side of the graph These lines can be
151. matically arranged from left to right but in the direction in which the line was drawn h 2 5 THE GRAPHICAL USER INTERFACE OF IDEA 20 Draw Rectangle For choosing an area within a picture one has to switch to Rectangle Draw Mode global switch see Sec 3 3 11 The selection can be done in the same way as you would draw a diagonal line in Line Draw Mode The selected area includes the drawn border lines As modification keys the CTRL key allows to draw squares and the SHIFT key allows only rectangles of size 2 x 2 a and b positive natural numbers Hold down these keys after setting the start point Refer to Status Bar see Sec 2 5 2 for feedback about coordinates Draw Crosshairs In Crosshairs mode global switch see Sec 3 3 12 the pixel at the intersection of a vertical and horizontal line crossing the whole picture is selected by left mouse click The crosshairs can be moved by holding down the left mouse button To select current pixel release mouse button As for other draw modes coordinates are simultaneously presented in the status bar Draw Polygon In Polygon mode global switch see Sec 3 3 13 a left mouse click selects a corner pixel of a polygon The last selection can be cancelled by pressing the right mouse button When all corners have been selected quit the process by a left button double click into the 2D Data field or Image At the position of the double click the x shaped start pixel for the
152. mographic algorithms must be used cf Sec 3 7 Collecting Data for Projections For further calculation the projections which correspond to a horizontal plane be tween the electrodes must be extracted from all integral 2D phase distributions cal 4 1 EXAMPLE 1 TOMOGRAPHIC RECONSTRUCTION 116 idea BEES File File Pool Edit View Filtering Abellnversion Tomography 2D FFT Phase Shift Phase Mask Information Window Help Su ea aaa eles elses mala AE pi PhaseniData 1Tomo0000 tom PhaseniWata 4 pha PhaseniData 1Tomo0001 tom PhaseniData 4 pha PhaseniData 1 Tomo0002 tom Phaseniata 3 pha PhaseniData 1Tomo0003 tom L PhasaniData 2 pha Phasen Data 1Tomo0004 tom Starting Point of Center Line x1 f 72 Yi fs Ending Point of Center Line x2 f 72 Ya f 32 Coordinate of Starting Point Relative to Line Center Multiline Graph Projections asc Reconstiuction Arc Reconstruction tor dx E 72 dy p tne Sepraton fF eves sey FohippiSeminanFFT TomographielPhasentDeta 1 pin F hinp Seminar FFT TomographieiPhaseniData 3 pin FahippiSeminaniFFT Tomographie Phasen Data 4 pjn Figure 4 3 Screenshot of IDEA showing how to choose projection data for tomographic recon struction The File Pool Phase fpl holds the phase data from which to retrieve projections In the Select Line s dialog the coordinates of the center
153. mply a handicap for connections to the border define a value here which will be multiplied to the cost calculated according to Eq 3 59 e Border Reduction Mode selection Keep Borderpixels within Neighbourhood of any Residuum Starting from any residuum a spiral scan searches for partners with oppo site sign If a border pixel is encountered before n to be defined below partners have been found it is kept for later processing of the cost matrix This is the most effective border reduction mode Keep Nearest Border Pixel to any Residuum Again by scanning in square spirals the nearest border pixels to all residues are located and kept for later processing of the cost matrix For a high density of residues this method is less effective than the neighbourhood scanning method Keep all Border Pixels Only recommended for small phase maps but the only way to be sure to obtain the true global cost minimum e Neighbours input field Here the number n of neighbours to scan for are defined in case the first of the order reduction modes from the radio box above has been selected e Set Branchcuts to Invalid checkbox Check this to set all pixels below a branchcut to invalid in the unwrapped phase 3 10 PHASE 103 Branchcut Minimum Cost Matching Invalids Substitute fp Border Costfactor p Border Reduction Mode Keep Borderpixels within Neighbourhood of any Residuum C Keep Nearest Borderpixels to any Residu
154. mum order lies between 5 and 15 Nevertheless orders up to 40 are allowed Though allowed by the algorithm it is not recommended to set order zero for the minimum order as this represents a constant added to the variation of f r This makes no sense as integral data at the border of any distribution must be zero the optical path is a tangent with infinitesimal length If measurement leads to such a result one should subtract this constant or bias with Edit 1D Data Remove Linear Tilt before Abel Inversion For nearly all distributions a minimum order of 1 is convenient 3 6 4 Abel Inversion Backus Gilbert Method In order to explain the Backus Gilbert Method of Abel Inversion it is necessary to give a short introduction of Inverse Theory 36 Suppose that f is an unknown vector that we plan to determine by some minimization principle Let A f gt 0 and Bl f gt 0 be two positive functionals of f so that we can try to determine f by either minimizing A f or minimizing B f Now suppose that we want to minimize A f subject to the constraint that B f have some particular value b The method of Lagrange multipliers gives the variation of LALA A B F 6 if A f AB F 0 3 10 where is a Lagrange multiplier yielding the one parameter family of solutions f A The functional A is a measure for the width of the so called resolution function or av eraging kernel It measures the agreement of a model to the da
155. n and r denote the coordinates of left and right border lines and y x is the value of the distribution at coordinate zi After calculation the red center line is shifted to m 3 3 EDIT 39 OO a b Figure 3 7 Removing linear tilt from a source distribution a Noisy data at outer regions can be clipped out by appropriately setting the border lines The gray area between these lines shows the linear tilt which is removed from the source data to get same height at position of the border lines The result b is clipped between borders during calculation whereas the position of the centerline in a is ignored Remove Linear Tilt All data below a straight line between intersections of distribution and border lines is subtracted from the 1D data The result is clipped to range between border lines see Fig 3 7 Left Side Only Creates symmetrical distribution by mirroring data between left border an center line to right side axis at center line see Fig 3 8 Right Side Only Creates symmetrical distribution by mirroring data between center and right border line to left side axis at center line see Fig 3 8 Average Left and Right Averages data between left border and center line with data from right side at same distance from center see Fig 3 9 If position of right border line truncates right side missing data is mirrored see Fig 3 10 Subtract Distribution Subtract a
156. n 46 the relation M 2nRAv 1 3 21 based on the sampling theorem provides a simple estimation of the number M of projections being necessary to obtain a tomographical reconstruction within an area of radius R that is characterized by the same bandwidth Av of spatial frequencies as the single projections 3 7 TOMOGRAPHY 68 Figure 3 24 Explanation of symbols for Radon Transformation and Inversion An inhomo geneous distribution f r p can be reconstructed from projection data representing path integrals h p 0 by using tomographic reconstruction algorithms which are numerical adaptations of the Radon Inversion Viewing directions corresponding to normal angles 9 0 7 2 and m are shown with small arrows 3 7 1 Get Single Projection Since phase data evaluated from interferometry is generally 2 dimensional one dimensional projections h p must be extracted from these data Selecting this menu entry extracts all data marked by a previously selected line which corresponds to the intersection of plane of evaluated phase and the reconstruction plane If no line was drawn in the 2D Data Field or Images a dialog pops up where starting point and endpoint coordinates must be defined This way single projections can be retrieved from 2D phase data To apply tomographic algorithms they must be collected in a Tomographic Input File see Sec 2 2 5 After creating an empty file File New Tomographic Input File it can be fil
157. n noticed which could not be solved e At one Windows 98 system it has been observed that the startpoint of a line and the corners of a polygon are not erased when a new drawing is started This did not occur at other PCs with the same operating system e On laptops the fonts in dialogs might appear too small e Rarely visualization of images or data fields are not actualized correctly and appear white With View Refresh the white picture can be repainted but win dows with this fault tend to inherit it to subsequently calculated data windows Chapter 2 Conventions and Definitions Before we started to implement specific algorithms and procedures for processing input data we had to spend much brain work about organization and logistics of all the data we would use for input and the data we would produce The result of these reflections are presented in this chapter To understand the partially arbitrary defined specific terms and the structure of data handling is essential for an effective use of IDEA 2 1 Definitions of IDEA Specific Terms As mentioned above we are restricted to 256 colour pictures for all data visualizations To distinguish the data from the visualization itself we defined specific terms The following specification shall give you an explanation of these terms as this is the key to understand this manual and the menu entries of IDEA 2 1 1 Palette A Palette contains information of all 256 colours used for
158. n of this window quits calculations as soon as the next iteration step will be finished just as if the any break off condition would have been fulfilled Therefore no data is discarded 3 7 7 Calculate Projection Data From any given square data field with odd side length a set of projections can be created In a small dialog two parameters must be defined e Size of Integrated 1D Data Defining a smaller number than width of 2D Data Field ignores outer rim of field during integration process as indices are taken relatively to center pixel e Number of Projections The number M of projection to calculate determines also the projection angles by Eq 3 27 Use this feature to compare your measured projections with those generate from your construction to get an idea about appropriate input values or to estimate error 3 8 2D FFT The two dimensional Fast Fourier Transformation 2D FFT is a powerful and pop ular tool the evaluate phase distributions from interferograms with implied carrier fringes It is well documented in literature e g 47 27 43 25 33 48 so only a short overview is given in this introduction The spatial intensity distribution of the fringe pattern determined by object s phase shift and the spatial carrier phase can be written as i r io r m r cos 2nrvo r r B r 3 32 where ig and m are the background and contrast functions Vo is the carrier frequency vector with components for z and
159. nates were selected in Crosshairs Draw Mode see Sec 2 5 1 before entering this menu refer to status bar coordinate section in Fig 2 2 the subtraction is performed immediately Else a dialog appears where you have to type in the six coordinates With the data at these locations a plane is calculated which is then subtracted point by point from the data field This can be very handy if a linear carrier fringe system is added to the object s phase change spatial heterodyning After calculating phase with 2D Fourier transform the object phase can be calculated by removing the linear tilt due to the linear carrier frequency This can be done here if areas with object phase shift equal to zero are known since the three points must be selected there Remove Fitted Linear Tilt Subtract a plane from 2D Data which is calculated from all masked data by a planar regression This method is more reliable than this in the previous section for data with significant noise level Resize Change width and height of a 2D Data field by setting the following parameters in the Resize dialog see Fig 3 3 5 3 3 EDIT 37 Resize 2D Data x Resize Mode Horizontal Factor Oo et et O m Vertical Factor ca v e yo OK Cancel Figure 3 6 Dialog for Resizing 2D Data Fields e Resize Mode Enlarge Zoom In by f 1 Shrink Zoom In by f 1 e Horizontal Factor Factor f for resizing
160. nce decreasing the values of window function near the Nyquist frequency reduces influence of high frequency noise in the reconstruction Therefore the choice of this parameter depends on data acquisition and on the type of the object under investigation 18 e Length Unit in pixels This parameter corresponds to 1 4 in Eq 3 25 and 3 23 It is recommended in 18 to set A to the reciprocal value of the sampling interval For reliable digitized data this means the Length Unit should always be 1 Anyway IDEA enables the user to recalculate projection data from the reconstruction to verify appropriate parameter setting Reading this you might get the impression that choosing parameters is quite a tricky thing In fact this cannot be denied Fortunately IDEA provides an algorithm which allows comparison of measured projection data with projection data derived from the reconstructed field So get some feedback by testing results with Tomography Calculate Projection Data documented in Sec 3 7 7 180 ds 4 0 1 M 1 3 27 Convolution x Size of Reconstructed 2D Data 391 Parameter for Hanning Window 0 54 Length Unit for Reconstruction f x a Figure 3 27 Dialog for tomographic reconstruction with the Convolution Method 3 7 TOMOGRAPHY 74 The computational simplicity and short processing time made the Convolution a pop ular and widely used method In 18 the author even goes so far to state tha
161. nces to other menu entries are not made by the number of the cor responding section in the manual but directly by the name of this entry in IDEA s menu It is then typed in italic fonts with submenus separated by a textbar If a button is used in IDEA for shortcuts to a menu entry it is shown at the left margin aside the menu name In chapter 4 some evaluation procedures are worked out as a kind of tutorial for the user 1 5 What s new in Version 1 5 1 5 1 New features e Image and Data Field windows now have scrollbars which eliminates system specific size restrictions related to the screen size As a consequence graph 1 5 WHAT S NEW IN VERSION 1 5 windows for 1 dimensional data can also be scrolled However pictures in a slide show are kept in full size mode A number of points can now be selected by a new draw feature and by the new polygon draw mode of course and x y and z values can be viewed and saved in Info Data at Selected Point s or even edited in Edit Edit Data at Selected Point s The main reason to add this however was to enable unwrapping procedures to start from points in regions isolated from each other The ability to draw polygons has been added therefore there is the new fea ture Mask Mask Polygon There is now the feature to make a One Step phase unwrapping subsuming the scan for the 27 jumps and the unwrap itself The famous branchcut phase unwrapping has been
162. ne Ele FilePool Edt View Fiteting Abelinvetsion Tomography 20 FFT PhaseShit Phase Mask Infomation Window Help aja 499 0 Ada alcala mal a ES Zoom 4 1 174 3911241 91E Ord 18546 241 56 213 51 5711 Figure 4 5 Screenshot of IDEA showing how to unwrap modulo 27 phase data PSI m2p obtained from 120 Three Frame Phase Shift technique Applying the 2D Scan method leads to the step function Order bmp needed for unwrapping A small rectangular area of Order bmp has been zoomed Zoom 4 1 to perform sub scans using the Order Dialog The unwrapped phase is displayed as Unwrapped pha The integral data along the line drawn in Unwrapped pha are shown as Unwrapped abl 4 2 3 Phase Unwrapping The same unwrap algorithm as in Sec 4 1 2 can be applied to the modulo 27 phase data PSI m2p In Fig 4 5 the resulting step function Order bmp is shown Please note that Window PSI m2p has been mirrored horizontally Mirror Horizontal in Fig 4 5 Since the algorithm was unable to determine the order for the white areas successive sub scans are necessary to determine the step function using the procedure described in Sec 3 10 5 For better resolution these sub scans can be performed in the Zoom Window Zoom 4 1 in Fig 4 5 created by View Zoom Selected Area 4 1 The unwrapped phase can be calculated with Phase Unwrap with Step Functi
163. ng Little Endian Byte Order 3 1 3 Close Close the actual Window If the data represented by the actual window were changed or created by calculation the user is asked if this data should be saved before closing On the other hand if data just were read from file and never changed it is closed immediately 3 1 4 Save Save data in current format with the filename in the title bar of the window Consider byte order settings in File Preferences Save File with Little Endian Byte Order 3 1 5 Save As Select new name and format in the standard file selector dialog of the platform in use Refer to Tab 2 3 for more details about suggested extensions Consider byte order settings in File Preferences Save File with Little Endian Byte Order 3 1 6 Convert Copy to Image 1 To convert a 2D Data Field to an Image the field s data is subdivided into 256 intervals All pixel values within interval i i 0 1 255 are then converted to the 8 bit value i 2 If a pure picture e g 256 colour Bitmap shall be converted to an Image one has to define how the gray level Z should be calculated from the 8 bit colour information R red G green and B blue in a dialog window In literature e g 29 the following expression is recommend I 77 R 151 G 28 B 256 This is the Standard option in the dialog shown in Fig 3 2 Apart from that you can select Red Green Blue and Custom The three text boxes
164. ng Problem Analysis for f Interpolation Method This window shows Data distributions belonging to the f Interpolation Method calculated by Abel Inversion Problem Analysis 1D Data distributions were extracted from the Multiline Window and zoomed by factor 2 Data fint Curvature dat shows the overall curvatures of Abel Inversions de pending from the number of polynomials order Equally organized Data fint Chi dat gives feedback about deviations from recalculated integral distribution to input data Finally the window Data fint Deviations dat shows a graph where deviations of two consecutive reconstructions with increased number of polynomials are shown Additionally all Abel inverted reconstructions are melted in a 2D Data window Data fint AllAbelInversion00 bin each forming a row from top to bottom of the Data Field In the protocol window the 9 smallest deviations of any inversion by f Interpolation to any result of Fourier Method are printed organized Data four Chi dat gives feedback about deviations from recalculated inte gral distribution to input data Finally the window Data four Deviations dat shows a graph where deviations of two consecutive reconstructions with increased parameter for maximum order are shown See Sec 3 6 5 for a more detailed descriptions of this distributions How can these graphs be interpreted Before this is demonstrated here it must be noted that the appearance of these dis
165. nomials are forced to have the same values at the previous and the next two zone separation points This way four polynomials are overlapping at one point For example the polynomial for P in Fig 3 22 must have the same values as all neighbouring polynomials at Ri 1 Ri41 and R 2 The calculation of the coefficients starts from the periphery at Po with assumption of P9 Ro 0 and P Ro 0 by approximating the analytical inversion to measured data using the least squares criterion With the smoothing and least square criterion the calculation of the next inner polynomial can be performed until the center is reached There the additional assumption Py _ is necessary to finish computation of the Abel inverted distribution For a more precise and mathematical explanation of this method refer to 38 3 6 ABEL INVERSION 61 Depending on setting of File Preferences Abel Inversion Strict Symmetry Checking you are warned if the integral 1D distribution does not fulfill all requirements for Abel Inversion The following small dialog appears before the f Interpolation is performed Abel Inversion f Interpolation x Number of Cubic Polynoms f w Deviation from Input Data I Show Radial Symmetric 2D Data Figure 3 23 Dialog for Abel Inversion by f Interpolation e Number of Cubic Polynomials Number M of Polynomials or zones respectively to be used for interpolation At least 2 are necessary to perform
166. o Show Branchcuts OK D Cancel Figure 3 37 Dialog for branchcut unwrapping applying nearest neighbour scan e Scan Start Y input field Define the y coordinate row of the starting point for the residuum search path if it has not been selected by the multiple point selection with the mouse e Set Branchcuts to Invalid checkbox Check this to set all pixels below a branchcut to invalid in the unwrapped phase If not checked a final unwrapping procedure will add a multiple of 27 to these pixels to minimize deviations from neighbours at either side of the branchcuts Regard this as a mere visual enhancement of the result as there are no criterions to select the best result at branchcut pixels e Show Residues in Extra Image checkbox Check this to create an image where the residues are represented by pixels with colour e g value depending on the sign blue pixels values 1 mark negative residues white pixels values 252 mark positive residues e Create Mask to Show Branchcuts checkbox With this box checked a mask is created in the data field with unwrapped phase data which covers the branchcuts excluding the residues The same is done in the image showing the residues if created After finishing calculation information about overall squared cutlength is given in the protocol window 3 10 10 2 Minimum Cost Matching Branchcuts The problem of optimizing for the minimal cut length is here approached by a method from g
167. obing beam with jt pixel in image vector hitting i element of y The vector e is referred to as error vector representing deviation of real data from the approximation leaving room for finding the best of many possi ble solutions of Eq 3 28 or in worst case for finding a solution at all Multiple solutions require criteria indicating which reconstruction x ought to be chosen as a solution of Eq 3 28 For the implemented Additive ART the criteria are defined by the Bayesian estimate and the estimated x is found using a iterative procedure called the relaxation method for inequalities The mathematical background of this approach would exceed the scope of this manual but a detailed description can be found in 18 So let s skip to the final relations used for the ART which are applied consecutively to all projections within an iteration z qe ae vel 3 29 ee aa ee 3 30 with ca DN ay e yy yi ria u 3 31 1 v r5 DE iS oe Here 1 is the index for a data element within a single projection and k the number of overall iteration The vector r includes all intersections of a beam on its way to the it data in the current projection with pixels included in x Therefore the vector product with x is the length of the beam s path through the whole reconstruction plane The denominator in Eq 3 30 includes the squared norm of r which is the sum off all squared element values and v is a cons
168. of Development When starting our first thoughts concentrated on Windows 95 and Windows NT as the most useful platforms due to their wide spread but unable to ignore the advantages of UNIX we decided to use a multi platform C class library providing Graphical User Interface GUI and other facilities We chose wxWindows 1 68e a free portable C GUI toolkit by Julian Smart from the Artificial Intelligence Applications Institute University of Edinburgh http www wxwindows org This public domain class library allowed us to get both Windows 95 NT and X Window executables with minor differences in program code and appearance of the interface Nevertheless this manual is referred mainly to the Windows 95 NT version as we assume this to be the more used one The core of most algorithms was overtaken from DOS software developed by several former members of the Institute for Experimental Physics Harald Philip initialized the use of the Fourier Techniques later expanded to two dimensions by Georg Pret zler and focused on Tomographical Reconstructions during his thesis Georg Pretzler also investigated the Abel inversion very carefully and developed an own method The algorithms used in IDEA are based on his code including all techniques recommended by his work The Phase Shifting was first used by Walter Fliesser and his experience influenced the implementation of this technique To compile all this work and knowl edge was the main task of the s
169. of the destination pixel yield the value for destination pixel This is a relatively fast and accurate technique More simple and considerably faster is nearest neighbour approximation which truncates the fractional address to the nearest integer pixel address As you can guess the drawback of this technique is inferior quality The called dialog requires following inputs fe 3 3 EDIT 43 e Scaling Method interlinked checkboxes Rescale using Scale Factor The contents of the following two text boxes are interpreted as factors and new height and width are calculated by multiplying entry in Horizontal by old width and the entry in Vertical by old height Rescale to total Width Height Entry in Horizontal is new width entry in Vertical is new height e Horizontal Factor or new width see Scaling Method e Vertical Factor or new height see Scaling Method e Apply Interpolation Checkbox If this box is activated interpolation is performed to get higher quality default Leave it deactivated for nearest neighbour approximation 3 3 10 Draw Line Activate this to globally switch draw mode to draw a line in active 2D Data or Image window see Sec 2 5 1 3 3 11 Draw Rectangle Activate this to globally switch draw mode to draw a rectangle in active 2D Data or Image window see Sec 2 5 1 3 3 12 Draw Crosshairs Activate this to globally switch draw mode to draw a rectangle in activ
170. oftware project The algorithms were revised as far as possible improved and submerged into a new environment by the author Peter Reit erer embedded the result into a well contrived graphical user interface and self written class library All our work was thoroughly supervised by Dr Jakob Woisetschl ger who kept us busy with his clear view for practical needs of an scientist working with interferometry We used the Borland C 5 0 Compiler for Windows 95 NT 2000 and the Gnu C Compiler 2 7 2 3 for X Window under Linux The code includes more than 78000 lines within 226 files written and tested within 2500 working hours 1 4 1 About the Manual Chapter 2 of this manual presents conventions and definitions we use with IDEA including terms file formats an some elements of the Graphical User Interface For understanding the principles of data handling in IDEA it is essential for the user to read this very carefully especially the section explaining the terms Chapter 3 deals with all menu entries of IDEA in the same structure as in the soft ware s menu bar The functionality of all items is explained sometimes with short mathematical background For further information references to literature are in cluded Screen shots of dialog windows help to familiarize with the software For some main menu entries covering more complex features a short overview is given before each of the subordinated items are explained in detail Casually refere
171. olation f xi n is compared to all results of Fourier inversions f x Nu by calculating overall deviation N d n Nu Y f x n f x Ni 1 Nu 3 18 i 1 and arranging all results d in a 2D Data field The vertical coordinates are chosen to be maximum orders of model function N leaving number n of poly nomials for horizontal coordinate Areas where this distribution is close to the minimum determine ideal data range of n and N Protocol Window entries Not only creation of all previously described windows are protocolled but also the 10 smallest deviations of Fourier Method results from f interpolation inversions 10 smallest values in third 2D Data window in previous section Sorted by the value of deviation d see Eq 3 16 the corresponding number of polynoms poly and the maximum order of model function ord are given The last column in the list shows the relative deviation from the smallest value of d 3 6 ABEL INVERSION 66 3 6 6 Radial Data Convolution Of course it is also possible to solve the Abel Inversion Problem by applying more common Tomography algorithms which are able to reconstruct even asymmetrical two dimensional distribution As described in the introduction to Sec 3 7 precision of reconstruction increases with the number of measured integral data distributions projections For radially symmetrical data projections should be identical for all directions In the input dialog
172. olution window the time to calculate the synthetic interferograms and the arctan operations is relatively small a signal to noise ratio about two times higher can be achieved However it is a fact that the absolute time consumption is quite high There are two dialogs for parameter definition The first one is mainly for the convo lution filter the second one is similar to the file selection dialog of the standard phase shifting methods The first one shown in Fig 3 9 8 requires the following inputs e Phase Shift Mode Select one of the phase shift techniques in the list They are the same as those implemented as standard techniques in the Phase Shift menu e Filter Mode Convolution with Unit Kernel Neighbourhood Mean The low pass filtering is an averaging process within a neighbourhood which corresponds to convolution with a kernel consisting exclusively of elements of value 1 see Sec 3 5 3 9 PHASE SHIFT 89 Convolution with User Kernel Define an arbitrary filter kernel With this selection the Continue button will bring up the dialog described in Sec 3 5 3 where all further filter options have to be defined Therefore the subsequent input input items are deactivated Filter Width Select the odd width of the filter kernel from the list As mentioned above the recommended value should be one quarter to one third of the smallest fringe separation Filter Height Select the odd height of the filter k
173. on In Fig 4 5 the final result for phase is shown as Unwrapped pha Using View Change Colour Palette we selected the Palette hot for the visualization in Fig 4 4 and in 4 5 cf Sec 3 4 5 The integral data along the line drawn in Unwrapped pha are obtained using Abel Inversion Get Integral Data and are shown as Unwrapped abl in Fig 4 5 4 3 Example 3 Abel Inversion 4 3 1 Background The third example demonstrates the evaluation of the radial phase shift within a candle flame The good radial symmetry of an undisturbed flame allows the appli cation of Abel Inversion to the integral phase shift yielding the corresponding radial distribution 4 3 EXAMPLE 3 ABEL INVERSION 119 idea MAE file File Pool Edit View Filtering Abelnversion Tomography 2D FFT Phase Shit Phase Mask Information Window Help Slt ela a 0 2 0 8 ee ma Sls E Elm 8 Integral Abel Candle abl of x 9 Abel Reconstruction Reconstruction abr ox 2D Data Rotate bin Mal Ed 4 Integral Abel Edited Data abl Lori OANE AE MO Integral Abel Deviation abl 17 53 o 103 206 te i e tos R Pos 0 0 103 0 1657 205 0 207x256 123 202 0 160559 Figure 4 6 Screenshot of IDEA demonstrating the application of Abel Inversion Window Can dle abl displays a one dimension
174. ontents of clipboard to active window Windows 95 98 NT only In the present version it is only possible to insert 1D Data into a 1D graph window or a grid window Previous contents are always overwritten so it might be wise to create 3 3 EDIT 34 an empty 1D Window see Sec 3 1 1 before importing data The size is automatically adapted to pasted data In all cases the 1D Data window must be open before the data is copied to clipboard in IDEA or any other application 3 3 3 Clip Copy contents of selected area to a new Image Picture or Data Field Selection is done by drawing a rectangle around the data to extract see Sec 3 3 11 3 3 4 Image Brightness Adjust the brightness of the active Image by adding or subtracting a constant value from all data with ranging between 0 and 255 this means all values above the upper or below the lower limit is set to the corresponding limit value Move the slider in the dialog by clicking on it holding the left mouse button and moving the mouse to either side For step by step movements click on the small arrow symbols left or right of the slider At the right side of the dialog a number represents the current relative brightness A value of 0 refers to subtraction of the maximum of the field all is black 100 to shifting the minimum to 255 all is white Therefore the initial relative brightness depends on values of maximum and minimum of the Image Contrast Manually Ad
175. onzales Cano and E Bernabeu Stable marriages algorithm for preprocessing phase maps with discontinuity sources Appl Opt 34 1995 no 23 5029 38 99 H T Goldrein R Cusack J M Huntley Improved noise immune phase unwrap ping algorithm Appl Opt 34 1995 no 5 781 89 99 P K Rastogi ed Digital speckle pattern interferometry and related techniques John Wiley amp Sons LTD 2001 83 D W Robinson and G T Reid Interferogram analysis Institute of Physics Publishing Bristol 1993 53 C Rodier and F Rodier Interferogram analysis using Fourier transform tech niques Applied Optics 26 1987 1668 73 77 78 U Schnars T M Kreis and W Jiptner Digital recording an numerical recon struction of holograms reduction of the spatial frequency spectrum Opt Eng 35 1996 no 4 977 982 R S Sirohi ed Speckle metrology Marcel Dekker Inc 1993 83 D L Snyder and J R Cox An overwiev of reconstruction tomography and limi tations imposed by infinite number of projections Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine Ter Pogossian et al ed Baltimore University Park 1977 67 M Takeda H Ina and K Kobayashi Fourier transform method of fringe pattern analysis for computer based topography and interferometry J Opt Soc Am 72 1982 156 160 77 C M Vest Holographic interferometry Wiley NewYork 1979 77 D Vukicevic T Neger H Jager J Woisetschlage
176. or Spiral Scan is active Sub scans calculate step functions in a selected area allowing elimination of propagated errors The following list shows how to perform sub scanning 1 Apply any scan method to modulo 27 data to obtain initial step function 2 Switch to Rectangle Draw Mode see Sec 3 3 11 and select an area where error propagation occurred by drawing a rectangle Tip source of error propagation should be just outside of selected area see also Sec 2 5 1 3 Select a starting point by right mouse click If no area has been selected before this opens a dialog where coordinates of upper left and lower right corner of rectangle must be defined Else a colour table pops up showing horizontal bars with colours of adjacent orders The mouse cursor is automatically positioned at the colour representing current order of starting point 4 From colour table select new or the same order of starting point by left mouse click on corresponding colour bar This starts the sub scan within the selected area using minimal phase jump value defined in Phase Jump Value for Sub Scan 3 10 6 Sub Scan Spiral Enabled Toggles Sub Scan Mode to Spiral Scan see Sec 3 10 5 3 10 PHASE 98 3 10 7 Add Step Function Select step function file 256 colour bitmap using the standard file selector The modulo 27 data Picture is then hidden by the step function Picture and title of window is changed too 3 10 8 Remove Step Function Rem
177. order line In case this position is not valid mouse move has no effect until the opposite border is in range of the graph To position a line at a certain coordinate write the coordinate into one of three text fields at the bottom of the window marked with L for left border C for center and R for right border Confirm your input by pressing ENTER key This shows the line at the specified location Contrary to integral 1D Data for tomography and Abel inversion simple line graphs have only a center line There is no possibility to define its location by typing the coordinate but it can be moved by mouse in the same way as described above 2 5 2 The Status Bar The Status Bar of the main window or active window in X is used to give the user information of the active window and some settings Its organization varies with data type and with mode of interaction see Sec 2 5 1 The bar is divided into four sections Corresponding to Fig 2 1 we enumerate them from left to right In section 1 a short description of the highlighted menu entry is shown If the platform in use is Windows 95 NT the help text of entries opening a submenu is not correctly shown in IDEA In fact you see help text of the previous valid entry instead Sorry for that but this is not within our responsibility Section 2 shows coordinate data of a line a rectangle or crosshairs used to select data In section 3 information about used draw and mask mo
178. ove step function from window This reveals the so far hidden modulo 27 phase data 3 10 9 Unwrap with Step Function With information of step function the continuous phase can be easily calculated apart from a constant offset This offset can be defined by the multiple points dialog see Sec 3 3 15 and Fig 3 12 where the coordinate s of the starting point s for the unwrapping procedure are shown In the column for the z values enter the phase offset s which will be added to the region specified by the corresponding starting point 3 10 10 Unwrap with Branchcut Method The main problem of the path dependent unwrapping techniques is the propagation of unwrap errors with each pass of the scan path For noisy data for example phase from partially decorrelated speckle interferograms it is quite usual that the phase order picture is a miserable mess with errors propagating from a net of sources across the whole area though visually the phase jumps are clearly to see It has been found 14 that the sources of the errors are local phase inconsistencies leading to different results for different scan paths This is demonstrated with the phase map in Fig 3 10 10 where such error sources are simulated Scanning along path 1 from point A to B reveals three phase discontinuities corresponding to an overall e g integrated change in phase order of X As s 3 see Eq 3 57 J Tr Path 2 back from B to A crosses only two discontinuitie
179. pe and Tab 2 3 for ID 2 2 6 Internal Format of Filter File This file type which includes all data for a filter kernel is completely in ASCII format with following structure FL Width Height Multiplier Divisor new line Data The common file ID FL is followed by the dimensions width and height of the kernel which have to be both odd Multiplier and Divisor define the constants D and M in Eq 3 1 used to normalize the results of convolution They are followed by the kernel elements in the next line which are given row by row 2 2 7 Internal Format of Colour Palette File It is possible to import a custom Colour Palette to IDEA Such a file has no header line and consists of the RGB values line by line in ASCII format each value smaller than 256 and separated by a blank or comma from its neighbour For example to import the gray scale palette the file to load must have one of the following structures 0 0 0 new line 1 1 1 new line 255 255 255 or 2 2 FILE TYPES AND FORMATS Table 2 1 Format of Frequency File Coordinates are given by the spatial frequencies fx in horizontal direction and fy in vertical direction The according number of periods in a Field of Nz pixels per row and Ny rows are also shown Indices min and max mean minimum and maximum of frequencies with Nyquist frequencies nyq regarded separately Negative horizontal frequencies are the complex conjugate to the related posi
180. ped pha is visualized with the hot Palette plane the resulting reconstructed field is shown as Arc Reconstruction tor in Fig 4 3 Using View Change Colour Palette we selected the Palette jet for the visualization cf Sec 3 4 5 For some planes the electrodes are also reconstructed since the data at their locations were set to zero in the projections Of course there are no sharp edges between data from the discharge and electrode regions due to the smoothing effects of the convolution method The reconstructions of the lower planes show the discharge climbing down the triangular cathode 4 2 Example 2 Phase Shifting 4 2 1 Experimental Background The second example demonstrates the evaluation of the radial phase shift within a Helium flow by classical interferometry For the set of phase shifted primary interfer ograms the modulo 27 phase distribution is calculated with three frame algorithm 4 2 2 Phase Shift Evaluation With the three interferograms Frame0 img Framel20 img and Frame240 img in Fig 4 4 it is possible to calculate the 2D phase distribution using Phase Shift Three Frame Technique 120 In Fig 4 4 the standard dialog Phase Shift with text boxes for filenames is shown The result is the phase distribution modulo 27 PSI m2p where data at locations with visibility lower than the previously defined 64 are masked 4 3 EXAMPLE 3 ABEL INVERSION 118 Or
181. pixel data extract line data and even to paint within a picture without destroying the pixel information behind the colour see mask Sec 2 1 5 You will find further information about these features in this chapter 2 5 1 Data Selection in Active Window Paint Mask To mark Image data or 2D data for example to exclude it from calculations or to set it to specific data use the right mouse button hold it down and draw within the window At every detected position of the mouse pointer a square or circle is drawn in mask colour Size and shape of these mask elements depend on the actual settings of mask pen width and mask pen style The data represented by the active picture is not affected by the mask which can be saved in an own mask file Len Draw Line After switching to Line Draw Mode global switch see Sec 3 3 10 one can select the data behind pixels on a line which can be drawn into the active Image or 2D data window To do that choose a starting point press the left mouse button to start drawing and move on to the desired endpoint By pressing the left mouse button again you select the end point of your line If CTRL key is held after starting the line drawing is restricted to vertical and horizontal lines Refer to Status Bar see Sec 2 5 2 for feedback about coordinates The position of line pixels are calculated from starting point and endpoint coordi nates using the Bresenham algorithm 12 Be aware that data is NOT auto
182. pping with the branch cut method role of phase field direction Appl Opt 39 2000 no 26 4802 16 99 G T Herman Image reconstruction from projections Academic Press New York 1980 57 67 72 73 74 75 76 K Hibino Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts J Opt Soc Am A 14 1997 no 4 918 87 A Le Hors Xpm Manual BULL France 1994 16 B R Hunt Matrix formulation of the reconstruction of phase values from phase differences J Opt Soc Am 69 1979 393 399 105 R Jones and C Wykes Holographic and speckle interferometry Cambridge Uni verstiy Press Cambridge 1983 83 B W Kernighan and D M Ritchie The C Programming Language second ed Prentice Hall International New Jersey 1998 17 D Kerr G H Kaufmann and G E Galizzi Unwrapping of interferometric phase fringe maps by the discrete cosine transform Applied Optics 35 1996 no 5 810 816 95 T Kreis Holographic interferometry Akademie Verlag Berlin 1996 77 T M Kreis and W Jiiptner Suppression of the de term in digital holography Opt Eng 36 1997 no 8 2357 60 M Kujawinska Spatial phase measurements methods Interferogram Analysis D W Robinson and G T Reid eds Institute of Physics Publishing 1993 pp 141 193 77 C A Lindley Practical image processing in c John Wiley amp Sons Inc New York 1991 42 T W Lipp Die grofe Welt der Grafikformate Synergy Ver
183. r and H Philipp Optical tomography by heterodyne holographic interferometry SPIE Institute Series 8 1990 160 193 67 D C Williams N S Nassar J E Banyard and M S Virdee Digital phase step interferometry a simplified approach Opt Las Tech 23 1991 147 50 84 J Woisetschlager H Jager H Philipp G Pretzler and T Neger Tomographic investigation of the particle density distribution of sodium atoms in a glow dis charge using holographic interferometry Phys Lett A 152 1991 42 46 67 113 I Yamaguchi and T Zhang Phase shifting digital holography Opt Lett 22 1997 no 16 1268 70
184. r left corner 3 8 13 Show Amplitude Calculates real amplitudes A fz fy from imaginary parts Im z y and real parts Re fx fy of complex amplitudes using the common relation for absolute values A fos fy Im fos fy Rel fos fy 3 37 Note Coordinates are not longer shown in the mode for frequency data where origin is at center pixel but in standard mode with origin at upper left corner 3 8 14 Show Phase Calculates phase fz fy from imaginary parts Im x y and real parts Rel fz fy of complex amplitudes using the common relation for absolute values z Im fe fy fx fy arctan Relfe fy 3 38 3 8 2D FFT 82 Note Coordinates are not longer shown in the mode for frequency data where origin is at center pixel but in standard mode with origin at upper left corner This can be used to evaluate phase from digital Quasi Fourier Holograms 3 9 PHASE SHIFT 83 3 9 Phase Shift Except for Fourier Transformation methods the Phase Shifting technique is a very common method to evaluate phase distribution from interferograms Here the modulation of intensity is determined during artificially shifting the phase of one beam in the interferometer with respect to the other by a well defined value Some examples for common phase shifting devices are moving mirrors tilted glass plates moving diffraction mirrors and rotating wave plates The intensity distribution of an interferogram can be written
185. r that is given in 3 In case the result is not zero the upper left of the 3 10 PHASE 99 renos SM E IS m Figure 3 36 Phase map showing phase inconsistencies that lead to a path dependent phase result when unwrapped with scanning methods Following path 1 from point A to B three phase discontinuities corresponding to an overall change of phase order s 3 are encountered Scanning from B back to A along path 2 however will reveal just two phase discontinuities with reversed sign e g s 2 Therefore the change in phase order is not zero when the circle path is closed but 1 This shows that the path includes a positive residuum marked by a circle which is the source of a phase discontinuity A second residuum can be detected when integrating the changes of phase orders from A to B following path 2 in reversed direction and closing the circle by scanning along path 3 giving s 1 Therefore this is a negative residuum Note that the direction in which the circle path is scanned must be consistent for each integration A closed path enclosing an equal number of positive and negative residues e g path 1 and 3 will give the ordinary result of s 0 The idea behind the branchcut unwrapping is to connect residues with opposite signs and prevent the subsequent path scanning procedures to cross the connection referred to as branchcut As the phase on both sides of the branchcut may differ by any value there are d
186. raph theory 6 Each branchcut is assigned a cost which is the squared distance between the connected pixels cij 21 25 yi yi 3 59 where x4 yi are coordinates of the negative residues and xj yj are the coordinates of the positive residues Taking the square is done just for the sake of computing speed as all operations can be done with integers The target is to minimize the overall cost that is the sum of all cij This is accomplished by the Hungarian algorithm from graph theory which is able to find the true global minimum from an iterative matrix operation The N x N matrix elements are given by the costs cj where indices 7 and j are obtained by enumerating separately positive and negative residues The matrix is referred to as cost matrix If the number of positive and negative residues are not 3 10 PHASE 102 equal so called virtual residues have to be added to get a square matrix as required by the Hungarian algorithm The cost to connect to a virtual pixel to another one is zero but infinity to a real residuum The problem is how to treat the border pixels Even for rather noisy phase data the are often more border pixels several thousands than residues As each border pixel should be allowed to connect to any residuum for finding the global minimum each of them has to be added to the list of positive and negative residues Then the matrix dimension N x N is dominated by the number of
187. rder lines The feature File Pool Subtract Every ith File has been added The files in a File Pool can be divided in subgroups of definable size and first or last file in such a subgroup is subtracted from the other files The counter of the files in File Pool can be changed by adding a constant num ber 1 5 WHAT S NEW IN VERSION 1 5 10 e Three methods to determine phase data for phase stepping speckle pattern in terferometry have been included which either require interferograms with sub traction fringes or just one interferogram with altered object phase and four phase shifted speckle pattern interferograms of the reference state of the object Phase Shift Speckle e Spatial Phase Shifting algorithms for pixel to pixel phase differences of 120 and 90 have been included Phase Shift Spatial Phase Shifting e In the new version not only real and imaginary part and amplitude can be extracted from Fourier space but also phase 2D FFT Show Phase 1 5 2 Amendments e Subtraction of modulo 27 data fields resulted in a wrong sign This has been corrected e Adding a constant value to modulo 27 data fields includes re mapping again to the interval 7 7 1 5 3 Known Issues IDEA is widely though not excessively tested The latest operating system for devel opment of the software has been Windows 2000 alternatively it has also been tested with Windows 98 and Windows XP Some problems have bee
188. rdinate of left x coordinate of border right border Mouse pointer coordinates uy 0 8 120 55 212 182 300x256 21 89 6 32 y value at x coordinate Multiline Graph x coordinate of center line Size of Graph width x height Value at x position of mouse pointer Pixel coordinates x y at mouse pointer Line at x 50 300x256 21 89 6 32 Figure 2 3 Contents of Status Bar for 1D Data Note the difference in Mouse Pointer Section For Multiline Graph the right value gives the y position of the active mouse pointer in the graph plane and is not related to any plotted curve For Size of Graph width x height Virtual y value at x position of mouse pointer the standard 1D Graph the y value on the curve at x position of the mouse pointer is shown Chapter 3 IDEA Menu Entries In this chapter the functions of all menu entries are explained in the same hierarchy as they appear in IDEA Descriptions of all dialog windows and input parameters are included In the text references to menu entries are written in the same style as paths but with vertical separation and emphasizing For example the submenu Save As of the main menu entry File is referred to as File Save As To avoid repeating descriptions some consecutive menu entries are comprised if dif ferences are marginal In
189. rhood window is close to a phase jump or containing one In the latter case even the likelihood of the center pixel being at the lower or higher side of the jump can be determined See Tab 3 1 for the details The dialog contains the following items for defining the filter parameters e Size the square neighbourhood from which the data to operate on is taken see Sec 3 5 4 3 5 FILTERING 54 Table 3 1 Determination of the most likely filter response by the adaptive median From the populations Np of the intervals h1 h2 and h3 of the histogram shown in the first column five cases are distinguished by the algorithm concerning the position of the filtering window relative to a phase jump For each case the most likely estimation for the phase is determined from the median of data within different intervals In the last column the conditions to distinguish the five cases by the values Np are given There V denotes the number of valid pixels within the filter window The other parameters 8 and y can be defined by the user recommended is to set 1 and 0 7 Histogram Filter window posi Median of Conditions tion ht ho h3 far from jump all data Nah gt BUNai Nna Toa ow Nn lt B Nm Nas h1 h2 h3 close to a phase jump h2 and h3 Pa a a by the high side y E Y h1 h2 T a Nn lt B Nm Nas
190. ring the time required for a whole phase shifting process as long as a set of phase shifted interferograms can be recorded at a stable reference state see the introduction to this section The procedure of the POD method is done in several steps which are 32 1 Record a set of phase shifted speckle interferograms at stable reference state of the object 2 Record a speckle interferogram of object after phase alteration 3 Subtract this from the phase shifted interferograms and take the modulus of the result to create subtraction fringe interferograms If time is no criterion square the difference instead of taking the modulus 4 Low Pass Filter the subtraction fringe interferograms by convolution 5 Apply the phase shifting algorithm which corresponds to the method used for the measurement in the reference state 3 9 PHASE SHIFT 88 Any of these steps can be applied with functions available in IDEA Nevertheless an extra menu entry has been dedicated to this method since the method has been further developed by embedding it into an iterative procedure 16 denoted here as IPOD Assuming that subtraction fringe interferograms have already been calculated corresponds to having step 3 in the upper procedure completed the major steps of the iteration are as follows 1 Set iteration counter to 0 2 Low pass filter the subtraction fringe interferograms by convolution 3 Calculate modulo 27 phase f x y by correspond
191. rmat may be found in 20 When opening a file with one of these formats the palette is checked If all entries are gray scales and are not more than 256 the file is automatically identified and opened as an Image Otherwise if at least one entry is not gray scale the file is identified as a Picture see Sec 2 1 2 2 3 HANDLING OF FLOATING POINT EXCEPTIONS 17 2 2 12 Saving ASCII data Any of the internal data can be saved in ASCII format Especially for 2D Data a decision must be made about the number of relevant digits To offer the user as much flexibility as possible IDEA accepts format strings equal to those used for all printf commands of ANSI C a detailed description of this format string can be found in various C C references e g in 23 For example to get float data with 6 relevant digits write 6g which is the default string For scientific notation and 8 digits you have to enter 8E These are simple examples but the format string of C offers you much more possibilities like adding signs preceding zeros forcing decimal etc But be aware that the format string is not checked by IDEA Incorrect inputs will likely result in corrupted ASCIT entries 2 3 Handling of floating point exceptions Depending on the settings made by the operating system the floating point unit FPU raises exceptions for specific operations according to IEEE floating point spec ification For instance Windows95 NT initial
192. rms of Considering that the phase result is a combination of the random speckle phase Y and the phase shift coming from the change of the object Ad the new expression can be written as 5 4 dan Y Ad arctan Ror 3 49 3 9 PHASE SHIFT 91 With Im as the modulation intensity the expressions a 2 24 2 I 21 2cos y 1 3 50 and b B 2 12 12 41 cosy 3 51 can be found depending just on cosw Eliminating Im by combining both equations leads to a quadratic in cos y with the solution aL Ivy a 2b2 cos h x y 12 yy 3 52 There is a total number of four solutions for as the arccos function yields two results due to the identity cosy cos w Defining the result of the arccos to be modulo r and with Eq 3 49 the result for the change of the object phase can be written as axrv a 2b Ad a y 1 2 3 4 Paulx y P 2 y 1 2 3 4 Pan x y arccos 35 3 53 To get an unambiguous result A x y is compared with the phase Adpop evaluated by the Phase of Difference method as suggested in 31 that is the one of the four results with the smallest deviation modulo 27 from Adpop x y is selected as the final result This is surely the weak point of the method as the result is governed by the reference phase Additionally by far the most computation time is taken up by the evaluation of the phase to compare Therefore the procedure can be regar
193. ront or behind the overtaken one You can choose between general binary or ASCII format or corresponding Raw data without Identification Code see Tab 2 3 In general you also have the Same format option which saves results in the same format as the source files For image data the section to the right is available where you have to define the desired file type for binary Images If ASCII Data or Raw ASCIT was selected as Saving Format the ASCII Format String must be defined in the lower right of the window see Sec 2 2 12 3 2 1 Add File s Add files from disk to File Pool by using the standard file selector of the platform in use for which Multi File Selection is activated In Windows 95 NT use the SHIFT or CTRL key in conjunction with the left mouse button to choose a group of files in the list of filenames In X Window Systems you can specify a wildcard Note The last selected filename appears always at the beginning of the text line showing the current selection located below the filenames list reversing the temporal order of your selection Be aware that the number of selected filenames is limited since the buffer for the corresponding characters is restricted by the Windows operating system Without warning message all filenames which did not find place in this buffer are simply ignored 3 2 FILE POOL 31 Saving Options For File Pool x Figure 3 4 Dialog for
194. s and s 2 therefore we end up at A with a different phase order than we have started with This is not valid for continual unwrapped phase and it is clear that a scanning unwrapping procedure can not handle this A spiralling path would pick up this error with each circle and spread it outwards However the connection of B to A along path 3 reveals an overall change in phase order of 3 Therefore by scanning from A along path 1 to B and back to A along path 3 the overall change in phase order is zero there is no violation of continuity any more This suggests that by preventing to cross the area between the two sources of phase discontinuities the inconsistency and subsequently the phase error spreading can be evaded Formulated mathematically the unwrapping error around a closed path is sz PEE yk B e k 1 ylk 1 s a 3 58 where W is the wrapped phase with values in the range r 7 x k and y k define the indices of a closed path parametrized by k and Z is the operator for rounding to the nearest integer with the constraint that 0 5 and 0 5 are both rounded to 0 Due to the Nyquist sampling theorem regular phase differences must lie in the range 1 r therefore in Eq 3 58 7 is used as a threshold e g phase jump The smallest closed path is given by connecting the centers of pixels forming a 2 x 2 square For such a path traversed clockwise s will always be either 1 0 or 1 a proof fo
195. s already familiar with these objects should have no difficulties handling Tomographic Input Files The main difference to a File Pool is that not only filenames are included but the projection data itself Therefore as for Slide Shows it is possible to add data directly from the desktop which have not been saved yet u L 3 7 TOMOGRAPHY 71 Add Single Projection To add projection data already opened on IDEA s desktop select this menu item to enter adding mode Windows 95 98 NT only In this mode the mouse cursor changes its shape to a little hand when it enters a graph window displaying projection data The menu item keeps checked as long as you either click the hand on the graph window and add it to the Tomographic Input File or reselect the menu item to end the adding mode compare procedure in Sec 3 3 7 Add Projection Files Add files from disk to Tomographic Input File by using the file selector for which Multi File Selection is activated In Windows 95 98 NT use the SHIFT or CTRL key in conjunction with the left mouse button to choose a group of files in the filenames list of the selector Note The last selected filename appears always at the beginning of the text line showing the current selection located below the filenames list reversing the temporal order of your selection Add n Projection Files Add a specific number of files from disk to Tomographic Input File by using an adapted file selector After d
196. s with resolutions of 1024 x 1024 pixels and higher another method insensitive to external disturbances have become widely used with speckle interferometry This technique is called spatial phase shifting and requires recording of just two interferograms from which phase and subsequently the phase difference is calculated 50 In principle a dense carrier fringe pattern see Sec 3 8 with three or four camera pixels per fringe is used With a mean speckle size of at least the same number of pixels the standard phase shift algorithms can be applied to a corresponding number of adjacent pixels to determine the phase In IDEA the algorithms for a pixel to pixel phase shift of 120 and 90 are implemented see Sec 3 9 11 and Sec 3 9 12 3 9 1 Three Frame Technique 90 At least three equations of from Eq 3 39 are required to get a solution for z y For a 90 1 2 3 2 5 2 it can be found I x y D z 2 x y zy The indices of intensities J correspond to the number of interferograms in order of acquisition Before calculation is started an input dialog as shown in Fig 3 30 appears It is the same dialog as for all other phase shifting techniques only the number of text boxes and browse buttons differs D x y arctan 3 40 e Image Files In the textboxes the paths to the interferograms with phaseshifts 7 a as written at the left side must be defined The structure of this part of the dialog
197. scan method which try to detect the 27 phase jumps between r and 7 by scan ning though the data field taking as much neighbouring data as possible into account to vote for a jump Obviously the accuracy of these methods is limited if wrapped phase data is noisy or undersampled Most disturbing errors in jump detection are propagated as the scan proceeds on its path through the field Unwrapping with these scan methods is performed in two steps 1 Starting from a position of noise free data the wrapped phase is scanned for phase jumps Data between jumps are set to an order value which increases for 3 10 PHASE 95 a positive jump and decreases for a negative jump The resulting field of order values is called Step Function 2 Multiply Step Function Values by 27 to obtain continuous phase uncertain only by a constant value As an alternative to the fast and popular scan methods more sophisticated methods are added unwrapping by the famous branchcut technique and with DCT Discrete Cosine Transform The former method detects error inducing locations in the phase map and connects them to each other by a branchcut which must not be crossed by a subsequent unwrapping with a simple path dependent method The DCT approaches the problem of unwrapping by solving the Poisson equation relating wrapped and unwrapped phase by two dimensional DCT 24 This way any path dependency is evaded and error propagation does no
198. se change Ad z y to be almost constant within this neighbourhood a standard least squares estimate can be calculated for tan A z y The result is D D2 AD2 D3 AD1 A l 3 55 A9 DAAD D D ADi AR where D lil D h s A lLoj Im 3 56 and the brackets denote the mean value in the pixel neighbourhood The phase error e g standard deviation of Ag depends on the standard deviation of the phase change Ag within the neighbourhood and can be calculated analytically This shows that problems arise in case of Ag is close to 7 As the standard deviation of the phase error in the neighbourhood is not known a priory it is difficult to quantify an overall phase error Simulation showed that for ideal speckle interferograms the result is visually better than these obtained with the iterative Phase of Difference method or this in section Sec 3 9 9 Therefore it should be smaller than 27 30 though in 9 it is stated that this procedure is more sensitive to noise and speckle decorrelation Compared to the other method the computation speed surely deserves to be qualified as fast The input dialog is similar to the file selection dialog of the standard phase shifting algorithms see Sec 3 9 1 and Fig 3 41 for explanation of the dialog elements There is just the addition of a radio box at the top of the dialog labelled Neighbour hood Definitions for Fit see Fig 3 33 a There the width and
199. spatial phase shifting is applied is interpreted as to mark data which shall not be taken into account for phase evaluation thus the result for those pixels and those at locations where the intensity data of the masked pixel is used for phase calculation is invalid NaN see Sec 2 3 The whole mask is automatically assigned to the resulting modulo 27 data 3 9 12 Spatial Phase Shifting 90 In principle the same procedure is done here as described in the previous section Sec 3 9 11 but here the phase shift between adjacent pixels in a row is assumed to be 90 Therefore the four frame algorithm in Eq 3 42 is applied to groups of four adjacent pixels in a row The phase evaluated by this expression is this corresponding to J4 which is the intensity value of the most left pixel in a group As the phase data field is kept at the same size as the interferogram there is always a three column border at the right side filled with invalids as no result can be calculated for those pixels 3 10 Phase This menu entry mainly deals with phase unwrapping algorithms Both Fourier Transform and Phase Shifting methods yield phase data modulo 27 since the arctan function is used for calculation So the phase data related to measured property is wrapped upon itself with a repeat distance of 27 The procedure to recover the absolute phase is called Phase Unwrapping IDEA covers two path dependent methods two directional scan and spiral
200. subsequent filtering To show the location of non values in data window we use a reserved colour with offset 254 entry number 255 in the palette of the representing picture It is similar to the mask colour but with reduced luminance 2 4 Input Macros and Operators In all text boxes for values input macros can be used see Tab 2 4 Be aware that use of min and max are restricted to dialogs which are directly connected to Images 2D or 1D Data fields Additionally for all value inputs the operators for division and multiplication can be used Macros and operators can be used simultaneously For the macro pi a preceding factor without an operator is treated as if multiplication would be in parenthesis e g 1 2pi 1 27 whereas 1 2 pi 1 2 7 2 4 INPUT MACROS AND OPERATORS 18 Table 2 3 File Types of IDEA Extension is the default file extension ID is Identification Code in the file header Class is the Format Class defined in IDEA see list in Sec 2 2 1 Extension ID Description Class binary ASCII binary ASCII Image Plain 8 bit Image Data internal img dat ig IG Image format Bitmap Standard Windows Graphics For bmp BM Image Picture mat Pixmap Standard X Window Graphics xpm ye Image Picture Format Format of Imaging Techn ITEX software pic IM Image
201. t unless evidence is available to the contrary in a particular application area it is more likely to be an appropriate method to use than any other method designed for the parallel mode of data collection Unfortunately in many cases Interferometrical Tomography seems to belong to those particular application areas where the Convolution reaches its limits The main reason is that the Convolution brings about problems if a too small number of projections is given a draw back from which complicated interfero metrical optical setups often suffer The corresponding reconstructions show strong oscillations at outer regions artefacts which can be minimized by increasing the number of projections by interpolation see Sec 3 7 4 though this will not enhance physical information Therefore IDEA includes also the algebraic reconstruction technique ART which gets along with a smaller number of projections 35 but requires much longer com putation times 3 7 6 ART The ART Algebraic Reconstruction Technique belongs to series expansion tech niques which have a completely different approach to solve the Radon Inversion The Radon Transform is developed into a linear equation system 18 Einstein nota tion used Yi Rijt j 3 28 with y denoting measured projection data merged into a vector x the data to re construct also regarded as a so called image vector and with Rij as the matrix containing intersections of pr
202. t appear localized as for scanning methods The advantage of this technique is that noise in wrapped data has less influence and that the whole unwrapping is performed in a single step no step function is created though the time consumed is rather high compared to path dependent methods Be aware that after application of any phase unwrapping algorithm true phase jumps in unwrapped phase data higher than 27 cannot be identified correctly since these are interpreted as changes of order 3 10 1 2D Scan Method As for all path dependent methods a starting point has to be defined where the scan starts For a phase map with several regions think of it as isles isolated from each other by areas of invalid data for each isle an own starting point can be selected This is done either by mouse interaction see Sec 3 3 14 or by the multiple points dialog see Sec 3 3 15 and Fig 3 12 The column through a starting point is regarded to consist of completely reliable data and is unwrapped from starting point to top and bottom after initializing the step function s everywhere with 127 to allow the maximum number of changes up and downwards within 8 bit data depth With these starting values all adjacent columns in the four quadrants are scanned up or downwards for jumps where both the order at horizontal and vertical inner neighbour is taken into account see Fig 3 34 De pending on difference of wrapped phase A0 to
203. t critical issue when importing data with third party software If the software you want to import data to does not accept invalids at all you have to preliminary substitute them within idea Edit 2D Data Substitute Invalid Values the same for Edit 1D Data Set Tabulator as Delimiter for ASCIT Export By default data is exported in ASCII format using separating spaces blanks between numbers Here one can replace the blanks by tabulators This is another feature added for the sake of compatibility with other software Other than the substitution of the string for invalid data this is not restricted to export in raw ASCII format since IDEA is capable of reading data with tabulator delimiters Load Preferences Load a previously created Preferences file cfg an optional ASCTI File containing all information about the settings in the Preferences menu to be actualized after loading During startup idea cfg is searched first in the current working folder then in the folder where idea exe is located The searching and finding process is reported in the protocol window Note the file in the current folder overrides any configuration file in the directory of idea eze Save Preferences During startup the optional file idea cfg is searched first in the current working folder then in the folder where idea exe is located The searching and finding process is reported in the protocol window All preferences settings are made according to
204. t of the sorted data set The difference of these values is then compared to the smooth limit values above indicate an edge to be preserved In this case the value at location of central pixel of mask is copied to filtered image without any further calculations In the dialog connected to this menu you have to define the size of the mask the number of repetitions to perform and the smooth limit mentioned above The check box Relative Smooth Limit must be activated if limit is given relative to dynamic range of the whole data field maximum minus minimum else the value is interpreted as absolute value For invalid values within a 2D Data field the rigourous filter mode is applied see Sec 3 5 1 1 3 5 6 Adaptive Median This filter is specific to modulo 27 phase data as it is designed to preserve the typical sawtooth edges in this data It utilizes also lowpass filtering by determining the median within a neighbourhood but here the filter response is adjusted to local factors depending on the signal characteristics 8 In principle the basic idea of the selective median is here developed in a much more sophisticated procedure Here not only the sequence of the sorted neighbourhood data is taken into account but their actual values A histogram of three intervals h1 h2 h3 separated at 7 Q a 7 is determined By comparing the number of data Nj i 1 2 3 in these intervals it is possible to determine wether the neighbou
205. ta When A by itself is minimized the agreement becomes impossibly good but the solution becomes unstable or wildly oscillating That is where B comes in It measures the smooth ness or stability of the desired solution and is called the stabilizing functional or regularizing operator The central idea in inverse theory is the prescription to minimize A AB for various values of 0 lt A lt oo along the so called trade off curve and then to settle on a best value of by one or another criterion ranging from fairly objective to entirely subjective The normalized Abel Integral equation reads h y D f r 2 7 _ where the un known function f r is to be determined from h y We substitute s y and 0 else a 0 lt s lt x r By introducing the response function p x s ya s and tak 3 6 ABEL INVERSION 63 ing into account that in practice only a discrete counts spectrum of N data points hi 7 h s ds can be observed we obtain Toca with ataye E m f Fe aya th ala f Seeds 8 11 In the Backus Gilbert technique 34 30 we seek a set of inverse response kernels q x such that N f z gt ala hi q x h 3 12 is the desired good statistical estimator of f a The functionals A and B are chosen as 36 A 9 al Wis 2 a z a x W x a z 3 13 B Var f x 79 ute Susto a afe S qC 3 14 where W x Lt 2 pi x p x da is the response matri
206. tant representing relation of variances of z and 3 7 TOMOGRAPHY 75 Algebraic Reconstruction Technique Ea Size of Reconstructed 2D Data 401 Maximum Number of Iterations fo Relaxation Parameter RP 0 05 RP Adjustment after Iteration Number 100 Minimum Correction Break off Condition Po Minimum iteration Error Break off Condition PO Size of Smoothing Matrix ft Selective Smoothing Parameter 9 fo TT Apply Zero Correction z ls Lower Limit of Reconstructed Values i 1000 Upper Limit of Reconstructed Values fi 000 Initialize with Precalculated Data Browse Ctomograptyiconvolution201 tor Figure 3 28 ART Dialog for input values e Finally A is the relaxation parameter and e a positive infinite small number For some application convergence was much faster when A was automatically set to a 1500 umm Ymax A 0 001 if AM lt 0 001 Mie 1 if AO gt 1 Obviously the choice of initialization of x has a large effect on the outcome of the iterative procedure especially if the number of iterations is limited In 18 it is recommended to set all data to the same value near to estimated average density of the field or to use the output of a reconstruction with Convolution As stated in 18 the ART is not an acceptable alternative to the convolution method if many projections with reliable data are available as expense of computation time is just too high However in situations where data collection g
207. ter s must be followed by a format string for ASCII files see Sec 2 2 12 in the same line The data itself whether ASCII or binary starts after a line feed For example the header of a phase field modulo 27 with 256 rows each row consisting of 512 data elements is M2 512 256 6g Apart from that internal formats we support three external formats for saving and loading Images See Sec 2 2 11 for further information 2 2 4 Internal Format of Frequency File The Frequency File contains the complex amplitudes of the spatial frequencies of an Image or 2D Data field The rather complicated structure of the data is due to efficient coding of the Fourier transform for optimization of speed and minimization of memory requirements The resulting Packed Order of the data is standard and well documented e g in 36 Its structure is shown in Tab 2 1 For ID refer to Tab 2 3 The data for the horizontal Nyquist frequencies is delivered separately in a vector by the algorithm This data is splitted into parts of the same length as one line Usually the last part has to be filled up to length of line with zeros These additional lines are appended to the packed field 2 2 5 Internal Format of Tomographic Input File This file type includes the one dimensional integral data of the different directions from which the reconstruction is calculated and the corresponding angles of these projections See Tab 2 2 for the structure of this file ty
208. th the rules of thumb we developed for our problems At first you have to define the maximum number of polynomials to be used for f Interpolation and the maximum order Ni of model function for the Fourier Method After that all f Interpolations from 2 up to the defined maximum number of polyno mials are calculated followed by serial application of the Fourier Method The used range of orders are 1 to 1 1 to 2 1 to 3 and so on until 1 to Ni is reached During the Inversion several parameters are calculated which can be used to evaluate the particular reconstruction problem They are represented in the protocol window in three 2D Data windows and one Multiline Window see Sec 3 12 3 Multiline Window The Multiline Window includes the following 6 graphs 1 Deviations for f Interpolation fint Deviations The idea behind this information is that in the vicinity of the ideal input pa rameter the overall difference between the reconstructions should be minimal This difference is calcuated by summing up squared deviations If too much polynomial are used the reconstructed noise will cause a rather high difference to the reconstruction with one more polynomial used For too few polynomi als the reconstruction is oversmoothed causing high difference to the next in general much better result In ideal case the resulting curve shows a fast decrease of the difference until a minimum is reached Then noise reconstruction be
209. tive frequencies and are therefore redundant For negative vertical frequencies indices amax and amin denote frequencies with maximum or minimum absolute value The second part of the table gives the vector with horizontal Nyquist frequencies which are added as additional rows last row completed by filling up with zeros if necessary The complex amplitude is saved with its Real Re and Imaginary Im parts both in binary double precision format 8 bytes Frequency Ex Dia fxitax Periods 0 1 1 fyg 0 Re Im Re Im Re Im ia 1 Re Im N fymax gt gt 1 En N aya 2 N fyamax ot 1 fy oia 1 Re Im Frequency fyg fytin fs ias a AY sent Periods 0 1 Wij Aw ma 1 ya de Re Im Re Im Re Im Table 2 2 Format of Tomographic Input File The effective length is the length of the shortest projection included and is used by algorithms From longer distributions data out of range at the sides is ignored fixed Location Repeats Data Format 1 line Identification Code ASCII Effective length of projections Number of Projections n 2 line Comment max 1024 byte read ASCII n lines Relative pathname of source projection file ASCII n times Projection angle new line ASCII Length of projection new line ASCII Projection Data binary he reconstru
210. tively non values see Sec 2 3 can be substituted with real values There are two modes for the substitution which can be selected at the top of the related dialog e Substitute by Spline Interpolation A cubic spline interpolation through all valid points fills the invalid holes in the data distribution e Substitute by Fixed Values Selecting this option activates the lower part of the dialog where values can be defined in the text boxes to substitute Infinity Infinity and Not a Number Here macros min and max are allowed see Sec 2 4 Shift Minimum Maximum Average to 0 Shift the current distribution of data to set new zero level Mirror Mirrors the whole 1D distribution by arranging data in reverse order Locations of center and borderlines have no effect Clip Creates a new 1D distribution from the data between the border lines including data at border Rescale Change number of data points representing the distribution After definition of the new number in a dialog the new distribution is calculated using Cubic Interpolation assuming derivative 0 at borders Be aware that rescaling symmetrical distributions may lead to loss of symmetry though not visible by eye Determine Center Estimates location of center by same relation as used for determining center of mass taking only data between border lines into account Here m is the coordinate of weighted center of distributio
211. tributions depend strongly on the basic shape of the integral data curve so the following interpretations might not be generally useful Basically the overall curvature of a reconstructed distribution should be not too high since then unwanted oscillations dominate the Abel Inversion The tendency of the distribution shows that this would require a small number for maximum order of model function But looking at the deviations from input data in Data four Chi dat one can see that this is contradictory to the desired approximation to measured in tegral data Therefore information of both curves must be taken into account The appropriate number of maximum order should be somewhere at the lower part of the knee in Data four chi dat and somewhere on the long low plateau in Data four Curvature dat Additionally increasing the maximum order by one should result in an Abel Inversion with just minor differences if the filter effect of the Fourier Method works effectively Therefore a single distinct minimum in Data four Deviations dat is a strong recommendation for the appropriate input parameter Unfortunately here 4 3 EXAMPLE 3 ABEL INVERSION 122 are two minima for maximum order 4 and 9 but both are not far below the noise amplitude For this particular example the final interpretation can be made as follows e The curve for curvature suggests maximum oder of model function N lt 10 e Deviations fro
212. ture Its 2 2 FILE TYPES AND FORMATS 12 palette consists of 252 gray scales from black RGB 0 0 0 to white RGB 255 255 255 Therefore not all available gray levels are used but this cannot be seen by the human eye Be aware that the Picture serves only as a visualization of the Image data to be processed and to be saved The remaining four palette entries are used for the mask underflow and overflow see below The visualization of the eight Bit data is done line by line from top to bottom If a bitmap is opened its palette is checked for non gray entries If such an entry is found the bitmap is not accepted as an Image and an error massage occurs If not the gray scale information is retrieved and then visualized by a gray scaled picture palette 2 1 4 2D Data or Data Field A Data Field is a two dimensionally arranged data in double precision format vi sualized by a picture The 252 colour nuances correspond to the same number of intervals between maximum and minimum of the Data Field As by Images the data is visualized line by line from top to bottom 2 1 5 Mask The user can interactively mark data within Images or Data Fields This can be done by paint brushing using a mask colour or by special algorithms masking certain data values Since masking is restricted to the visualization level original data are not harmed The mask can be saved separately see Sec 2 2 8 Sec 3 11 2 and Sec 3 11 3 In each palette the 25
213. um C Keep all Borderpixels Number of Neighbours F IV Set Branchcuts to Invalid I Show Residues in Extra Image J Create Mask to Show Branchcuts OK i Cancel Figure 3 38 Dialog for branchcut unwrapping applying minimum cost matching algorithm If not checked a final unwrapping procedure will add a multiple of 27 to these pixels to minimize deviations from neighbours at either side of the branchcuts Regard this as a mere visual enhancement of the result as there are no criterions to select the best result at branchcut pixels e Show Residues in Extra Image checkbox Check this to create an image where the residues are represented by pixels with colour e g value depending on the sign blue pixels values 1 mark negative residues white pixels values 252 mark positive residues e Create Mask to Show Branchcuts checkbox With this box checked a mask is created in the data field with unwrapped phase data which covers the branchcuts excluding the residues The same is done in the image showing the residues if created After finishing calculation information about overall squared cutlength is given in the protocol window 3 10 10 3 Local Minimum Cost Matching Branchcuts As the global minimum cost matching algorithm is too slow for a large number of phase residues and requires a huge amount of memory for the cost matrix another method to set the branchcuts has been coded by the author to provide a relatively
214. uniform phase shifts given that there are no higher order phase errors The required input parameters are set in a dialog similar to this in Fig 3 30 but with four text boxes and browse buttons For detailed explanations see Sec 3 9 1 Be warned that visibility or modulation respectively to be compared with defined values are calculated for phase shifts a 90 3 9 7 6 Frame with Nonlinearity Correction The following equation suggested in 19 I 612 1713 171 I arctan v3 51 612 1745 1714 615 515 3 46 L 2613 2513 2514 2615 4 Ig derived for a 60 and i 5 2 3 2 1 2 1 2 3 2 5 2 is insensitive to quadratic phase shift errors and nonuniform spatial phase shift This means that if the actual phase shift a can be written as a function of the unperturbed phase shift a in the form of a polynom a all et eal 3 47 this algorithm not only compensates for the error coefficients and e2 but also for spatial variation of the phase shift across the aperture However errors due to random noise is 1 65 times higher than in standard methods 3 9 8 Speckle Phase of Difference As the other speckle methods implemented in IDEA the phase of difference method POD allows to determine phase from a single speckle interferogram recorded after or during phase alteration by the object under investigation Therefore it is applicable to objects which are not stable du
215. ur Chi dat gives feedback about deviations from recalculated integral distribution to input data Finally the window Data four Deviations dat shows a graph where deviations of two consecutive reconstructions with increased parameter for maximum order are shown Additionally all Abel inverted reconstructions are melted in a 2D Data window Data Four AllAbellInversion00 bin each forming a row from top to bottom of the Data Field Above the window Data fint four Deviations00 bin shows overall deviation from each inversion obtained from f Interpolation to every result from Fourier Method of the reconstruction is shown as window Integral abl in Fig 4 6 The deviation from measured and transformed data is shown in the graph Deviation abl Finally for visualization the 2D distribution is shown in the 2D Data window Rotate bin in Fig 4 6 Problem Analysis To help the user finding appropriate parameters for Abel Inversion Abel Inver sion Problem Analysis was implemented see Sec 3 6 5 The screenshot in Fig 4 7 shows windows with the 1D Data distributions belonging to the Fourier Method which were extracted from the original Multiline Window and zoomed by factor 5 Additionally all Abel inverted reconstructions from minimum order 1 to maximum order 1 40 are shown in the 2D Data window Data Four AllAbelInversion00 bin each forming a row from top to bottom of the Data Field Above the window Data
216. us filter mode can not be applied to 2D Data fields containing invalid data see Sec 3 5 1 1 3 5 2 High Pass 3 x 3 This filter eliminates low frequency components from slowly varying characteristics of an image such as overall contrast and average intensity It highlights fine detail and sharpens the image The Filter Kernels for high pass are characterized by negative coefficients in the outer periphery and positive coefficients in the center Due to the negative elements the result R in Eq 3 1 may be also negative which is not valid for an image So negative values are set back to zero 3 5 3 User Kernel Whereas the other menus provide direct access to commonly used filter techniques with specific Kernels here not so common and even user defined Kernels can be created modified loaded and applied The according dialog is shown in Fig 3 19 its interaction elements are described below e Filter Kernel In the list you can select a predefined filter Kernel in the list to load it It is shown at the left side of the dialog but cannot be edited create a new one by selecting New Kernel axa of size a or by selecting User Defined to enable editing of a previously loaded predefined kernel New Kernels are initialized with zeros Kernels with integer elements are faster but also floating point elements can be defined 3 5 FILTERING 52 Filter Kernel New Kernel 7x7 gt z
217. valuation with 2D FFT method This menu entry is exactly the same as Edit 2D Data Remove Linear Tilt see Sec 3 3 5 It was duplicated to provide a shortcut in an environment where many users will search for such a procedure 3 10 16 Remove Fitted Linear Phase Shift As with the previous menu item this subtracts a plane function from 2D phase data However here the plane is calculated by planar regression of the phase values taken from all masked and valid pixels This way the plane is quite independent from noise At least three valid masked pixels lying not on a straight line are required for the algorithm to work 3 11 Mask The mask see also Sec 2 1 5 and Sec 2 5 1 is a tool to interactively mark data in an Image or any 2D Data Field without changing it For some applications marked data can be excluded from calculation e g unwrapping by scanning operations or it can be set to specific data which allows editing of a field Most important a filter mask is required to apply image filtering or phase evaluation by 2D FFT This menu comprises all available commands for setting mask draw mode as well as managing setting and editing mask data 3 11 1 Copy Mask Copy mask from other window B into current active window A The procedure is similar to that in Sec 3 3 7 Only mask in range is copied so A and B may have different size 3 11 2 Add Mask Loads a mask from file using standard file selector and add it to current act
218. x and Sij is the covariance matrix If one can neglect off diagonal elements covariances as when the errors on the h s are independent then 6 07 is diagonal We introduce the integrals of the response kernels R fo pi x dx for each data point The integrals R and W can be calculated analytically The functions q x are now determined by the principle of minimizing A AB q x W x AS q x subject to the constraint that E q x R q x R 1 For any particular data set h set of measurements h the solution f x is obtained as 30 a h W 2 AS 7 R I R Wera R 3 15 If you select the Menu Item Abel Inversion Backus Gilbert Method a dialog pops up where you can set the following parameters e Tradeoff Parameter 0 We do not minimize the functional A AB but rather the functional A cos 27 B sin 270 The valid range for is the interval 0 1 A value of 6 1 corresponds to A oo The higher the value of 0 the more the reconstructed data are smoothed From our experience useful values for 9 are 1078 1075 Relative Error Assumed as Constant If checked then it is assumed that o n otherwise 0 1 The covariance matrix is assumed to be diagonal in both cases e Show Integral of Reconstructed Data Check this to show not only the Abel reconstruction f a a is Cartesian co ordinate with left border of distribution at 0 substituting r but also an additional graph of the analytical
219. y direction and 2 y is the required phase function The frequency vo is introduced e g by tilting a mirror within one arm of the interferogram If the greatest gradient from the object phase is less than the spatial carrier phase and vo is less than half of the sampling frequency Nyquist condition then the phase distribution can be determined in both magnitude and sign The relation in Eq 3 32 can be transformed to ilr io r e r e o c r e 270 3 33 where 1 id c r mrje 3 34 The asterisk superscript denotes a complex conjugate Fourier transforming x y yields ilr ig v E v vo amp v vo 3 35 This reveals the purpose of applying a fringe carrier system Disturbing changes in background intensity io are of low frequencies By applying a comparable high carrier frequency vo the information of the fringe system is separated from disturbing low frequencies in the Fourier domain shifting it to the vicinity of vo The first approach to calculate the phase distribution r is straight forward 1 Determine the domain vw vo of the interferogram around the carrier fre quency 3 8 2D FFT 2 Set all frequencies outside this domain to zero 3 Transfer the domain centered around v towards the origin zero frequency which removes the carrier to be skipped for algorithm in IDEA 4 Determine inverse Fourier transform to the resulting frequency field yieldin
220. ymmetric anode which is located 4mm above the triangular shaped cathode Zero Padding and Masking of Interferogram The 2D FFT algorithm requires dimensions of the Image which are powers of two see Sec 3 8 This is not the case in our example Hence we zero pad the Image Original img in Fig 4 using 2D FFT Zero Padding Since the shadows of the elec trodes contain no information these areas are masked using the methods explained in Sec 3 11 The result can be saved to a file Object msk The final interferogram with the mask overlay is shown in Fig 4 as Padded img 2D FFT and Filtered Backtransformation Now the interferogram can be forward transformed with 2D FFT algorithm see Sec 3 8 using 2D FFT Forward FFT Window 2D FFT frq in Fig 4 shows ab 113 4 1 EXAMPLE 1 TOMOGRAPHIC RECONSTRUCTION 114 OLA Eie FilePool Edit View Filtering AbeHnversion Tomography 2D FFT Phase Shift Phase Mask Information Window Help Sal aja MES a Joa Ses Mise aaa aa ala O Frequency 2D FFT fig BE E 6 Phase Mod 2P Back FFT m2p lol 5 Phase Camerpha ixi Frequency Masked frg BEES 129 129 129 129 129 129 1x1 257x257 R 513 31 60 42 Figure 4 1 Screenshot of IDEA showing different processing steps for application of 2D FFT The original object interferogram Original img has to be zero padded Padded img As result of 2D FFT a field of complex amplitud

User Manual for IDEA 1.5 - optics.tugraz.at

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