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IDL Wavelet Toolkit User's Guide

1. WV_PLOT_MULTIRES Run the multiresolution analysis GUI WV_TOOL_DENOISE Run the wavelet de noising GUI Table 4 1 Widget Commands and Tools Wavelet Transform The following table describes the wavelet transform commands Command Description WV_CWT Compute the continuous wavelet transform of an array WV_DENOISE Denoise an array using the discrete wavelet transform WV_DWT Compute the discrete wavelet transform of an array WV_PWT Compute the partial wavelet transform of a vector Table 4 2 Wavelet Transform Commands List of Commands by Functionality IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference Wavelet Functions 59 The following table describes the built in wavelet functions Command Description WV_FN_COIFLET Construct coiflet wavelet coefficients WV_FN_DAUBECHIES Construct Daubechies wavelet coefficients WV_FN_GAUSSIAN Construct the Gaussian wavelet function WV_FN_HAAR Construct Haar wavelet coefficients WV_FN_MORLET Construct the Morlet wavelet function WV_FN_PAUL Construct the Paul wavelet function WV_FN_SYMLET Construct symlet wavelet coefficients Table 4 3 IDL Wavelet Toolkit Wavelet Basis Functions List of Commands by Functionality 60 Chapter 4 IDL Wavelet Toolkit Reference WV_APPLET The WV_APPLET procedure runs the IDL Wavelet Toolkit graphical user interface
2. Logio Pm into the range 32 255 Values greater than zero but less than 10 Log o P are set equal to 32 4 Set all values removed by the filter to zero 0 Display the image using a grayscale color palette Denoise Tool IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 43 Using the above method all retained coefficients will appear in the image shaded from dark gray 32 to white 255 Coefficients that have been removed will be black Coefficient Power This graph shows the wavelet power for each coefficient sorted into decreasing order and scaled so that the total power is 100 The wavelet power is also shown as a cumulative plot where each point represents the sum of all of the previous points Both curves are plotted on a logarithmic x axis so that the largest coefficients are easily visible The dashed line shows the current cutoff value that you have selected Wavelet Options You can change the current wavelet family or the order Since all of the denoise options remain constant you can compare the effects of different wavelet orders and families Denoise Options Cumulative Power The slider bar allows you to set the cutoff threshold for cumulative power Coefficients to the right of the dotted line in the Coefficient Power graph will be excluded The Coeffs box is adjusted accordingly Note At low cumulative power you may notice that the slider adjusts itself in uneven increments
3. WV_DWT The WV_DWT function returns the multi dimensional discrete wavelet transform of the input Array The transform is done by WV_PWT using a user inputted wavelet filter The length of each dimension of Array must be either a power of two 2 or must be less than four 4 The transform is not computed over dimensions of lengths less than four 4 but is computed over all other dimensions for example the wavelet transform of an array of size 3 256 is computed over each 1 256 column vector WV_DWT is based on the routine wtn described in section 13 10 of Numerical Recipes in C The Art of Scientific Computing 2nd ed Cambridge University Press and is used by permission Syntax Result WV DW T Array Scaling Wavelet Ioff Joff DOUBLE INVERSE N LEVELS value Return Value The result is an output array of the same dimensions as Array containing the discrete wavelet transform over each dimension Arguments Array The input vector or array The length of each dimension must be either less than four 4 or a power of two 2 Scaling A vector of scaling father coefficients of length N Wavelet A vector of wavelet mother coefficients of length N WV DWT IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 75 loff An integer that specifies the support offset for Scaling To center the scaling function over each point in Array set Ioff to N 242 Joff An integer that s
4. Noe The IDL Wavelet Toolkit must be licensed on your system to be able to use this procedure Syntax WV APPLET Input ARRAY array GROUP_LEADER widget_id NO_SPLASH TOOLS string array WAVELETS string or string array Arguments Input Input can be either a string giving the name of a IDL Wavelet Toolkit save file or a one or two dimensional array of data If Input is not specified then the sample file wv_sample sav is opened If Input is set to null string then the IDL Wavelet Toolkit is started with an empty dataset Keywords ARRAY Set this keyword to a one or two dimensional array of data to be imported into the IDL Wavelet Toolkit upon startup If argument nput is set to a filename then ARRAY will be added to the list of variables GROUP LEADER The widget ID of an existing widget that serves as group leader for the newly created widget When a group leader is killed for any reason all widgets in the group are also destroyed A given widget can be in more than one group The WIDGET CONTROL procedure can be used to add additional group associations to a widget For more information see WIDGET CONTROL DL Reference Guide It is not possible to remove a widget from an existing group WV APPLET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 61 NO SPLASH If this keyword is set then the splash screen will not be displayed on startup TOOLS A scalar string
5. Notice the large peak near Scale 256 sec This is primarily due to the discontinuity that occurs when the dataset is wrapped around from the end back to the beginning Move the Order slider bar from 6 to 4 to make the peak more narrow You can use your mouse to rotate Zoom in or out or move the plot 5 To find the chirp peaks select the Zero Phase Lines check box 6 Now deselect the 3D check box to view the surface from above Scale sec 0 100 200 300 400 500 Figure 3 1 The Wavelet Power Spectrum of the Chirp Signal Wavelet Power Spectrum IDL Wavelet Toolkit Chapter 3 Theory and Examples 51 Denoise Background One of the most useful applications of wavelet analysis is to remove unwanted noise from a dataset This noise could be due to measurement errors or instrument noise In image processing the noise might be small scale features or artifacts You could try to remove noise from the signal by using a low pass or band pass Fourier filter There are two problems with this approach 1 You need to carefully choose the width and shape of your filter both to avoid removing too much of your signal and to decrease ringing from peaks and discontinuities and 2 In many cases the noise is white in other words it is distributed across all frequencies or spatial scales Wavelet analysis on the other hand offers a scale independent and robust method to filter out noise The basic technique in
6. when computing the wavelet transform while nonorthogonal wavelets will have some overlap nonzero correlation Using an orthogonal wavelet you can transform to wavelet space and back with no loss of information Nonorthogonal wavelet functions tend to artificially add in energy due to the overlap and require renormalization to conserve the information In general discrete wavelets are orthogonal while continuous wavelets are nonorthogonal Symmetry This flag describes the symmetry of the wavelet function about the midpoint Symmetric wavelets show no preferred direction in time while asymmetric wavelets give unequal weighting to different directions Compact Support This value measures the effective width of the wavelet function A narrow wavelet function such as the Daubechies order 2 compact support 3 is fast to compute but the narrowness in time implies a very large width in frequency Conversely wavelets with large compact support such as the Daubechies order 24 compact support 47 are smoother have finer frequency resolution and are usually more efficient at denoising IDL Wavelet Toolkit Wavelet Viewer 30 Chapter 2 Using the IDL Wavelet Toolkit Vanishing Moments An important property of a wavelet function is the number of vanishing moments which describes the effect of the wavelet on various signals A wavelet such as the Daubechies 2 with vanishing moment 2 has zero mean and zero linear t
7. After choosing LEE the file using the Select Import File dialog you can specify the particular format for the ASCII TEMPLATE dialog See ASCII TEMPLATE IDL Reference Guide for more information The ASCII TEMPLATE routine handles ASCII files consisting of an optional header of a fixed number of lines followed by columnar data The procedure consists of three steps 1 Define Data Type Range Specify whether the data is in fixed width columns or separated by commas or spaces The first 50 lines are displayed Choose the first line of data and click on the Next button 2 Define Fields Choose the number of fields per line and then click Next gt 3 Field Specification You can change the names and data types for the various fields The Field names can also be changed once the data is imported into the Toolkit Click on the Finish button to import the data into the Wavelet Toolkit Once the data is successfully imported you can change the default names for the variable Title Variable etc Independent Variable For ASCII files with multiple columns if the first column is determined to be monotonically increasing in value or is assigned the field name TIME within the ASCII TEMPLATE then it is assumed to be the independent variable In this case the remaining columns are then imported as the dependent variables Note You may change the name TIME after the data has been imported into the Wa
8. WV_FN_HAAR and WV_FN_SYMLET IDL Wavelet Toolkit WV DENOISE 70 Chapter 4 IDL Wavelet Toolkit Reference Order The order number or parameter for the wavelet function given by Family If not specified the default for the wavelet function will be used Note If you pass in a DENOISE STATE structure then Family and Order may be omitted In this case the values from DENOISE STATE are used Keywords COEFFICIENTS Set this keyword to a scalar specifying the number of wavelet coefficients to retain in the filtered wavelet transform This keyword is ignored if keyword PERCENT is present CUTOFF Set this keyword to a named variable that upon return will contain the cutoff value of wavelet power that was used for the threshold DENOISE STATE This is both an input and an output keyword If this keyword is set to a named variable then on exit DENOISE STATE will contain the following structure Tag Type Definition FAMILY STRING Name of the wavelet function used ORDER DOUBLE Order for the wavelet function DWT FLT DBLARR Discrete wavelet transform of Array WPS FLT DBLARR Wavelet power spectrum equal to IDWTI 2 SORTED FLT DBLARR Percent normalized WPS sorted CUMULATIVE FLT DBLARR Cumulative sum of SORTED Table 4 4 The Structure Tags for DENOISE STATE WV DENOISE IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 71 Tag Type Definition COEFFICIENTS LONG Number of coefficient
9. 0 255 quantize the 24 bit colors down to 256 colors or split the three channels into separate red green and blue images WAV Audio Files KS Select this menu item or button to import a WAV RIFF audio file as a one dimensional vector The file must be in uncompressed PCM format Multiple channels are imported as separate variables one for each channel IDL Command Line You can also import data directly from the IDL gt command prompt using the WV IMPORT DATA command WV IMPORT DATA variable where variable is either a data vector or array or a structure of data tags see WV IMPORT DATA on page 96 for tag information Importing Data IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 27 If there is more than one Wavelet Toolkit applet currently running then variables are entered into the one that was most recently active IDL Wavelet Toolkit Importing Data 28 Chapter 2 Using the IDL Wavelet Toolkit Wavelet Viewer The Wavelet Viewer is accessible from the Visualize menu or button and can also be started from the IDL gt command prompt using the WV_CW_WAVELET function wid WV CW WAVELET For more information see WV CW WAVELET on page 62 The Wavelet Viewer consists of a graph of the currently selected wavelet function a selection area for the wavelet function and an information area shown in the following figure al View Wavelets i DI x Family Daubechie
10. Architecture IDL Wavelet Toolkit This chapter discusses the following topics Starting the Toolkit 12 Menu Description 0 eem 14 Preferentes occo eda ted hea Ghee ad 18 Luatuset VIGNWEE io seo serbe escas 20 bhuporimg Data ccc ctw dawekiadaa wes 25 IDL Wavelet Toolkit Wavelet Viewer cus seed eme 28 Wavelet Power Spectrum 32 Multiresolution Analysis 39 Denoue Tool wee ie ex Ee 41 Adding User Tools 005 45 11 12 Chapter 2 Using the IDL Wavelet Toolkit Starting the Toolkit To start the IDL Wavelet Toolkit type the following at the IDL command prompt wv_applet This action compiles the wv applet routines and starts up the main window shown in the following figure For other startup options see WV APPLET on page 60 The window consists of Menu Items the Toolbar the Dataset Viewer and a Status Bar at the bottom Menu Items The menu items located at the top of the IDL Wavelet Toolkit window allow you to perform various actions These menu items are described in the next section Toolbar The toolbar is divided into five sections File Import Edit Visualize and Help The toolbar buttons allow you to easily access various menu items When you position the mouse pointer over a toolbar the Status Bar displays a description of its function Dataset Viewer The variables contained in your dataset are displayed in the dataset table describe
11. The black regions of the Wavelet Coeffs plot shows the discarded coefficients The percent difference between the original and filtered image is about 6 Examining the filtered image you will notice that much of the speckling around the outside is now gone In addition some of the small scale features and low contrast regions within the image have been diminished Finally the dotted line on the Cumulative Power graph indicates that although you are only retaining 12 596 of the information you are preserving almost 10046 of the variance or power IDL Wavelet Toolkit Denoise 54 Chapter 3 Theory and Examples Multiresolution Analysis Background The wavelet transform can be thought of as a band pass filter where the location and width in Fourier space depends on the wavelet scale Larger scales imply a lower frequency and small bandwidth In computing the wavelet transform you change from small scales to larger scales At each stage you can stop and compute the inverse wavelet transform using the remaining coefficients while setting the small scale coefficients to zero You can then build up a series of smooth or low passed detailed or band passed or rough high passed versions of your original data Method Details on computing the multiresolution analysis can be found in Lindsay et al 1996 Example Use the Mantle convection dataset that is included in the Wavelet sample file This dataset contains an image of
12. convection within the Earth s mantle Try the following steps 1 Select the Convection dataset and start up the Multiresolution Analysis viewer using either the Visualize Menu or the Toolbar button 2 As you progress from top to bottom the wavelet scale increases in powers of two At the smallest scale most of the image is still in the Smooth image Notice that the Rough image contains only the edges or discontinuities which the small scales can pick out 3 Change to the Haar wavelet and observe the different structure of the images Multiresolution Analysis IDL Wavelet Toolkit Chapter 3 Theory and Examples 55 Bibliography Daubechies I 1992 Ten Lectures on Wavelets Society for Industrial and Applied Mathematics 357 pp Donoho D L and I M Johnstone 1994 Ideal spatial adaptation by wavelet shrinkage Biometrika 81 425 455 Hubbard B B 1998 The World According to Wavelets 2nd ed A K Peters Wellesley Mass 331 pp Lindsay R W D B Percival and D A Rothrock 1996 The discrete wavelet transform and the scale analysis of the surface properties of sea ice IEEE Trans Geosci Remote Sens 34 771 787 Mallat S 1989 Multiresolution approximation and wavelets Trans Amer Math Soc 315 69 88 Press W H S A Teukolsky W T Vetterling and B P Flannery 1992 Numerical Recipes in C The Art of Scientific Computing 2nd ed Cambridge University Press 994 pp Torrence C
13. has multiplied by Sqrt 2 and for some orders the coefficients are reversed Coefficients for orders 11 15 are from http www isds duke edu brani filters html Version History 5 3 Introduced WV_FN_SYMLET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 95 See Also WV_DWT WV_FN_COIFLET WV_FN_DAUBECHIES WV_FN_HAAR IDL Wavelet Toolkit WV_FN_SYMLET 96 Chapter 4 IDL Wavelet Toolkit Reference WV_IMPORT_DATA The WV_IMPORT_DATA procedure allows the user to add a variable to the currently active WV_APPLET widget from the IDL gt command prompt Note The IDL Wavelet Toolkit must be licensed on your system to be able to use this procedure Syntax WV_IMPORT_DATA Data MESSAGE_OUT string PARENT variable Arguments Data A one or two dimensional array of data or a structure containing the data Keywords MESSAGE_OUT A scalar string giving a message to be output to the WV_APPLET message bar PARENT A long integer specifying the ID of the WV_APPLET widget in which to import the data The default is the most recently active WV_APPLET widget Examples To import a 1D or 2D array directly into the active WV_APPLET widget WV_IMPORT_DATA Array To import a data structure WV_IMPORT_DATA DATA PTR_NEW Array SOURCE Chandra X Ray Observatory TITLE Cygnus X 1 X Ray Image VARIABLE Cygnus X 1 UNITS Intensity WV IMPORT DATA IDL Wave
14. higher power of two greater than 2N Padding with zeroes prevents wraparound of the Array and speeds up the fast Fourier transform Note Padding with zeroes reduces but does not eliminate edge effects caused by the discontinuities at the start and end of the data SCALE Set this keyword to a named variable in which to return the scale values used for the continuous wavelet transform The SCALE values range from START SCALE up to START SCALE2 NSCALE 1 DSCALE START SCALE Set this keyword to a scalar value giving the starting scale in non dimensional units The default is 2 which gives a starting scale that is twice the spacing between Array elements Reference Torrence and Compo 1998 A Practical Guide to Wavelet Analysis Bull Amer Meteor Soc 79 61 78 Example Assume we have monthly random data n 500 dt 1d 12 time sampling seed 999 data RANDOMN seed n time 1960 dt FINDGEN n Compute the wavelet transform and the power wave WV CWT data Morlet 6 PAD SCALE scales IDL Wavelet Toolkit WV CWT 68 Chapter 4 IDL Wavelet Toolkit Reference wavePower ABS wave 2 Convert scales to time units scales dt Contour visualization ICONTOUR wavePower time scales Y LOG YRANG C_VALUE FINDGEN 7 1 FILL RGB TABLE 39 YTITLE Scale years VIEW TITLE Wavelet Power Insert a legend tool ITGETCURRENT TOOL 0oTool void oTool gt DoAction
15. include e Students introduction to wavelets graphical analysis e Engineers data analysis signal processing data compression e Scientists data analysis filtering and denoising cross wavelet e Computer scientists image compression speed of operations e Mathematicians explore wavelet families test out new analysis techniques Applications Examples of specific applications are What Is the IDL Wavelet Toolkit IDL Wavelet Toolkit Chapter 1 Introduction to the IDL Wavelet Toolkit 7 e Time series analysis time scale power spectrum noise filtering multiresolution analysis e Self similar series fractals long memory processes e Turbulence detection of coherent structures e Signal processing filtering and denoising Image processing edge detection compression enhancement Features The IDL Wavelet Toolkit has the following features Wavelet Applet The Toolkit Applet lets you manage your projects import data and wavelets visualize the results and add your own user tools Continuous Wavelet Transform Allows you to compute the continuous wavelet transform on one dimensional vectors This routine is written in IDL pro code Discrete Wavelet Transform Allows you to compute the discrete wavelet transform partial or full on multi dimensional data These routines are written in C and contained in the IDL wavelet dim Wavelet Functions The Toolkit comes with several wavelet fu
16. look at a magnetic resonance image MRI of the brain and use the Denoising widget tool to filter out unwanted speckles and compress the size of the image Try the following steps 1 In WV_APPLET choose File gt Import Image File and navigate to the examples data directory in the IDL distribution 2 Import the file mr brain dcm The file should contain a 256 x 256 unsigned integer UINT image 3 In the Dataset Viewer change the Title field to MRI Brain Image and the Variable field to Brain 4 Select the Brain dataset and start up the Denoise tool from the Tools Menu You should see the Denoise widget with the threshold set to 10046 and all coefficients retained 5 Setthe ff coeffs threshold to 8192 points You should then see a view similar to that of the following figure Denoise IDL Wavelet Toolkit Chapter 3 Theory and Examples 53 ai Denoise Mr_brain Iof x Original data Filtered data W avelet options Family Daubechies 2 Cumulative power zz t coeffs 8192 Hard threshold C Soft threshold Wavelet coeffs black 0 Coefficient power rms difference 6 44 difference 6 09 E an x z 5 amp an z Results 5 amp Threshold 506 A 4 4n of coeffs 12 5 a E 6 10 100100002 ee Cuellicien Close Ready Figure 3 2 The Denoise Widget for the MRI Brain Scan Notice that you have retained 12 5 of the coefficients and have discarded 87 5
17. or vector of strings giving the names of user defined functions to be included in the WV APPLET Tools menu The actual function names are constructed by removing all white space from each name and attaching a prefix of WV TOOL WAVELETS A scalar string or vector of strings giving the names of user defined wavelet functions to be included in WV APPLET The actual function names are constructed by removing all white space from each name and attaching a prefix of WV EN Examples WV APPLET TOOLS Renormalize My Tool The above statement will start up the Wavelet Toolkit and add the user tools Renormalize and My Tool to the Tools menu When these are selected the actual functions that will be called are WV TOOL RENORMALIZE and WV TOOL MYTOOL Version History 5 3 Introduced See Also WV_CW_WAVELET WV_IMPORT_DATA WV_IMPORT_WAVELET WV_PLOT3D_WPS WV_PLOT_MULTIRES WV_TOOL_DENOISE IDL Wavelet Toolkit WV_APPLET 62 Chapter 4 IDL Wavelet Toolkit Reference WV_CW_WAVELET The WV_CW_WAVELET function is a compound widget that lets the user select and display wavelet functions WV_CW_WAVELET is accessible from the Visualize Menu of WV_APPLET Noe The IDL Wavelet Toolkit must be licensed on your system to be able to use this function Syntax Result WV_CW_WAVELET Parent DISCRETE NO_COLOR NO DRAW WINDOW TITLE string UNAME string UVALUE value VALUE structure W
18. 050 1 Yname STRING Y Yunits STRING pixels Ystart STRING 0 Dy STRING 1 Xoffset LONG 0 0 Xcount LONG 16384 256 Xstride LONG 1 2 Yoffset LONG 0 Ycount LONG 256 Ystride LONG 2 Source STRING wavelet data hello wav wavelet data IEEEFtest tif Notes STRING Voice saying hello IEEE test image Table 2 1 Data fields in the Dataset Viewer Xname This string is the name of the independent variable for the first data dimension X and is used to label the x axis The default is the null string IDL Wavelet Toolkit Dataset Viewer 22 Chapter 2 Using the IDL Wavelet Toolkit Xunits This string gives the units of X The default is the null string Xstart This string gives the value of the first X coordinate The default is 0 Xstart can contain complicated mathematical expressions although the result must be a scalar number Dx This string gives the sampling interval between the X coordinates The default is 1 Dx can contain complicated mathematical expressions although the result must be a scalar number Yname This string is the name of the independent variable for the second data dimension Y and is used to label the y axis for a one dimensional variable this is actually equivalent to the name of the dependent Variable The default is the null string Yunits This string gives the units of Y The default is the null string Ystart This string gives the value of the firs
19. 54 PRUE NY Lu oid damdotcaeiade ae Baie d Rd tdt dade eie Res 55 Chapter 4 IDL Wavelet Toolkit Reference rr nsa 57 list of Commands by Functionality cuc ee Geor rte ee ie ies 58 WY APPLET 2 auentanre e RR ame DEPO IO OR OD P als 60 WY A iiri 62 FE EWP c E 66 WY LENO E 69 LOGEI gum 74 WOW BN COIBLET 2x2idonnceett onde e i OR PE HR a reo Pen 78 WV EN BAUBECHTBS ino ere eH te Renten e ei rete vedo nx denne 80 WY BN DGADSSDUN e a eei eir PE anr certius trente a energia preste NEn eun teu ont 82 Lai c TA c EE 85 WY EN MOERELHT ssa cease asda ti ie te ed ace en ie ecd e ue ince al Yee e 87 WY PN PAVE 90 WY BN OSYMLET 2e EPI th ie e e RA addins RE RE ees 93 WW AIPORT DATA iuuenem trei ete bie tb eee eyes E EEEE 96 WY IMPORT WAYELET 2 2 eienceree toe nm et od tb pi ei i re ine 99 LIN IPSI WPS ERES 101 WY PLOT MULTIBES oeiee sirita na NEE tn tte hemi ada atone AEE EN 104 OW a E E E A T 107 WY TOOL DENISE inedite net Ree ettet ente aee pire dieta 109 e 113 Contents IDL Wavelet Toolkit This chapter discusses the following topics What Is the IDL Wavelet Toolkit 6 IDL Wavelet Toolkit Architecture 9 IDL Wavelet Toolkit 5 6 Chapter 1 Introduction to the IDL Wavelet Toolkit What Is the ID
20. AL is set SPATIAL Set this keyword to return the wavelet function in real space The default is to return the wavelet function in Fourier space WAVELET Set this keyword to a named variable in which to return the wavelet function IDL Wavelet Toolkit WV FN PAUL 92 Chapter 4 IDL Wavelet Toolkit Reference Reference Torrence and Compo 1998 A Practical Guide to Wavelet Analysis Bull Amer Meteor Soc 79 61 78 Examples Plot the Paul wavelet function at scale 100 n 1000 pick a nice number of points info WV FN PAUL 6 100 n SPATIAL WAVELET wavelet plot float wavelet THICK 2 oplot imaginary wavelet Now plot the same wavelet in Fourier space info WV FN PAUL 6 100 n FREQUENCY frequency WAVELET wave fourier plot frequency wave fourier xrange 0 2 0 2 thick 2 Version History 5 4 Introduced See Also WV CWT WV EN GAUSSIAN WV EN MORLET WV FN PAUL IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 93 WV FN SYMLET The WV FEN SYMLET function constructs wavelet coefficients for the Symlet wavelet function Noe The Symlet wavelet for orders 1 3 are the same as the Daubechies wavelets of the same order Syntax Result WV FN SYMLET Order Scaling Wavelet Ioff Joff Return Value The returned value of this function is an anonymous structure of information about the particular wave
21. AVELETS string array Return Value The returned value of this function is the widget ID of the newly created widget Arguments Parent The widget ID of the parent widget Omit this argument to created a top level widget Keywords DISCRETE Set this keyword to include only discrete wavelets in the list of wavelet functions Set this keyword to zero to include only continuous wavelets The default is to include all available wavelets NO_COLOR If this keyword is set the wavelet functions will be drawn in black and white NO DRAW WINDOW If this keyword is set the draw window will not be included within the widget WV CW WAVELET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 63 TITLE Set this keyword equal to a scalar string containing the title of the top level base TITLE is not used if the wavelet widget has a parent widget If it is not specified the default title is Wavelets UNAME Set this keyword to a string that can be used to identify the widget in your code You can associate a name with each widget in a specific hierarchy and then use that name to query the widget hierarchy and get the correct widget ID To query the widget hierarchy use the WIDGET_INFO function with the FIND BY UNAME keyword See WIDGET INFO IDL Reference Guide for more information The UNAME should be unique to the widget hierarchy because the FIND BY UNAME keyword returns the ID of the first widget with the specified nam
22. CONTROL IDL Reference Guide for more information It is not possible to remove a widget from an existing group Note The following keywords are ignored if nput is a filename This includes the SURFACE STYLE TITLE UNITS XTITLE XUNITS YTITLE and YUNITS keywords SURFACE STYLE Set this keyword to an integer specifying the initial style to use for the three dimensional surface Valid values are TITLE 0 Off 1 Points 2 Mesh 3 Shaded 4 XZ lines 5 YZ lines 6 Lego 7 Lego fill A scalar string giving the label to be used for the widget The default is WPS UNITS A scalar string giving the units of Array WV PLOT3D WPS IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 103 XTITLE A scalar string giving the label to be used for the first dimension XUNITS A scalar string giving the units of X YTITLE A scalar string giving the label to be used for the y axis for a 1D vector or for the second dimension for a 2D array YUNITS A scalar string giving the units of Array for a 1D vector or the units of Y for a 2D array Widget Keywords Accepted The WV_PLOT3D_WPS function also accepts the following WIDGET_BASE keywords DISPLAY_NAME EVENT_FUNC FRAME KBRD_FOCUS_EVENTS KILL_NOTIFY MODAL NOTIFY_REALIZE RESOURCE NAME SCR_XSIZE SCR YSIZE SPACE TLB FRAME ATTR TRACKING EVENTS UNITS XOFFSET XSIZE YOFFSET YSIZE See WIDGET BASE IDL Reference Guide for
23. EVELS keyword or WV PWT Construct a random vector n 1024 x randomn s n info WV FN DAUBECHIES 2 wavelet Take the wavelet transform but stop at level 3 WV DWT scaling ioff joff IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 77 wv_dwtpartial WV_DWT x wavelet scaling ioff joff N_LEVELS 3 First level of the pyramid algorithm wv_levell WV_PWT x wavelet scaling ioff joff w_scalingl wv level1 0 n 2 1 Left scaling half w_waveletl wv_levell n 2 Right wavelet half Second level of the pyramid algorithm wv_level2 WV_PWT w_scalingl wavelet scaling ioff joff w scaling2 wv_level2 0 n 4 1 Left scaling half w_wavelet2 wv_level2 n 4 Right wavelet half Third level of the pyramid algorithm wv_level3 WV_PWT w_scaling2 wavelet scaling ioff joff Verify that using WV DWT with N LEVELS 3 is the same as calling WV PWT three times wv partiall123 wv level3 w wavelet2 w waveletl1 print MAX ABS wv dwtpartial wv partiall23 IDL prints 0 000000 Version History 53 Introduced 6 1 Added N LEVELS keyword See Also WV CWT WV PWT WTN IDL Wavelet Toolkit WV DWT 78 Chapter 4 IDL Wavelet Toolkit Reference WV_FN_COIFLET The WV EN COIFLET function constructs wavelet coefficients for the coiflet wavelet function Syntax Result WV FN COIFLET Order Scaling W
24. GET_VALUE value to read the current wavelet To change the current wavelet use the command WIDGET_CONTROL id SET_VALUE value In both cases value is an anonymous structure FAMILY ORDER 0 where FAMILY is a string containing the name for example Daubechies and ORDER is a variable giving the order number Depending on the family ORDER can be of type Integer or Double See Creating a Compound Widget Chapter 2 Widget Application Programming for a more complete discussion of controlling compound widgets using WIDGET_CONTROL and WIDGET_INFO Widget Events Returned by the WV_CW_WAVELET Widget This widget generates event structures each time the family or order is changed The event structure has the following definition Event ID 0L TOP 0L HANDLER 0L FAMILY ORDER 0 The ID field is the widget ID of the WV_CW_WAVELET widget The TOP field is the widget ID of the top level widget HANDLER is the widget ID of the widget handler The FAMILY field contains the family name The ORDER field contains the order number and can be an Integer or a Double depending on the family Version History 5 3 Introduced WV CW WAVELET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 65 See Also WV_FN_COIFLET WV_FN_DAUBECHIES WV_FN_GAUSSIAN WV_FN_HAAR WV_FN_MORLET WV_FN_PAUL WV_FN_SYMLET IDL Wavelet Toolkit WV CW WAVELET 66 Chapter 4
25. IDL Wavelet Toolkit User s Guide IDL Version 7 0 opyright ITT Visual Information Solutions 1107IDL70WV Restricted Rights Notice The IDL IDL Analyst ENVI and ENVI Zoom software programs and the accompanying procedures functions and documentation described herein are sold under license agreement Their use duplication and disclosure are subject to the restrictions stated in the license agreement ITT Visual Information Solutions reserves the right to make changes to this document at any time and without notice Limitation of Warranty ITT Visual Information Solutions makes no warranties either express or implied as to any matter not expressly set forth in the license agreement including without limitation the condition of the software merchantability or fitness for any particular purpose ITT Visual Information Solutions shall not be liable for any direct consequential or other damages suffered by the Licensee or any others resulting from use of the software packages or their documentation Permission to Reproduce this Manual If you are a licensed user of these products ITT Visual Information Solutions grants you a limited nontransferable license to reproduce this particular document provided such copies are for your use only and are not sold or distributed to third parties All such copies must contain the title page and this notice page in their entirety Export Control Information This software and its
26. IDL Wavelet Toolkit Reference WV_CWT The WV_CWT function returns the one dimensional continuous wavelet transform of the input array The transform is done using a user inputted wavelet function Syntax Result WV_CWT Array Family Order DOUBLE DSCALE scalar NSCALE scalar PAD SCALE variable START_SCALE scalar Return Value The result is a two dimensional array of type complex or double complex containing the continuous wavelet transform of the input Array Arguments Array A one dimensional array of length N of floating point or complex type Family A scalar string giving the name of the wavelet function to use for the transform Order The order number or parameter for the wavelet function given by Family Keywords DOUBLE Set this keyword to force the computation to be done in double precision arithmetic DSCALE Set this keyword to a scalar value giving the spacing between scale values in logarithmic units The default is 0 25 which gives four subscales within each major scale WV CWT IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 67 NSCALE Set this keyword to a scalar value giving the total number of scale values to use for the wavelet transform The default is log N START_SCALE DSCALE 1 PAD Set this keyword to force Array to be padded with zeroes before computing the transform Enough zeroes are added to make the total length of Array equal to the next
27. Introduced See Also WV_DWT WV_FN_COIFLET WV_FN_DAUBECHIES WV_FN_SYMLET WV_FN_HAAR IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 87 WV FN MORLET The WV FN MORLET function constructs wavelet coefficients for the Morlet wavelet function In real space the Morlet wavelet function consists of a complex exponential modulated by a Gaussian envelope g iV pe exp i k x s exp x 52 2 where s is the wavelet scale k is a non dimensional parameter and x is the position Syntax Result WV FN MORLET Order Scale N DOUBLE FREQUENCY zvariable SPATIAL WAVELET variable Return Value The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING Morlet ORDER_NAME STRING Parameter ORDER_RANGE DBLARR 3 3 24 6 Valid orders first last default ORDER DOUBLE The chosen Order DISCRETE INT 0 O2continuous 1 discrete ORTHOGONAL INT 0 0 nonorthogonal 1 orthogonal SYMMETRIC INT 1 O asymmetric 12symm SUPPORT DOUBLE Infinity Compact support width MOMENTS INT 1 Number of vanishing moments REGULARITY DOUBLE Infinity Number of continuous derivatives E_FOLDING DOUBLE SQRT 2 Autocorrelation e fold distance FOURIER_PERIOD DOUBLE Ratio of Fourier wavelength to scale Table 4 10 Structure Tags for Result IDL Wavelet Toolkit WV FN
28. L Wavelet Toolkit The IDL Wavelet Toolkit consists of a set of graphical user interfaces GUI and IDL routines for wavelet analysis of multi dimensional data Motivation Wavelet analysis is becoming a popular technique for data and image analysis By decomposing a signal using a particular wavelet function one can construct a picture of the energy within the signal as a function of both spatial dimension or time and wavelet scale or frequency The wavelet transform is used in numerous fields such as geophysics seismic events medicine EKG and medical imaging astronomy image processing and computer science object recognition and image compression The technique is flexible and robust yet it is fast enough to be used in real time image processing A set of standard wavelet techniques have been developed which make it possible for the average user to apply the wavelet method with confidence Recent advances in significance testing and cross wavelet analysis have also enhanced the acceptability of wavelet analysis within the scientific community Nevertheless the calculation of the wavelet transform and the display of the output requires considerable experience Users The IDL Wavelet Toolkit is designed for a wide audience ranging from the casual user who wishes to explore the possibilities of wavelet analysis to the scientist or engineer who wants to produce robust and complex results Potential users and their applications
29. MORLET 88 Chapter 4 IDL Wavelet Toolkit Reference Arguments Order A scalar that specifies the non dimensional order parameter for the wavelet The default is 6 Scale A scalar that specifies the scale at which to construct the wavelet function N An integer that specifies the number of points in the wavelet function For Fourier space SPATIAL 0 the frequencies are constructed following the FFT convention e For N even 0 1 N 2 N N 2 2N 1 2 N 2 QQN 1 N e For N odd 0 1 N 2 N N 1 2N N 1 2N 1 N For real space SPATIAL the spatial coordinates are N 1 2 N 1 2 Noe If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords DOUBLE Set this keyword to force the computation to be done in double precision arithmetic FREQUENCY Set this keyword to a named variable in which to return the frequency array used to construct the wavelet This variable will be undefined if SPATIAL is set SPATIAL Set this keyword to return the wavelet function in real space The default is to return the wavelet function in Fourier space WAVELET Set this keyword to a named variable in which to return the wavelet function WV_FN_MORLET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 89 Reference Torrence and Compo 1998 A Practical Guide to Wavelet Analysis Bull Amer
30. Meteor Soc 79 61 78 Examples Plot the Morlet wavelet function at scale 100 n 1000 pick a nice number of points info WV_FN_MORLET 6 100 n SPATIAL WAVELET wavelet plot float wavelet THICK 2 oplot imaginary wavelet Now plot the same wavelet in Fourier space info WV FN MORLET 6 100 n FREQUENCY frequency WAVELET wave fourier plot frequency wave fourier xrange 0 2 0 2 thick 2 Version History 5 4 Introduced See Also WV_CWT WV_FN_GAUSSIAN WV_FN_PAUL IDL Wavelet Toolkit WV FN MORLET 90 Chapter 4 IDL Wavelet Toolkit Reference WV FN PAUL The WV FN PAUL function constructs wavelet coefficients for the Paul wavelet function In real space the Paul wavelet function is proportional to the complex polynomial 1 i x s m 1 where s is the wavelet scale m is a non dimensional parameter and x is the position Syntax Result WV FN PAUL Order Scale N DOUBLE FREQUENCY variable SPATIAL WAVELET variable Return Value The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING Paul ORDER_NAME STRING Parameter ORDER_RANGE DBLARR 3 1 20 4 Valid orders first last default ORDER DOUBLE The chosen Order DISCRETE INT 0 O2continuous 1 discrete ORTHOGO
31. NAL INT 0 0 nonorthogonal 1 orthogonal SYMMETRIC INT 1 O asymmetric 12symm SUPPORT DOUBLE Infinity Compact support width MOMENTS INT 1 Number of vanishing moments REGULARITY DOUBLE Infinity Number of continuous derivatives E_FOLDING DOUBLE 1 sqrt 2 Autocorrelation e fold distance FOURIER_PERIOD DOUBLE Ratio of Fourier wavelength to scale Table 4 11 Structure Tags for Result WV FN PAUL IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 91 Arguments Order A scalar that specifies the non dimensional order for the wavelet The default is 4 Scale A scalar that specifies the scale at which to construct the wavelet function N An integer that specifies the number of points in the wavelet function For Fourier space SPATIAL 0 the frequencies are constructed following the FFT convention e For N even 0 1 N 2 N N 2 2N 1 2 N 2 QQN 1 N e For N odd 0 1 N 2 N N 1 2N N 1 2N 1 N For real space SPATIAL the spatial coordinates are N 1 2 N 1 2 Nell uA If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords DOUBLE Set this keyword to force the computation to be done in double precision arithmetic FREQUENCY Set this keyword to a named variable in which to return the frequency array used to construct the wavelet This variable will be undefined if SPATI
32. Operations Insert Legend 20 0 25 Gl Version History 5 4 Introduced See Also WV_DWT WV EN GAUSSIAN WV_FN_MORLET WV FN PAUL WV CWT IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 69 WV_DENOISE The WV_DENOISE function uses the wavelet transform to filter or de noise a multi dimensional array WV_DENOISE computes the discrete wavelet transform of Array and then discards wavelet coefficients smaller than a certain threshold WV_DENOISE then computes the inverse wavelet transform on the filtered coefficients and returns the result Syntax Result WV_DENOISE Array Family Order COEFFICIENTS value CUTOFF variable DENOISE_STATE variable DOUBLE DWT_FILTERED variable PERCENT value THRESHOLD value WPS_FILTERED variable Return Value The result is an array of the same dimensions as the input Array If Array is double precision or DOUBLE is set then the result is double precision otherwise the result is single precision Arguments Array A one dimensional array of length N of floating point or complex type Family A scalar string giving the name of the wavelet function to use for the transform WV_DENOISE will construct the actual function name by removing all white space and attaching a prefix of WV_FN_ Note WV_DENOISE may only be used with discrete wavelets such as WV_FN_COIFLET WV_FN_DAUBECHIES
33. Return Value The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING Haar ORDER_NAME STRING Order ORDER_RANGE INTARR 3 1 1 1 Valid order range first last default ORDER INT 1 DISCRETE INT 1 O2continuous 1 discrete ORTHOGONAL INT 1 0 nonorthogonal 1 orthogonal SYMMETRIC INT 0 O asymmetric 1 symm 2 near symm SUPPORT INT 1 Compact support width MOMENTS INT 1 Number of vanishing moments REGULARITY DOUBLE Od Number of continuous derivatives Table 4 9 Structure Tags for Result IDL Wavelet Toolkit WV_FN_HAAR 86 Chapter 4 IDL Wavelet Toolkit Reference Arguments Order A scalar that specifies the order number for the wavelet The default is 1 Scaling On output contains a vector of double precision scaling father coefficients Wavelet On output contains a vector of double precision wavelet mother coefficients loff On output contains an integer that specifies the support offset for Scaling Joff On output contains an integer that specifies the support offset for Wavelet Note _ _ S SS SS _ SS _ SEO ESS If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords None Reference Daubechies I 1992 Ten Lectures on Wavelets SIAM Version History 5 3
34. This is designed so that at least one additional coefficient is either discarded as the slider moves left or retained as the slider moves right These jumps in power correspond to the discrete steps in the coefficient power graph Number of Coefficients You can specify the exact number of coefficients that you wish to retain The cumulative power slider bar will be adjusted accordingly Hard Threshold The hard threshold removes all discarded wavelet coefficients by setting them to zero and computing the inverse wavelet transform For details see Denoise on page 51 IDL Wavelet Toolkit Denoise Tool 44 Chapter 2 Using the IDL Wavelet Toolkit Soft Threshold The soft threshold also sets all discarded wavelet coefficients to zero However it also linearly reduces the magnitude of the each retained wavelet coefficient by an amount equal to the largest discarded coefficient For details see Denoise on page 51 Results Window This text window contains the following output results Threshold The threshold is the actual wavelet power in the variable s units squared that is used for the cutoff value Percent of Coefficients This is the percent number of coefficients used in the reconstruction The smaller the percent coefficients the more efficient the filter RMS Difference This is the root mean square difference between the original data upper left plot and the filtered data upper right plot in the varia
35. Toolkit Chapter 4 IDL Wavelet Toolkit Reference 99 WV_IMPORT_WAVELET The WV_IMPORT_WAVELET procedure allows the user to add wavelet functions to the currently active IDL Wavelet Toolkit s Note Any widgets that are currently active will not have access to the new wavelet functions until they are restarted Note The IDL Wavelet Toolkit must be licensed on your system to be able to use this procedure Syntax WV_IMPORT_WAVELET Wavelet RESET Arguments Wavelet A string scalar or vector giving the names of the wavelet functions The actual function names are constructed by removing all white space from each name and attaching a prefix of WV_FN_ Keywords RESET If set then remove all user defined wavelets from memory If Wavelet is also specified then the new wavelets will be appended onto the built in wavelets Version History 5 3 Introduced IDL Wavelet Toolkit WV IMPORT WAVELET 100 Chapter 4 IDL Wavelet Toolkit Reference See Also WV_APPLET WV_CW_WAVELET WV IMPORT WAVELET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 101 WV PLOT3D WPS The WV PLOT3D WPS function runs the graphical user interface for three dimensional visualization of the wavelet power spectrum WV PLOT3D WPS is accessible from the Visualize Menu of WV APPLET Note __ sssssssssSsSsssSsSSSSSSSSSSSSSSSSSS The IDL Wavelet Toolkit must be licensed on your system to be able to use this fun
36. Y An optional vector of uniformly spaced values giving the location of points along the second dimension of Array The default is 0 1 2 Ny 1 where Ny is the size of the second dimension IDL Wavelet Toolkit WV TOOL DENOISE 110 Chapter 4 IDL Wavelet Toolkit Reference Keywords GROUP_LEADER The widget ID of an existing widget that serves as group leader for the newly created widget When a group leader is killed for any reason all widgets in the group are also destroyed A given widget can be in more than one group The WIDGET_CONTROL procedure can be used to add additional group associations to a widget See WIDGET CONTROL IDL Reference Guide for more information It is not possible to remove a widget from an existing group TITLE A scalar string giving the label to be used for the widget The default is Denoise UNITS A scalar string giving the units of Array XTITLE A scalar string giving the label to be used for the first dimension XUNITS A scalar string giving the units of X YTITLE A scalar string giving the label to be used for the y axis for a 1D vector or for the second dimension for a 2D array YUNITS A scalar string giving the units of Array for a 1D vector or the units of Y fora 2D array Widget Keywords Accepted The WV TOOL DENOISE function also accepts the following WIDGET BASE keywords DISPLAY NAME EVENT FUNC FRAME KBRD FOCUS EVENTS KILL NOTIFY MODAL NOTIFY REALI
37. ZE RESOURCE NAME SCR XSIZE SCR YSIZE SPACE TLB FRAME ATTR WV TOOL DENOISE IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 111 TRACKING_EVENTS UNITS XOFFSET XSIZE YOFFSET YSIZE See WIDGET BASE IDL Reference Guide for more information Version History 53 Introduced See Also WV APPLET WV DENOISE IDL Wavelet Toolkit WV TOOL DENOISE 112 Chapter 4 IDL Wavelet Toolkit Reference WV TOOL DENOISE IDL Wavelet Toolkit A adding Wavelet Toolkit tools 45 B bandpass multiresolution plots 39 C cascade plot See multiresolution analysis coefficient power plot 43 coiflet See wavelet functions compact support 29 compress save files See preferences IDL Wavelet Toolkit confirm exit See preferences continuous wavelet transform 29 48 66 contours in wavelet power spectrum 37 copyrights 2 cumulative power plot 43 D datasets mathematical expressions 23 selecting variables 24 variable information fields 20 Daubechies See wavelet functions default directory See preferences denoising techniques coefficient power plot 43 coefficient threshold 43 113 114 color scaling 42 cumulative power threshold 43 denoise tool 41 hard thresholding 43 51 MRI 52 soft thresholding 44 52 theory 51 wavelet coefficient method 42 WV_DENOISE function 69 WV_TOOL_DENOISE function 109 detail multiresolution plots 39 DIALOG_READ_IMAGE S
38. ake the surface Flat or change to a Gray palette Wavelet Power Spectrum IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 37 Contour Lines You can choose to include contour lines at the top of the plot the bottom or three dimensional Color Contours You can also put color contours at the top bottom or 3D The color contours can be either open or filled The color palette is the same as that used for the surface plot Tip To produce a shaded surface with contours make the surface Shaded set the Gray button and select 3D color contours Significance The statistical significance of each point in the wavelet power spectrum can be plotted as a three dimensional sheet or as contours on the top bottom or 3D Points in the wavelet power spectrum that lie above the sheet or within the contours are said to be significant at the xx level where xx is your chosen percentage You can choose the significance level as 10 5 1 or 0 1 Note The significance level is given by the chi square function with one degree of freedom for real wavelet functions or two degrees of freedom for complex wavelets such as the Morlet This significance is relative to the wavelet power spectrum of a random dataset assuming Gaussian white noise Power Display The graphics window contains the three dimensional image and a color palette If you move the mouse cursor over points in the image the c
39. aling Wavelet Ioff Joff compute coefficients here find support moments and regularity info family Spline order name Order order range 1 5 1 order order discrete 1 orthogonal 1 IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 31 symmetric 0 support support moments moments regularity regularity RETURN info END 2 Save this function ina file wv n spline pro thatis accessible from your current IDL path 3 Now start the Wavelet Toolkit with your new wavelet function WV APPLET WAVELETS Spline Or if you are already running the Wavelet Toolkit WV IMPORT WAVELET Spline Your new wavelet function should appear in the list of current wavelet functions and should be accessible from any of the wavelet tools IDL Wavelet Toolkit Wavelet Viewer 32 Chapter 2 Using the IDL Wavelet Toolkit Wavelet Power Spectrum The wavelet transform converts the data array into a series of wavelet coefficients each of which represents the amplitude of the wavelet function at a particular location within the array and for a particular wavelet scale The Wavelet Power Spectrum viewer shown in the following figure allows you to visualize the wavelet power as a three dimensional surface plot where the height of the surface represents the magnitude of the wavelet coefficients El WPS Sine wave increasing frequency Ioj x File Edit Vie
40. and Dy it is highly recommended that whenever possible you enter mathematical expressions rather than converting to numbers For example in the above table the sampling rate for hello wav is 22050 Hz One could have entered Dx as 0 00004535 rather than 1d0 22050 Nevertheless the latter is not only more accurate limited only by your computer s precision but is also much more informative Note that the 1d0 forces the computation to be done in double precision You may also enter IDL functions in these strings For example if your X coordinate was in Julian days starting from say 29 February 2000 you could set Xstart JULDAY 2 29 2000 IDL Wavelet Toolkit Dataset Viewer 24 Chapter 2 Using the IDL Wavelet Toolkit Selecting Variables To select a particular variable for visualization or some other action click the mouse on any field for that variable or click the mouse on the Table row label to highlight the entire row To select multiple variables for deletion click the mouse on any field and drag down to select the list of variables or click once on the row label scroll down and hold the lt Shift gt key while clicking on the last row label Dataset Viewer IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 25 Importing Data You can import data in multiple file formats into the IDL Wavelet Toolkit ASCIl Files 8 Select this menu item or button to import data from an ASCII text file
41. and G P Compo 1998 A practical guide to wavelet analysis Bull Amer Meteor Soc 79 61 78 IDL Wavelet Toolkit Bibliography 56 Chapter 3 Theory and Examples Bibliography IDL Wavelet Toolkit This reference lists the following topics List of Commands by Functionality 58 WY APPLET Lucus teeta Re 60 WV CW WAWELET eese 62 Dope s PS 66 WV DENOISE ueeeese 69 WW DWE Seien hee AS ERE 74 WV EN COIFPLET seeer nm 78 WV FEN DAUBECHIES 80 WV FEN GAUSSIAN sesss 82 WV FEN HAAR 0 020020 85 IDL Wavelet Toolkit WV EN OMORLET 2eee em 87 WV EN PAUL be eek tak SR ERE 90 v BEN SYMLBT 12r ed Le 93 WV IMPORT DATA 96 WV IMPORT WAVELET 99 WV PLOT3D WPS eee cer ces 101 WV PLOT MULTIRES 104 LR d C CP 107 WV TOOL DENOISE 109 57 58 Chapter 4 IDL Wavelet Toolkit Reference List of Commands by Functionality Widget Commands and Visualization Tools The following table describes the widget and visualization tools Command Description WV_APPLET Run IDL Wavelet Too interface Ikit GUI graphical user WV CW WAVELET Compound widget to display and select wavelets WV IMPORT DATA Import data from the 1 DL gt command prompt WV_IMPORT_WAVELET Import wavelet functions into the current applet WV_PLOT3D_WPS Run the wavelet power spectrum GUI
42. associated documentation are subject to the controls of the Export Administration Regulations EAR It has been determined that this software is classified as EAR99 under U S Export Control laws and regulations and may not be re transferred to any destination expressly prohibited by U S laws and regulations The recipient is responsible for ensuring compliance to all applicable U S Export Control laws and regulations Acknowledgments ENVI and IDL are registered trademarks of ITT Corporation registered in the United States Patent and Trademark Office ION ION Script ION Java and ENVI Zoom are trademarks of ITT Visual Information Solutions Numerical Recipes is a trademark of Numerical Recipes Software Numerical Recipes routines are used by permission GRG2 is a trademark of Windward Technologies Inc The GRG2 software for nonlinear optimization is used by permission NCSA Hierarchical Data Format HDF Software Library and Utilities Copyright 1988 2001 The Board of Trustees of the University of Illinois All rights reserved NCSA HDFS Hierarchical Data Format 5 Software Library and Utilities Copyright 1998 2002 by the Board of Trustees of the University of Illinois All rights reserved CDF Library Copyright O 2002 National Space Science Data Center NASA Goddard Space Flight Center NetCDF Library Copyright 1993 1999 University Corporation for Atmospheric Research Unidata HDF EOS Library Copyrigh
43. aussian ORDER_NAME STRING Derivative ORDER_RANGE DBLARR 3 Valid orders first last default ORDER DOUBLE The chosen Order DISCRETE INT 0 02continuous 1 discrete ORTHOGONAL INT 0 0 nonorthogonal 1 orthogonal SYMMETRIC INT 1 0O asymmetric 1 symm SUPPORT DOUBLE Infinity Compact support width MOMENTS INT 1 Number of vanishing moments REGULARITY DOUBLE Infinity Number of continuous derivatives E_FOLDING DOUBLE SQRT 2 Autocorrelation e fold distance FOURIER_PERIOD DOUBLE Ratio of Fourier wavelength to scale Table 4 8 Structure Tags for Result WV FN GAUSSIAN IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 83 Arguments Order A scalar that specifies the non dimensional order parameter for the wavelet The default is 2 Scale A scalar that specifies the scale at which to construct the wavelet function N An integer that specifies the number of points in the wavelet function For Fourier space SPATIAL 0 the frequencies are constructed following the FFT convention For N even 0 1 N 2 N N 2 2N 1 2 N 2 QN 1 N e For N odd 0 1 N 2 N N 1 2N N 1 2N 1 N For real space SPATIAL the spatial coordinates are N 1 2 N 1 2 Noe If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords DOUBLE Set this keyword to force th
44. avelet loff Joff Return Value The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING Coiflet ORDER NAME STRING Order ORDER RANGE INTARR 3 1 5 1 Valid order range first last default ORDER INT The chosen Order DISCRETE INT O2continuous 1 discrete ORTHOGONAL INT 1 0 nonorthogonal 1 orthogonal SYMMETRIC INT 2 O asymmetric 1 symm 2 near symm SUPPORT INT 6 Order 1 Compact support width MOMENTS INT 2 Order Number of vanishing moments REGULARITY DOUBLE The number of continuous derivatives Table 4 6 Structure Tags for Result Arguments Order A scalar that specifies the order number for the wavelet The default is 1 WV FN COIFLET IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 79 Scaling On output contains a vector of double precision scaling father coefficients Wavelet On output contains a vector of double precision wavelet mother coefficients loff On output contains an integer that specifies the support offset for Scaling Joff On output contains an integer that specifies the support offset for Wavelet Note If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords None Reference Coefficients are from Daubechies I 1992 Ten Lectures on Wavelet
45. avelet Power Spectrum IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 35 Wavelet Options You can change the current wavelet family or the order The plot will be automatically updated Noe For two dimensional input data only the discrete wavelet functions are available View Options 3D Turn this button off to rotate the image so it appears flat Turn this button on to rotate the image to a three dimensional perspective For vector data this button also controls whether the data series and global wavelet plot are flat or vertical Note The surface will remain three dimensional only the viewpoint is changed Color Bar Turn this button off to remove the color bar at the bottom Turn this button on to restore the color bar Data Plot One dimensional only Turn this button off to remove the data series plot at the back Turn this button on to restore the plot Global One dimensional only Turn this button off to remove the plot of the global wavelet Turn the button on to restore the plot Zero Phase Lines Complex wavelet functions only Turn this button on to add the zero wavelet phase lines to the surface plot Energy Scaling These buttons control the scaling of the wavelet magnitude in the Z direction IDL Wavelet Toolkit Wavelet Power Spectrum 36 Chapter 2 Using the IDL Wavelet Toolkit Power The power is the absolute value squared of the wavelet coefficients The height of each
46. avelet coefficients Each wavelet coefficient represents the closeness of the fit or correlation between the wavelet function at a particular size and a particular location within the data array By varying the size of the wavelet function usually in powers of two and shifting the wavelet so it covers the entire array you can build up a picture of the overall match between the wavelet function and your data array Since the wavelet functions are compact hence the term wave let the wavelet coefficients only measure the variations around a small region of the data array This property makes wavelet analysis very useful for signal or image processing the localized nature of the wavelet transform allows you to easily pick out features in your data such as spikes for example noise or discontinuities discrete objects in for example astronomical images or satellite photos edges of objects etc The localization also implies that a wavelet coefficient at one location is not affected by the coefficients at another location in the data This makes it possible to remove noise of all different scales from a signal simply by discarding the lowest wavelet coefficients For a general introduction to the wavelet transform and its applications see Hubbard 1998 Method The IDL Wavelet Toolkit uses the continuous and discrete wavelet transforms Details on the discrete wavelet transform can be found in Daubechies 1992 and Malla
47. ble units A smaller number implies a more accurate reconstruction Percent Difference This is the percent difference between the original and filtered data and is equal to 100 x RMS difference StdDev where StdDev is the standard deviation of the original data The smaller the percent difference the more accurate the reconstruction Function Call The text under Function Call contains the actual IDL code used to call the WV DENOISE function See WV_DENOISE on page 69 to copy this code into your own programs to call the denoise function directly Denoise Tool IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 45 Adding User Tools You can extend the capabilities of the IDL Wavelet Toolkit by adding your own user defined tool functions These wavelet functions should follow the same calling mechanism as the built in tool functions such as WV TOOL DENOISFE on page 109 In addition your tool function should begin with the prefix wv_tool_ 1 Let s say you want to add a wavelet tool called Edge Detect that uses the wavelet transform to detect edges in images To do this first create a tool function that accepts a data array and possibly other variable parameters FUNCTION wv tool edgedetect Array 1D vector or 2D array iX X coordinates of array Y Y coordinates of array GROUP LEADER group leader TITLE title UNITS units XTITLE xtitle XUNITS xunits YTITLE ytitle YUNITS
48. contains a vector of double precision wavelet mother coefficients loff On output contains an integer that specifies the support offset for Scaling Joff On output contains an integer that specifies the support offset for Wavelet Note If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords None Reference Coefficients for orders 1 10 are from Daubechies I 1992 Ten Lectures on Wavelets SIAM p 195 Note that Daubechies has multiplied by Sqrt 2 Coefficients for orders 11 24 are from http www isds duke edu brani filters html Version History 5 3 Introduced See Also WV DWT WV FN COIFLET WV_FN_HAAR WV_FN_SYMLET IDL Wavelet Toolkit WV FN DAUBECHIES 82 Chapter 4 IDL Wavelet Toolkit Reference WV_FN_GAUSSIAN The WV_FN_GAUSSIAN function constructs wavelet coefficients for the Gaussian wavelet function In real space the Gaussian wavelet function is proportional to the m th order derivative of a Gaussian exp x2 2 The Gaussian second derivative x2 1 exp x2 2 is often referred to as the Marr wavelet Syntax Result WV FN GAUSSIAN Order Scale N DOUBLE FREQUENCY variable SPATIAL WAVELET variable The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING G
49. ction Syntax Result WV PLOT3D WPS Array X Y GROUP LEADER widget id SURFACE STYLE zvalue TITLE string UNITS string XTITLE string XUNITS string YTITLE string YUNITS string Return Value The returned variable is the Widget ID of the newly created widget Arguments Input Input must be either a string giving the name of a file to open or a one or two dimensional array of data If set to a string the file must contain a valid WV PLOT3D WPS saved state X An optional vector of uniformly spaced values giving the location of points along the first dimension of Input The default is 0 1 2 Ny 1 where Ny is the size of the first dimension This argument is ignored if nput is a filename Y An optional vector of uniformly spaced values giving the location of points along the second dimension of Input The default is 0 1 2 Ny 1 where Ny is the size of the second dimension This argument is ignored if Input is a filename IDL Wavelet Toolkit WV PLOT3D WPS 102 Chapter 4 IDL Wavelet Toolkit Reference Keywords GROUP_LEADER The widget ID of an existing widget that serves as group leader for the newly created widget When a group leader is killed for any reason all widgets in the group are also destroyed A given widget can be in more than one group The WIDGET_CONTROL procedure can be used to add additional group associations to a widget See WIDGET
50. d in Dataset Viewer on page 20 Status Bar The Status Bar displays descriptions of the Toolbar buttons and the status of various actions such as Open Import and Save The Status Bar also provides warnings if for example you select Visualize without selecting a variable Starting the Toolkit IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 13 gl Wavelet wv sample sav CE x ajae eae x wes v Chip 2 Convection m LNEZE BYTE 248 248 Convection X convec dat Sine wave increasing exponer Earth mantle convection Figure 2 1 The main Wavelet Toolkit window IDL Wavelet Toolkit Starting the Toolkit 14 Chapter 2 Using the IDL Wavelet Toolkit Menu Description The main window has five items File Menu Edit Menu Visualize Menu Tools Menu and Help Menu Each menu and its submenus is described below File Menu The File menu accesses and manipulates files New Applet This menu item or button starts a new Wavelet Toolkit applet with an empty dataset pes Open Dataset This menu item or button closes the current dataset and allows you to open up a different dataset Wavelet datasets have the default filename suffix sav and are written in IDL SAVE format For more information see SAVE IDL Reference Guide If the previous dataset has not been saved then you will be prompted to save the previous datas
51. duction to the IDL Wavelet Toolkit 9 IDL Wavelet Toolkit Architecture File Organization The Toolkit consists of the following components e Source pro files in the wavelet directory e A bitmaps subdirectory with button bitmaps e The data subdirectory with sample data files The Online help manual in the IDL help directory The DLM Dynamically Loadable Module in the IDL bin directory Note You are encouraged to view the source files for details on implementation and technique You are also welcome to modify the source files however it is strongly encouraged that you copy the files to your own directory first By modifying the IDL PATH variable you can ensure that your routines are compiled first See PATR IDL Reference Guide for more information Structure The IDL Wavelet Toolkit consists of three layers The topmost layer is the Wavelet Applet which allows you to import data and wavelet functions and access various visualization and tool routines The middle layer is the set of compound widgets and widget tools for visualization and analysis These tools are accessible both from the Wavelet Applet and from your own routines The lowest layer are the wavelet API application programming interface that consist of the wavelet functions the wavelet transform and the import data routine IDL Wavelet Toolkit IDL Wavelet Toolkit Architecture 10 Chapter 1 Introduction to the IDL Wavelet Toolkit IDL Wavelet Toolkit
52. e UVALUE Set this keyword equal to the user value associated with the widget VALUE Set this keyword to an anonymous structure of the form FAMILY ORDER 0d representing the initial value for the widget WAVELETS A scalar string or vector of strings giving the names of user defined wavelet functions to be included in WV CW WAVELET The actual function names are constructed by removing all white space from each name and attaching a prefix of WV FEN Widget Keywords Accepted The WV CW WAVELET function also accepts the following WIDGET BASE keywords ALIGN BOTTOM ALIGN CENTER ALIGN LEFT ALIGN RIGHT ALIGN TOP DISPLAY NAME FRAME GROUP LEADER KBRD FOCUS EVENTS MAP NOTIFY REALIZE RESOURCE NAME SCR XSIZE SCR YSIZE SPACE TLB FRAME ATTR TRACKING EVENTS UNITS XOFFSET XSIZE YOFFSET YSIZE See WIDGET BASE IDL Reference Guide for more information IDL Wavelet Toolkit WV CW WAVELET 64 Chapter 4 IDL Wavelet Toolkit Reference Keywords to WIDGET_CONTROL and WIDGET_INFO The widget ID returned by most compound widgets is actually the ID of the compound widget s base widget This means that many keywords to the WIDGET_CONTROL and WIDGET_INFO routines that affect or return information on base widgets can be used with compound widgets In addition you can use the GET_VALUE and SET_VALUE keywords to WIDGET_CONTROL to obtain or set the value of the wavelet Use the command WIDGET_CONTROL id
53. e computation to be done in double precision arithmetic FREQUENCY Set this keyword to a named variable in which to return the frequency array used to construct the wavelet This variable will be undefined if SPATIAL is set SPATIAL Set this keyword to return the wavelet function in real space The default is to return the wavelet function in Fourier space WAVELET Set this keyword to a named variable in which to return the wavelet function IDL Wavelet Toolkit WV_FN_GAUSSIAN 84 Chapter 4 IDL Wavelet Toolkit Reference Reference Torrence and Compo 1998 A Practical Guide to Wavelet Analysis Bull Amer Meteor Soc 79 61 78 Examples Plot the Gaussian wavelet function at scale 20 n 1000 pick a nice number of points info WV FN GAUSSIAN 2 20 n SPATIAL WAVELET wavelet plot wavelet Now plot the same wavelet in Fourier space info WV FN GAUSSIAN 2 20 n FREQUENCY frequency WAVELET wave fourier plot frequency wave fourier xrange 0 2 0 2 thick 2 Version History 54 Introduced See Also WV CWT WV FN MORLET WV FN PAUL WV FN GAUSSIAN IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 85 WV_FN_HAAR The WV_FN_HAAR function constructs wavelet coefficients for the Haar wavelet function Note The Haar wavelet is the same as the Daubechies wavelet of order 1 Syntax Result WV_FN_HAAR Order Scaling Wavelet loff Joff
54. ee importing discrete wavelet transform 29 48 74 drag quality 34 E e folding time 30 energy scaling 35 G Gaussian See also wavelet functions H Haar See wavelet functions high pass multiresolution plots 39 l IDL Wavelet Toolkit main window 12 menus 14 status bar 12 toolbar 12 image compression See denoising techniques images Index in wavelet toolkit 26 importing adding wavelet functions 99 data from command line 96 IDL command line data 26 structure tags 97 user defined wavelet functions 30 WAV audio files 26 L legalities 2 lego style surface 36 localization of wavelet functions 48 low pass multiresolution plots 39 M Macintosh using mouse with wavelet toolkit 37 Mallat See pyramidal algorithm Marr wavelet See WV_FN_GAUSSIAN func tion mathematical expressions See datasets Morlet See wavelet functions MRI denoising technique 52 multiresolution analysis 104 about 54 using 39 N noise removal See also denoising techniques nonorthogonal wavelet functions 29 O orthogonal wavelet functions 29 IDL Wavelet Toolkit P parameters passing to Wavelet Toolkit functions 45 partial wavelet transform See wavelet trans form passing parameters 45 Paul See wavelet functions PCM format 26 percent difference 44 plotting multiresolution analysis 39 104 wavelet power spectrum 32 101 preferences compress save files 18 confirm exit 18 cu
55. ence IMSL is a trademark of Visual Numerics Inc Copyright O 1970 2006 by Visual Numerics Inc All Rights Reserved Other trademarks and registered trademarks are the property of the respective trademark holders PAR BS AL Chapter 1 Introduction to the IDL Wavelet Toolkit 5 What Ts The IDL Wavelet Fool 2 u onec sues ertet tet rhetor oer 6 IDL Wavelet Toolkit Architecture i e merit reete torte ferre eere dn 9 Chapter 2 Using the IDL Wavelet Toolkit Linien eint rptu iud ani uiu muk mark dne 11 rata Nc E 12 Memi MOSS CHILO ED M 14 PBN Seas oy coca A dsc E ped taeda dee teaah E 18 Dalaset VISWB aietenidtenetei reor Ir OR PH PP AER EDU calepsecegusastancaspacersteeaaees 20 Importing Data 12 1 RR Mte Ee E M eas 25 Di 1 EE 28 Wavelet Power DESDE 1er ter erret een rece t gebe e re ore eor e ded EN DE R Nen 32 Multiresolution Analysis 1 eerie rene eene te eter de ee coe rana aiii 39 Derose Toal M 41 IDL Wavelet Toolkit 3 Addins i dn 45 Chapter 3 Theory and AIS siisii annsna CHEVER CHF MED EA Id RD iaasa 47 It E o a es 48 Wavelet Power Sperry ERE 49 TICS 21er eerie iere levee cdc cused ta ENTE E UR Te que eo a Ure e Tel cath i en e E eu ue ue Ue ERE cia daouendages 51 Mul resolunon SOULS em Loiret rettet ete teet e PUT eterne e d eca coco pd opera
56. et first Le Save Select this menu item or button to save the current dataset and preferences If the dataset has not yet been saved then you are prompted for a filename with the Save As dialog Save As This menu item allows you to choose a new filename for the current dataset using the Save As dialog and then saves the dataset to this file Import Select this menu item to import an external data file into the current dataset Details on allowable file formats and import options can be found in Dataset Viewer on page 20 You can also import data from the current IDL session using the WV IMPORT DATA procedure Preferences This menu item opens up a Preferences dialog in which you can customize your interaction with the Wavelet Toolkit The Default button restores the built in default Menu Description IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 15 options for all of the preferences The OK button keeps all of the changes to Preferences The Cancel button discards all of the changes Exit This menu item will close the current Wavelet Toolkit applet Other Wavelet applets either started from the command line or via the New Applet menu item are unaffected If you have made changes to the current dataset then you will be prompted to save the dataset before exiting Edit Menu The Edit Menu manipulates the Dataset Viewer duu Move Variable Left Select this menu item or button to mo
57. gets in the group are also destroyed A given widget can be in more than one group The WIDGET_CONTROL procedure can be used to add additional group associations to a widget See WIDGET CONTROL IDL Reference Guide for more information It is not possible to remove a widget from an existing group TITLE A scalar string giving the label to be used for the widget The default is MRes UNITS A scalar string giving the units of Array XTITLE A scalar string giving the label to be used for the first dimension XUNITS A scalar string giving the units of X YTITLE A scalar string giving the label to be used for the y axis for a 1D vector or for the second dimension for a 2D array YUNITS A scalar string giving the units of Array for a 1D vector or the units of Y fora 2D array Widget Keywords Accepted The WV PLOT MULTIRES function also accepts the following WIDGET BASE keywords DISPLAY NAME EVENT FUNC FRAME KBRD FOCUS EVENTS KILL NOTIFY MODAL NOTIFY REALIZE RESOURCE NAME SCR XSIZE SCR YSIZE SPACE TLB FRAME ATTR IDL Wavelet Toolkit WV PLOT MULTIRES 106 Chapter 4 IDL Wavelet Toolkit Reference TRACKING_EVENTS UNITS XOFFSET XSIZE YOFFSET YSIZE See WIDGET BASE IDL Reference Guide for more information Version History 5 3 Introduced See Also WV APPLET WV PLOT MULTIRES IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 107 WV_PWT The WV PWT function ret
58. idal algorithm For a one dimensional vector with 16 elements one level of the pyramid appears below Array elements 0 L 25 3 4 55 6 7 8 9 10 11 12 13 14 15 Xu No Y E s0 d0 si d1 s2 d2 s3 d3 s4 d4 s5 d5 s6 d6 s7 d7 where Si and Di are the scaling and wavelet coefficients and i represents the position The wavelet coefficients are stored in Result in the following order Result s0 sl s2 s3 s4 s5 s6 s7 dO dil d2 d3 d4 d5 d6 da7 Version History 5 3 Introduced See Also WV_DWT WV_PWT IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 109 WV TOOL DENOISE The WV TOOL DENOISE function runs the graphical user interface for wavelet filtering and denoising WV TOOL DENOISE is accessible from the Tools Menu of WV APPLET Note The IDL Wavelet Toolkit must be licensed on your system to be able to use this function Syntax Result WV TOOL DENOISE Array X Y GROUP LEADER widget id TITLE string UNITS string XTITLE string XUNITS string YTITLE string YUNITS string Return Value The returned variable is the Widget ID of the newly created widget Arguments Array A one or two dimensional array of data to be analyzed X An optional vector of uniformly spaced values giving the location of points along the first dimension of Array The default is 0 1 2 Nx 1 where Ny is the size of the first dimension
59. let Tag Type Definition FAMILY STRING Symlet ORDER_NAME STRING Order ORDER_RANGE INTARR 3 1 15 4 Valid order range first last default ORDER INT The chosen Order DISCRETE INT 1 O2continuous 1 discrete ORTHOGONAL INT 1 Oz2nonorthogonal 1 orthogonal SYMMETRIC INT 2 O asymmetric 12symm 2 near symm SUPPORT INT 2 Order 1 Compact support width MOMENTS INT Order Number of vanishing moments REGULARITY DOUBLE The number of continuous derivatives Table 4 12 Structure Tags for Result IDL Wavelet Toolkit WV_FN_SYMLET 94 Chapter 4 IDL Wavelet Toolkit Reference Arguments Order A scalar that specifies the order number for the wavelet The default is 4 Scaling On output contains a vector of double precision scaling father coefficients Wavelet On output contains a vector of double precision wavelet mother coefficients loff On output contains an integer that specifies the support offset for Scaling Joff On output contains an integer that specifies the support offset for Wavelet Note SS _ SEO ESS If none of the above arguments are present then the function will simply return the Result structure using the default Order Keywords None Reference Coefficients for orders 1 10 are from Daubechies I 1992 Ten Lectures on Wavelets SIAM p 198 Note that Daubechies
60. let Toolkit Chapter 4 IDL Wavelet Toolkit Reference 97 If Data is a structure then it must include at the very least a DATA tag containing a pointer to a one or two dimensional array Recognized tags are shown in the following table Tags other than these will be quietly ignored Tag Type Definition DATA PTR gt Array Pointer to data array TITLE STRING Long name of data variable VARIABLE STRING Short name of data variable UNITS STRING Units for variable XNAME STRING Name of X coordinate XUNITS STRING Units for X coordinate XSTART STRING Starting value for X coord DX STRING Sampling rate for X coord YNAME STRING Name of Y coordinate YUNITS STRING Units for Y coordinate YSTART STRING Starting value for Y coord DY STRING Sampling rate for Y coord XOFFSET LONG Starting index of X coord to use XCOUNT LONG Number of X coords to use XSTRIDE LONG X sampling interval to use YOFFSET LONG Starting index of Y coord to use YCOUNT LONG Number of Y coords to use YSTRIDE LONG Y sampling interval to use SOURCE STRING Filename or contact info NOTES STRING Miscellaneous notes COLORS PTR gt Bytarr 3 256 Pointer to color table for Data Table 4 13 Tags Recognized by WV_IMPORT_DATA IDL Wavelet Toolkit WV IMPORT DATA 98 Version History Chapter 4 IDL Wavelet Toolkit Reference 5 3 Introduced See Also WV_APPLET WV IMPORT DATA IDL Wavelet
61. lly calculate the X and Y stride values by dividing the length of vector arrays by the Vector stride factor and each dimension of two dimensional arrays by the Array stride factor After the data is imported you may change the X and Y stride values on the Dataset Viewer The minimum stride factor is 2 Preferences IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 19 Tip AJ To force the stride values to always be set to 1 when importing data set the stride factors to a value larger than the maximum dimension for your data Defaults Button Press this button to restore all of the preferences to their default settings IDL Wavelet Toolkit Preferences 20 Chapter 2 Using the IDL Wavelet Toolkit Dataset Viewer Your dataset can consist of several different variables each with a different data format The Dataset Viewer located in the middle of the Wavelet Toolkit applet allows you to organize and manipulate your dataset The variables are assigned a number and a name derived from the Variable name You can sort the variables using the Move Variable Left and Move Variable Right buttons Variable Information Each variable contains a one dimensional vector or two dimensional array of data values The data values can be of any numeric type such as BYTE INTEGER FLOAT etc The variable also has several descriptor fields which you can modify described below and summarized in the table below To modif
62. lts Threshold 783 of coeffs 10 2 rms difference 4 19 Scale power pf 2 e Power 55 a difference 4 84 Cumulative power 5c O 100 200 200 400 add 1 10 100 lirne sec Coellicien Close Figure 2 5 The Denoise Tool IDL Wavelet Toolkit Denoise Tool 42 Chapter 2 Using the IDL Wavelet Toolkit File Menu Open State This menu item opens a previously saved state file into a new window Save State This menu item saves the current state of the Denoise Tool into a file Close This menu item closes the Denoise Tool viewer Original Data This window displays a graph of the original one dimensional vector or two dimensional image For images all values are converted to an intensity 0 255 and a grayscale color palette is used Filtered Data This window displays the data after filtering using the wavelet function and options given on the right For images all values are converted to an intensity 0 255 and a grayscale color palette is used Wavelet Coefficients The filtered coefficients are displayed as a two dimensional image using a logarithmic energy scaling The method is as follows 1 Find the maximum value P of the original unfiltered wavelet power absolute value squared of the wavelet coefficients 2 Square the filtered wavelet coefficients to get wavelet power then take the base 10 logarithm of each 3 Scale this logarithmic power from the range 10 Logj0 P
63. more information Version History 5 3 Introduced See Also WV_APPLET IDL Wavelet Toolkit WV PLOT3D WPS 104 Chapter 4 IDL Wavelet Toolkit Reference WV PLOT MULTIRES The WV PLOT MULTIRES function runs the graphical user interface for multiresolution analysis WV PLOT MULTIRES is accessible from the Visualize Menu of WV APPLET Noe The IDL Wavelet Toolkit must be licensed on your system to be able to use this function Syntax Result WV_PLOT_MULTIRES Array X Y GROUP_LEADER widget_id TITLE string UNITS string XTITLE string XUNITS string YTITLE string YUNITS string Return Value The returned variable is the Widget ID of the newly created widget Arguments Array A one or two dimensional array of data to be analyzed X An optional vector of uniformly spaced values giving the location of points along the first dimension of Array The default is 0 1 2 Nx 1 where Ny is the size of the first dimension Y An optional vector of uniformly spaced values giving the location of points along the second dimension of Array The default is 0 1 2 Ny 1 where Ny is the size of the second dimension WV PLOT MULTIRES IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 105 Keywords GROUP_LEADER The widget ID of an existing widget that serves as group leader for the newly created widget When a group leader is killed for any reason all wid
64. n analysis of the currently selected variable You can also start the viewer using the WV PLOT MULTIRES function from the IDL command prompt The Multiresolution viewer is described in Multiresolution Analysis on page 39 Tools Menu The Tools Menu contains built in and user defined tools Denoise This menu item starts the widget for denoising filtering and compression of the currently selected variable You can also start the viewer from the IDL gt command prompt by using the WV TOOL DENOISE function The Denoise tool is described in Denoise Tool on page 41 Other user tools If you have added other tools then they will be displayed here The currently selected variable will be passed to the tool function See Adding User Tools on page 45 Help Menu The Help Menu provides various help functions IDL Help This menu item will start up the IDL Online Help manual T IDL Wavelet Toolkit Help This menu item or button will start up the online help manual for the IDL Wavelet Toolkit Menu Description IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 17 Wavelet Readme This menu item will display the Readme file included with the Toolkit Wavelet Release Notes This menu item will display the Release Notes file included with the Toolkit About IDL Wavelet Toolkit Select this menu item to display information about the current version of IDL and the IDL Wavelet Toolkit IDL Wavelet Toolkit Men
65. nctions that are accessible both inside the Applet and from your own programs You can easily add your own wavelet functions to the Toolkit 3D Wavelet Power Spectrum Callable from within the Applet and from your own programs the visualizer plots the wavelet power as a three dimensional surface with optional contour lines You can rotate translate and find the power at a particular location Multiresolution Analysis Stand alone or callable from the Applet this routine produces plots for the smooth low pass detail band pass and rough high pass components of your data IDL Wavelet Toolkit What Is the IDL Wavelet Toolkit 8 Chapter 1 Introduction to the IDL Wavelet Toolkit Denoise Tool This widget tool enables you to denoise your vector or image array by thresholding hard or soft either by cumulative power or coefficient number Dataset Viewer Manage the datasets within each project by importing new data viewing data values and customizing the data fields Import Data You can import data from a variety of file formats ASCII binary image BMP JPEG PNG PPM SRF TIFF DICOM and WAV audio Image files can be either indexed color 8 or 16 bit or TrueColor 24 bit You can also import data directly from the IDL gt command prompt User Tools You can extend the functionality of the IDL Wavelet Toolkit by adding your own tools What Is the IDL Wavelet Toolkit IDL Wavelet Toolkit Chapter 1 Intro
66. nslate the complicated 3D geometry produced by IDL object graphics into equivalent VRML code Print This menu item will output the image to a printer Close This menu item closes the Wavelet Power Spectrum viewer IDL Wavelet Toolkit Wavelet Power Spectrum 34 Chapter 2 Using the IDL Wavelet Toolkit Edit Menu Undo This menu item will undo the previous rotation scaling or translation of the model Copy To Clipboard This menu item makes a copy of the current graphics image and places it on the system clipboard View Menu Color Table Selecting this item brings up the XLOADCT color table editor You can then choose different color tables for the graphics image See XLOADCT IDL Reference Guide for more information Drag Quality This submenu has three different settings that affect the drawing speed during object manipulations e Low only the axes are exposed for graphics manipulation such as rotation and translation e Medium low resolution graphics are used for graphics manipulation e High full resolution is used for all graphics manipulations Wavelet Options If you select this menu item the Wavelet Options panel will be hidden Select this menu item again to show the panel View Options If you select this menu item the View Options panel will be hidden Select this menu item again to show the panel Help Menu This menu contains Help items for the Wavelet Power Spectrum and for IDL W
67. o zero 1 Soft threshold The soft threshold sets all DWT i with magnitude less than T to zero and also linearly reduces the magnitude of the each retained wavelet coefficient by T Positive coefficients are set equal to DWT i T while negative coefficients are set equal to DWT i T Table 4 5 THRESHOLD Values WPS FILTERED Set this keyword to a named variable in which the filtered wavelet power spectrum will be returned Examples Remove the noise from a 128 x 128 image image dist 128 5 randomn 1 128 128 Keep only 100 out of 16384 coefficients denoise WV DENOISE image Daubechies 2 COEFF 100 DENOISE STATE denoise state window xsize 256 ysize 155 tvscl image 0 tvscl denoise 1 xyouts 64 196 5 5 Image Filtered device align 0 5 charsize 2 print Percent of power retained denoise state percent IDL prints Percent of power retained 93 151491 Change to a soft threshold use DENOISE STATE to avoid re computing denoise2 WV DENOISE image COEFF 100 DENOISE STATE denoise state THRESHOLD 1 WV DENOISE IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 73 Image Filtered Figure 4 1 Example of De Noising an Image Version History 5 4 Introduced See Also WV_DWT WV_TOOL_DENOISE IDL Wavelet Toolkit WV DENOISE 74 Chapter 4 IDL Wavelet Toolkit Reference
68. pecifies the support offset for Wavelet To center the wavelet function over each point in Array set Joff to N 242 Keywords DOUBLE Set this keyword to force the computation to be done in double precision arithmetic INVERSE If set the inverse transform is computed By default the forward transform is computed N LEVELS Set this keyword to the number of wavelet levels to compute in the pyramid algorithm starting with the smallest wavelet scale and progressing to larger scales If this keyword is not set or is set to zero then all wavelet levels in the pyramid algorithm are computed Method and Result Format The WV DWT function computes the wavelet coefficients using the pyramidal algorithm Mallat 1989 One Dimensional Vector For a one dimensional vector the pyramid appears below Array elements 0 1 2 3 4 5 T7 8 29 10 11 12 13 14 15 Nd Nu Nod Yy Nog Ne s0 d0 1 d1 s2 d2 3 d3 s4 d4 s5 d5 s6 d6 s7 d7 N N N N Nick sd D0 S1 D1 S2 D2 S3 D3 IDL Wavelet Toolkit WV_DWT 76 coefficient Si and wavelet coefficient 1 letters s S S and Result do dl So S1 So Do d2 d3 d4 Two Dimensional Array Do d5 Di d6 Chapter 4 IDL Wavelet Toolkit Reference DO D1 a7 D2 Nu 1 D1 At each level of the hierarchy the WV_PWT function is used to compute the scaling Di where i repre
69. point measures the contribution to the total energy This scaling emphasizes large peaks and sharp discontinuities and de emphasizes low amplitude background noise Magnitude The magnitude is the absolute value of the wavelet coefficients and provides a measure of the relative amplitude of each point This scaling reduces the weighting given to large peaks and can bring out finer detail features Decibels The power can also be displayed in decibels normalized relative to the mean of the wavelet power spectrum Since decibels are a logarithmic scale the smallest wavelet coefficients are given just as much weight as the largest coefficients This scaling is most useful for data that contain a broad range of energy or that contain a single sharp spike embedded in small amplitude noise db Cutoff You can specify the lower cutoff for the Decibel plot The default is 50 db Surface Style There are seven different surface plots from which to choose e Points places colored dots at each location height e Mesh creates an unfilled surface plot e Surface creates a shaded filled surface e XZ Lines draws lines parallel to the X axis one for each Y location e YZ Lines draws lines parallel to the Y axis one for each X location e Lego draws a lego block plot with mesh sides e Lego filled draws a lego block plot with solid sides You can also use the buttons to remove or add a Skirt around the surface m
70. r For a vector such as a time series the coefficients of wavelet power can be rearranged to yield a two dimensional picture where the first dimension is the independent variable e g time and the second dimension is the wavelet scale e g 1 frequency Two dimensional Array The wavelet transform of a 2D array is also two dimensional and is arranged so that the smallest scales are in the upper right quadrant assuming that index 0 0 is in the lower left Example Use the Chirp dataset that is included in the Wavelet sample file This dataset contains a time series with a sine wave that has an exponentially increasing frequency You can use the Multiresolution Analysis viewer to examine the time series IDL Wavelet Toolkit Wavelet Power Spectrum 50 Chapter 3 Theory and Examples Try the following steps 1 Tip From the main window select the Chirp dataset and start the Wavelet Power Spectrum viewer using either the Visualize Menu or the Toolbar button The WPS can be seen under Wavelet Power Spectrum on page 32 Select the Morlet wavelet function from the Family dropdown box You should be able to see the exponential increase in frequency as a band of high power extending from left to right and ranging from about Scale 256 sec near the beginning to Scale 16 sec near the end of the time series To bring out the features more clearly change the Energy Scaling dropdown item from Power to Magnitude
71. r 32 zero phase lines 35 wavelet toolkit importing data 25 status bar 12 wavelet transform continuous 29 48 66 discrete 29 48 74 Index partial 107 wavelet widget commands 58 WV_APPLET procedure 60 WV_CW_WAVELET function GET_VALUE 64 reference 62 SET_VALUE 64 widget events generated 64 WV_CWT function 66 WV DENOISE function 69 WV_DWT function 74 WV_FN_COIFLET function 78 WV FN DAUBECHIES function 80 WV FEN GAUSSIAN function 82 WV FEN HAAR function 85 WV FN MORLET function 87 WV EN PAUL function 90 WV EN SYMLET function 93 WV IMPORT DATA procedure 96 WV IMPORT WAVELET procedure 99 WV PLOT MULTIRES function 104 WV PLOT3D WPS function 101 WV PWT function 107 WV TOOL DENOISE function 109 Z zero phase lines 35 IDL Wavelet Toolkit
72. rend When the Daubechies 2 wavelet is used to transform a data series both the mean and any linear trend are filtered out of the series A higher vanishing moment implies that more moments quadratic cubic etc will be removed from the signal Regularity The regularity gives an approximate measure of the number of continuous derivatives that the wavelet function possesses The regularity therefore gives a measure of the smoothness of the wavelet function with higher regularity implying a smoother wavelet e Folding Time Continuous Wavelets Only The e folding time is a measure of the wavelet width relative to the wavelet scale s Using the wavelet transform of a spike the e folding time is defined as the distance at which the wavelet power falls to 1 e 2 where e 2 71828 Larger e folding time implies more spreading of the wavelet power User Defined Wavelets You can easily extend the IDL Wavelet Toolkit by adding more wavelet functions These wavelet functions should follow the same calling mechanism as the built in wavelet functions such as WV_FN_DAUBECHIES on page 80 In addition your wavelet function should begin with the prefix wv_fn_ 1 Wavelet Viewer Let s say you would like to add a wavelet function called Spline giving the Daubechies Spline wavelets To do this first create a wavelet function to return the wavelet coefficients and the information structure FUNCTION wv_fn_spline Order Sc
73. rrent directory selection 18 default directory 18 restoring defaults 19 stride factor 18 pyramidal algorithm result format 75 returning 107 R regularity of wavelet functions 30 remember current directory See preferences RMS difference 44 root mean square difference 44 rough multiresolution plots 39 S scaling functions 28 starting wavelet toolkit 12 statistical significance testing 48 stride factor See preferences structure tags IDL Wavelet Toolkit 115 importing wavelet data 97 surfaces style 36 symlet See wavelet functions symmetry of wavelet functions 29 T toolkit structure 9 tools adding 45 denoise function 109 denoise tool 41 user defined 16 45 trademarks 2 TrueColor 24 bit images 26 U user defined tools 45 V vanishing moments 30 variable information wavelet dataset viewer 20 variable selection See datasets viewing wavelet functions 28 W WAV audio files 26 wavelet commands functions 59 transform 58 widgets 58 wavelet functions coiflet 78 Index 116 compact support 29 Daubechies 80 family 64 Gaussian 82 Haar 85 Morlet 87 nonorthogonal 29 order 64 orthogonal 29 Paul 90 regularity 30 symlet 93 symmetry 29 user defined 30 99 vanishing moments 30 viewing 28 wavelet power spectrum See also WV_PLOT3D_WPS function energy scaling 35 plotting method 49 rotation translation stretching 37 theory 49 viewe
74. s 2 1 5 Scaling Compact support 3 Vanishing moments 2 Regularity 0 55 Figure 2 2 The Wavelet Viewer Wavelet and Scaling Functions The wavelet consists of two components the scaling function which describes the low pass filter for the wavelet transform and the wavelet function which describes the band pass filter for the transform Changing Wavelets The droplist contains the names of all currently available wavelets The Family refers to the overall properties of the wavelet while the Order determines the particular wavelet within each family Wavelet Viewer IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 29 Wavelet Information After you select a wavelet family and order the following information will be displayed Discrete Continuous Discrete wavelet functions are used with the discrete wavelet transform which provides the most compact representation of the data The discrete transform is very fast and is best suited for image processing filtering and large arrays Continuous wavelet functions are used to approximate the continuous wavelet transform which provides a highly redundant transformation of the data The continuous wavelet transform is much smoother than the discrete transform and is better suited for time series analysis on small arrays less than 20000 data points Orthogonal Nonorthogonal Orthogonal wavelet functions will have no overlap with each other zero correlation
75. s SIAM p 261 Note that Daubechies has divided by Sqrt 2 and the coefficients are reversed Version History 5 3 Introduced See Also WV_DWT WV_FN_DAUBECHIES WV_FN_HAAR WV_FN_SYMLET IDL Wavelet Toolkit WV_FN_COIFLET 80 Chapter 4 IDL Wavelet Toolkit Reference WV_FN_DAUBECHIES The WV_FN_DAUBECHIES function constructs wavelet coefficients for the Daubechies wavelet function Syntax Result WV_FN_DAUBECHIES Order Scaling Wavelet Ioff Joff Return Value The returned value of this function is an anonymous structure of information about the particular wavelet Tag Type Definition FAMILY STRING Daubechies ORDER NAME STRING Order ORDER_RANGE INTARR 3 1 24 2 Valid order range first last default ORDER INT The chosen Order DISCRETE INT 1 O2continuous 1 discrete ORTHOGONAL INT 1 O2nonorthogonal 1 orthogonal SYMMETRIC INT 0 O asymmetric 12symm 2 near symm SUPPORT INT 2 Order 1 Compact support width MOMENTS INT Order Number of vanishing moments REGULARITY DOUBLE The number of continuous derivatives Table 4 7 Structure Tags for Result Arguments Order A scalar that specifies the order number for the wavelet The default is 2 WV FN DAUBECHIES IDL Wavelet Toolkit Chapter 4 IDL Wavelet Toolkit Reference 81 Scaling On output contains a vector of double precision scaling father coefficients Wavelet On output
76. s retained PERCENT DOUBLE Percent of coefficients retained Table 4 4 The Structure Tags for DENOISE_STATE Continued Note If the DOUBLE keyword is set then the arrays will be of type double Upon input if DENOISE_STATE is set to a structure with the above form then DWT WPS SORTED and CUMULATIVE will not be recomputed by WV_DENOISE This is useful if you want to make multiple calls to WV_DENOISE using the same Array Warning ___ No error checking is made on the input values The values should not be modified between calls to DENOISE_STATE DOUBLE Set this keyword to force the computation to be done using double precision arithmetic DWT_FILTERED Set this keyword to a named variable in which the filtered discrete wavelet transform will be returned PERCENT Set this keyword to a scalar specifying the percentage of cumulative power to retain Note LL If neither COEFFICIENTS nor PERCENT is present then all of the coefficients are retained i e no filtering is done IDL Wavelet Toolkit WV DENOISE 72 Chapter 4 IDL Wavelet Toolkit Reference THRESHOLD Set this keyword to a scalar specifying the type of threshold The actual threshold T is set using COEFFICIENTS or PERCENT Possible values are Value Description 0 Hard threshold this 1s the default The hard threshold sets all wavelet coefficients with magnitude less than or equal to T t
77. sents the position The D D represent increasing scale The wavelet coefficients are stored in Result in order from largest scales to smallest D3 For a two dimensional Array the wavelet transform is computed using the pyramidal algorithm along each dimension The wavelet coefficients are stored in order with the largest scales in the 0 0 position As an example for an 8 x 8 Array the Result is an 8 x 8 array with the following structure 0 0 LE AOBO AOB1 AODO AOD1 A0d0 A0d1 A0d2 A083 ue ur E ME 1B0 1B1 LDO LD1 Ldo Ld1 Ld2 Ld3 COBO COB1 CODO COD1 codo cod1 Cc0d2 c0d3 C1 i QQ anaa BO B1 1D0 1D1 1d0 d1 182 1d3 cOBO c0B1 cODO cOD1 codo c0d1 c0d2 c0d3 Q qgqaaa 1B0 1B1 1D0 1D1 1d0 1d1 1d2 1d3 c2BO c2B1 c2D0 c2D1 c2d0 c2di c2d2 c2d3 c3BO c3B1 c3D0 c3D1 c3d0 c3d1 c3d2 c3d3 l l l l l 11 Here A and B represent the scale coefficients for the first and second dimensions respectively The c and 1 D represent the largest scale wavelet coefficients for the first and second dimensions respectively while c and d represent the small scale wavelet coefficients Subscripts 0 1 2 3 denote the position of the wavelet within the image Example The following example shows how to compute the first three levels of the pyramid algorithm using either the N L
78. t 1996 Hughes and Applied Research Corporation SMACC Copyright 2000 2004 Spectral Sciences Inc and ITT Visual Information Solutions All rights reserved This software is based in part on the work of the Independent JPEG Group Portions of this software are copyrighted by DataDirect Technologies 1991 2003 BandMax Copyright O 2003 The Galileo Group Inc Portions of this computer program are copyright 1995 1999 LizardTech Inc All rights reserved MrSID is protected by U S Patent No 5 710 835 Foreign Patents Pending Portions of this software were developed using Unisearch s Kakadu software for which ITT has a commercial license Kakadu Software Copyright 2001 The University of New South Wales UNSW Sydney NSW 2052 Australia and Unisearch Ltd Australia This product includes software developed by the Apache Software Foundation www apache org MODTRAN is licensed from the United States of America under U S Patent No 5 315 513 and U S Patent No 5 884 226 FLAASH is licensed from Spectral Sciences Inc under a U S Patent Pending Portions of this software are copyrighted by Merge Technologies Incorporated Support Vector Machine SVM is based on the LIBSVM library written by Chih Chung Chang and Chih Jen Lin www csie ntu edu tw cjlin libsvm adapted by ITT Visual Information Solutions for remote sensing image supervised classification purposes IDL Wavelet Toolkit Copyright O 2002 Christopher Torr
79. t 1989 A good introduction to the DWT and multiresolution analysis is given in Lindsay et al 1996 The DWT routines are based on the routines described in section 13 10 of Numerical Recipes in C The Art of Scientific Computing 2nd ed Cambridge University Press and are used by permission An introduction to the continuous wavelet transform for time series analysis can be found in Torrence and Compo 1998 along with a discussion of statistical significance testing Wavelet Transform IDL Wavelet Toolkit Chapter 3 Theory and Examples 49 Wavelet Power Spectrum Background The wavelet coefficients yield information as to the correlation between the wavelet at a certain scale and the data array at a particular location A larger positive amplitude implies a higher positive correlation while a large negative amplitude implies a high negative correlation A useful way to determine the distribution of energy within the data array is to plot the wavelet power equivalent to the amplitude squared By looking for regions within the Wavelet Power Spectrum WPS of large power you can determine which features of your signal are important and which can be ignored Method Given the wavelet transform W of a multi dimensional data array A where i 0 N 1 is the index and N is the number of points then the Wavelet Power Spectrum is defined as the absolute value squared of the wavelet coefficients IW One dimensional Vecto
80. t Y coordinate The default is 0 Ystart can contain complicated mathematical expressions although the result must be a scalar number Dy This string gives the sampling interval between the Y coordinates The default is 1 Dy can contain complicated mathematical expressions although the result must be a scalar number Xoffset The offset along the first data dimension at which to start The default is OL Xcount The number of data points to use along the first data dimension The default is the size of the first dimension Dataset Viewer IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 23 Xstride The sampling interval along the first data dimension The default is 1L Yoffset This long integer gives the offset along the second data dimension at which to start The default is OL Ycount This long integer gives the number of data points to use along the second data dimension The default is the size of the second dimension Ystride This long integer gives the sampling interval along the second data dimension The default is 1L Source This string describes the original source or location of the data The default is either the full filename if the data was from a file or Imported if the data was from the IDL gt command prompt Notes You can enter miscellaneous information into the Notes string The default is the null string Mathematical Expressions For Xstart Dx Ystart
81. t and width for exporting and printing Export Postscript Export the image to a postscript file Printer Setup This menu item allows you to set up the printer via the Printer Dialog Print This menu item prints the image Close This menu item closes the Multiresolution viewer Wavelet Options You can change the current wavelet family or the order The plot will be updated automatically IDL Wavelet Toolkit Multiresolution Analysis 40 Chapter 2 Using the IDL Wavelet Toolkit 182 E EOM 182 O 100200300400500 Smooth low pass Details dnd pass Rough high pass Wl FA KM I8 34 AANA EK IM ste E77 Fil sh Ere EM Figure 2 4 Multiresolution Analysis of the Chirp Variable Ll Scale Multiresolution Analysis IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 41 Denoise Tool You can use the Denoise Tool to explore different techniques for removing noise and compressing data using the wavelet transform The Denoise Tool is shown in the following figure The plots and options are described below al Denoise Sine wave increasing frequency Original data Filtered data Wavelet options 400 410 Family Daubechies 4 a mi B a a Order 2 2n E an o o Denoise options 33 mu o 10 Cumulative power 100200300 400 504 100200300 400300 Time sec T Time sec coeffs Be Wavelet coeffs black 0 Coefficient power 100 d d C Soft threshold a Resu
82. u Description 18 Chapter 2 Using the IDL Wavelet Toolkit Preferences The Preferences dialog under the File Menu allows you to set various default preferences and options for the currently active dataset Noe The Preferences are saved within each dataset rather than in a separate preferences file each dataset can therefore have its own set of preferences Note however that opening a new dataset may change the current preferences These new preferences will remain in effect until changed either via the Preferences window or by opening a different dataset Default Directory Set this option to your working directory The Wavelet Toolkit will start all file open or save dialogs in this directory This directory may be overridden if Remember Current Directory is set Remember Current Directory Set this option to cause the Wavelet Toolkit to store the directory selected within any file open or save dialogs and to use this directory for future dialogs If this option is not set the Default Directory will be used Confirm Exit If this option is set the Wavelet Toolkit will ask you for confirmation when you exit the Toolkit Compress Save Files Set this option to use file compression when saving dataset files Compressed files will occupy less disk space than uncompressed files but may be slower to save and open Stride Factor When importing large data arrays the IDL Wavelet Toolkit will automatica
83. urns the partial wavelet transform of the input vector A The transform is done using a user inputted wavelet filter WV PWT is called by WV DWT WV PWT is based on the routine pwt described in section 13 10 of Numerical Recipes in C The Art of Scientific Computing 2nd ed Cambridge University Press and is used by permission Syntax Result WV PWT A Scaling Wavelet Ioff Joff DOUBLE INVERSE Return Value The result is an output vector of the same length as A containing one stage of the pyramidal algorithm Mallat 1989 Arguments A The input vector The length must be either less than four 4 or a power of two 2 Scaling A vector of scaling father coefficients of length N Wavelet A vector of wavelet mother coefficients of length N loff An integer that specifies the support offset for Scaling To center the scaling function over each point in Array set Ioff to N 242 Joff An integer that specifies the support offset for Wavelet To center the wavelet function over each point in Array set Joff to N 242 IDL Wavelet Toolkit WV PWT 108 Chapter 4 IDL Wavelet Toolkit Reference Keywords DOUBLE Set this keyword to force the computation to be done in double precision arithmetic INVERSE If set the inverse transform is computed By default the forward transform is computed Method and Result Format The WV_PWT function computes the wavelet coefficients for one level of the pyram
84. urrent location and power will be displayed in the Status Bar Rotation Translation Stretching To rotate the image click on the image while holding down the left mouse button and drag the mouse pointer to rotate the image about the midpoint To translate the image click on the image while holding down the right mouse button on the Macintosh hold down the command key also and drag the mouse pointer IDL Wavelet Toolkit Wavelet Power Spectrum 38 Chapter 2 Using the IDL Wavelet Toolkit To stretch the image click on the image while holding the middle mouse button on Windows hold down the Ctrl key also on Macintosh hold down the Option key Drag the mouse pointer right left to stretch shrink in the X direction drag the pointer up down to stretch shrink in the Y direction Wavelet Power Spectrum IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 39 Multiresolution Analysis Multiresolution Analysis uses the wavelet transform to decompose a data series in a cascade from the smallest scales to the largest At each scale there are three components the Smooth or low pass filtered data series the Details or band pass data series and the Rough or high pass For one dimensional vectors this can be viewed as a hierarchy of x y plots as shown in the following figure For two dimensional arrays the multiresolution analysis gives a series of images File Menu Page Setup This menu item sets up the page heigh
85. ve the currently selected variable to the left mu Move Variable Right Select this menu item or button to move the currently selected variable to the right crt View Data Values D10 This menu item or button displays the values for the currently selected variable X Delete Variable Select this menu item or button to delete the currently selected variable or variables You are asked for confirmation before the variables are removed Visualize Menu The Visualize Menu contains methods to graphically display and manipulate the wavelet transform ull Wavelets This menu item or button starts up the wavelet compound widget which allows you display the available wavelet functions and their properties You can also start the wavelet viewer using the WV_CW_WAVELET function from the IDL gt command prompt The wavelet widget is described in Wavelet Viewer on page 28 IDL Wavelet Toolkit Menu Description 16 Chapter 2 Using the IDL Wavelet Toolkit e Wavelet Power Spectrum This menu item or button starts the three dimensional viewer for the wavelet power spectrum using the currently selected variable You can also start the viewer using the WV PLOT3D WPS function from the IDL command prompt For more information see WV PLOT3D WPS on page 101 The wavelet power spectrum viewer is described in Wavelet Power Spectrum on page 32 E Multiresolution Analysis This menu item or button starts the viewer for multiresolutio
86. velet Toolkit Two Dimensional Arrays By default each column within the file will be imported into the IDL Wavelet Toolkit as a separate variable To import a two dimensional array of data you should IDL Wavelet Toolkit Importing Data 26 Chapter 2 Using the IDL Wavelet Toolkit use the Group All button within the ASCII TEMPLATE dialog to connect all of the columns into one field Binary Files Ld Select this menu item or button to import data from a binary data file After choosing the file using the Select Import File dialog you can specify the particular format for the file using the BINARY TEMPLATE dialog See BINARY TEMPLATE IDL Reference Guide for more information The BINARY TEMPLATE routine handles raw binary files consisting of headers and multiple data fields The dialog consists of a Binary Template window where you can define various fields within the file Each field will be imported into the Wavelet Toolkit as a separate variable Image Files Select this menu item or button to import an image file The function DIALOG READ IMAGE is used to select the image file For files with multiple images you can choose the particular image you wish to import See DIALOG READ IMAGE IDL Reference Guide for more information For TrueColor 24 bit images you will then be asked how you wish to convert the three channels into a single two dimensional image You have the option to scale the data into an intensity from
87. volves computing the wavelet transform of your data and then decreasing or discarding the smallest wavelet coefficients The inverse transform of these coefficients will then be a filtered version of your data Method We assume that you have computed the wavelet transform W of a multi dimensional data array A where i 0 N 1 is the index and N is the number of points You then compute a threshold level Wo This threshold level can be based on the percent of wavelet power that you wish to retain the number of coefficients or some other method Suggestions for choosing the threshold are given in Donoho and Johnstone 1994 Wavelet coefficients smaller than this threshold are discarded while those above are retained There are two methods for thresholding Hard threshold The hard threshold removes all discarded wavelet coefficients by setting them to zero and computing the inverse wavelet transform This can be defined as w Wi W gt Wo 0 W sW IDL Wavelet Toolkit Denoise 52 Chapter 3 Theory and Examples where Wi is the wavelet coefficient and Wo is the chosen threshold level Soft Threshold The soft threshold also sets all discarded wavelet coefficients to zero However it also linearly reduces the magnitude of the each retained wavelet coefficient by an amount equal to the largest discarded coefficient i e sgn W W Wo W gt Wo 0 Wi lt Wo W 1 where sgn W is the sign of Wi Example We will
88. w Help Wavelet Options Family Morlet bd X mi 2 Order 4 View ptions Iv 3D MV Data Plot ColorBar Global Zero Phase Lines Energy scaling Power mj ja Surface iMi Skit T Flat Gray Contour lines Of v Colo Of v Filed Significance or 01 z Power at 419 0 8 500 119094 119094 Figure 2 3 The Wavelet Power Spectrum 3D viewer Wavelet Power Spectrum IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 33 File Menu Open State This menu item opens a previously saved state file into a new window Save State This menu item saves the current state of the Wavelet Power Spectrum into a file Export To e Bitmap File Windows only The bitmap file saves the current image as a bitmap e Vector Metafile Windows only The vector metafile produces a scalable image file but may not be able to accurately reproduce the 3D geometry e Bitmap Pict Macintosh only The bitmap pict saves the current image as a bitmap e Bitmap Postscript The bitmap postscript format saves the current image as a bitmap e Vector Postscript The vector postscript format takes less disk space than bitmap and is scalable but may not be able to accurately reproduce the 3D geometry e VRML The Virtual Reality Markup Language produces a three dimensional output file suitable for web publication Noe It is not always possible to tra
89. y a field double click with the left mouse button on the field After editing the field press the lt Return gt key to keep your changes or click outside of the table to discard your changes Type This string shows the numeric type and the array size of the data It is not modifiable by the user Title This string contains the overall name of the variable The Title field is used to label the Wavelet Power Spectrum and Multiresolution widgets The default is the null string Variable This string provides a short name for the variable The Variable is used to label plots and for the labels in the Dataset Viewer For a one dimensional vector e g a time series the Variable is equivalent to Ytitle The default is either the name of the import file or Data if imported from the IDL gt command prompt Units This string gives the units of the variable and is used to label various plots For a one dimensional vector e g a time series the Units is equivalent to the Yunits The default is the null string Dataset Viewer IDL Wavelet Toolkit Chapter 2 Using the IDL Wavelet Toolkit 21 Field Type Example 1D vector Example 2D array Title STRING Wave audio recording IEEE Test Image Variable STRING Channell IEEEtest Units STRING Amplitude intensity Xname STRING Time x Xunits STRING seconds pixels Xstart STRING 0 0 Dx STRING 1d0 22
90. yunits XOFFSET xoffset YOFFSET yoffset start the edge detection applet return the Widget ID for the applet RETURN wID END 2 Save this function in a file wv tool edgedetect pro that is accessible from your current IDL path 3 Now start the Wavelet Toolkit with your new wavelet function WV APPLET TOOLS Edge Detect Your new tool should appear in the Tools Menu The actual function name is constructed by removing all white space from the name and attaching a prefix of WV TOOL Note __ sSsSssssssssSSSSSSSSSSSSSSS S At a minimum your tool function must accept a data Array All other parameters such as X and Y and keywords GROUP LEADER TITLE etc are optional The IDL Wavelet Toolkit will pass in only those parameters and keywords that are usable by your tool function IDL Wavelet Toolkit Adding User Tools 46 Chapter 2 Using the IDL Wavelet Toolkit Adding User Tools IDL Wavelet Toolkit This chapter discusses the following topics Wavelet Transform cele 48 Multiresolution Analysis 54 Wavelet Power Spectrum 49 Bibliography 40 55 PONG 222 leillel i 0 RD ER tia 51 IDL Wavelet Toolkit 47 48 Chapter 3 Theory and Examples Wavelet Transform Background Wavelet analysis is a technique to transform an array of N numbers from their actual numerical values to an array of N w

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