C++Wavelets: A User's Guide S. E. FERRANDO, L. A. KOLASA
Contents
1. OF MATHEMATICS PHYSICS AND COMPUTER SCIENCE RYERSON POLYTECHNIC UNIVERSITY TORONTO ONTARIO M5B 2K3 CANADA E mail address ferrando acs ryerson ca E mail address lkolasa acs ryerson ca E mail address natasak home com2. so Interval Input 10 or Interval Input new 10 In each instance we have a pointer Input to an array of 10 default intervals When the time comes one may use the Set member function to fill out a desired Interval Ultimately all of the Interval constructors call the Set member function and when a default Interval already exists the Set function is the correct one to use in order to fill it out for example Input 6 Set 0 1023 or Input 4 Set 1 1022 data The first line above fills out the seventh element of the arrray Input with an array of 1024 zeros indexed from 0 to 1023 The second line fills out the fifth element of Input with the 1024 elements of the array pointed to by data and indexes them from 1 to 1022 When using new as in the above example one is obliged to use delete but as with all C Wavelets data structures memory management is transparant and self contained One need only understand how to employ the basic constructors to bring a single data structure into existence the rest of memory management is done behind the scenes When you become familiar with C Wavelets the header files should be enough to refresh your memory REFERENCES 1 F Carrano P Helman R Verofff Data Abstraction and Problem Solving in C Addison Wesley 1998 2 H Schildt Teach Yourself C McGraw Hill 1998 3 M Wickerhauser Adaptive Wavelet Analysis From Theory to Software A K Peters 1994 DEPARTMENT
3. to C can be found in 2 Chapters 2 6 contain all that is necessary to know when using C Wavelets In particular we like the memory management features of C We shall explain our data structures below it suffices to say right now that in C Wavelets the data structures of wavelet analysis are invoked and destroyed almost as easily as native data structures i e integers floats We take advantage extensively of the feature in C of overloaded operators and function in C Wavelets Again this is done so as to make writing programs as natural and transparent as possible Otherwise we are not object oriented For us wavelet analysis is a process and the algorithms are paramount For example we have often been criticized for not using the private data member capabilities of C We avoid this because we are interested in speed and ease of programming to us private data implies costly member functions to get your hands on it We trust the integrity of our library and the intelligence of our users Should data be inadvertently modified this would be a mistake but we prefer to allow you that possibility On the other hand we have taken great care in writing the algorithms using essentially the features of C which allow for faster running code such as pointer Research partially supported by NSERC 2 S E FERRANDO L A KOLASA AND N KOVACEVIG arithmetic Therefore some of the code itself may be difficult to read we ask
4. C Wavelets A User s Guide S E FERRANDO L A KOLASA AND N KOVA EVI ABSTRACT We expalin the technical features of C Wavelets and how to use the software Presantly this users manual is still being written Anyone compotent in wavelet analysis and C should be able to use this software presently The header files and the demos supplied with this package should be enough to get one started 1 INTRODUCTION C Wavelets is a collection of C C based functions for performing some of the basic functions of wavelet analysis wavelet analysis and synthesis wavelet packet analysis and synthesis Our objective is to provide the serious programmer with ready to use functions that may be employed in applications or programs which call for wavelet analysis While there are many good packages available for performing wavelet based analysis e g Matlab some applications must be written in a high level programming language Often one has to perform a great many calculations and the overhead associated with software packages can be prohibitively expensive What is needed then is a library of useful functions written in a high level programming language ready available at ones finger tips We have chosen C as our high level language C programmers howerer should not despair though most of wavelet based analysis is algorithmic and we use only as much C so as to make code writing as transparent as possible A good introduction
5. eper into the stack it is more and more to our advantage to avoid having the clutter of long parameter lists the baggage of the Interval class carries many important parameters Admittedly the length member is redundant but we use it often enough that it deserves its place as a data member The origin data member is important and must be handled carefully it is used to point to the actual data If we only wish to access or modify the data without concern for speed or its exact memory location then the easiest thing to do is to use the overloaded operator For example Input 2 2 1 Input 1 0 3 Input 3 9 4 From this example we see that by design the indexing of the original sequence corresponds to the indexing used by the operator exactly how a mathematician thinks Thus if we desire to change the value of z from 1 1 to 2 9 we write Input 1 2 9 Of course the data represented by x is the fourth data member in our data array but mathematically we think of it as having index equal to 1 Our programming corresponds to our thinking this is by design But a price must be paid for this The way this feature has been implemented is by pointer offset Input origin is not a pointer to the beginning of the data array i e v2 in this case Input origin points to xo In general when an Interval is constructed origin is pointing to the beginning of the data array But then it is modified origin beg Thi
6. only that you trust us when we say that it has been tested We have striven for speed in writing this code not readability C Wavelets allows you to perform the one dimensional Discrete Wavelet Trans form DWT using orthogonal or bi orthogonal wavelelets in a periodic or an ape riodic setting Discrete Wavelet Packet Analysis along with the best wavelet basis algorithm of Coifman and Wickerhauser 3 is also available This too comes in a periodic and an aperiodic version We assume some familarity on the reader s part with wavelet based analysis But as there may be some ambiguity as to what the DWT and DWPA actually are we outline here what they are in C Wavelets The soul of discrete wavelet analysis is convolution decimationand its adjoin t 3 is an excellent reference we merely highlight those details germane to C Wavelets Let X x be a sequence of real numbers which represents your data or a signal to be processed let G g be a fixed given filter Then the convolution decimation operator defined by G acts on X in the following way GX X 9 82i 5 j and its adjoint is given by G Xj J gai jai 2 THE DATA CLASSES In this section we describe the classes that are used in C Wavelets The classes that appear in C Wavelets are 1 Interval Array TreePer ArrayTreeA per BinTree HedgePer HedgeAper QMF PQMF GPQMF We discuss the Interval class in the most detail It is the most common clas
7. s and the simplest it is all that is needed to perform the DWT o bw 2 1 Interval The Interval class is an array of reals with a few extras Typically an Interval is used as the input and the output when performing wavelet analysis Input data a sequence of floating point numbers 2 are stored in an array The main difference between a C array and an invterval is that the indexing of an interval does not have to start with zero As noted earlier indexing is important in the wavelet theory The data members of the interval class are Interval origin Interval beg Interval end Interval length Suppose we have a sequence of reals 2 1 0 3 o 0 0 z1 1 1 2 0 2 3 9 4 C 4 Wavelets A User s Guide 3 and we store them in an Interval called Input Then Input beg is the integer 2 Input end is the integer 3 and Input length Input end Input beg 1 which is the number of elements in the data sequence Aside from the fact that it is crucial to keep track of Input beg and Input end when performing convolution decimation there are other instances where these data members are useful The following example shows how to write a for loop involving an Interval for i Input beg i lt Input end i This relieves us from having to pass as a parameter to functions the beginning and ending indices of a sequence The length data member is also useful in this regard As we get deeper and de
8. s then is how the operator works Input j Input origin j Should a pointer to the physical beginning of the data be needed the proper reference is Input origin Input beg This is very important if you work with sequences which mathematically speaking are not indexed starting from zero la the C C convention In practice however it is often convenient or desirable to have Input beg 0 in this case of course Input origin is a pointer to the actual beginning of the array and no confusion results One notable instance when Input beg 0 is in the periodic version of the DWT where we may assume this without loss of generality For this common setting when the number of data points must be a power of 2 we have special constructors Interval p data or Interval p which construct an interval indexed from 0 to 2 1 filled from data or if data is not given filled with zeros The more generic constructor is Interval a b data or Interval a b 4 S E FERRANDO L A KOLASA AND N KOVACEVIG This constructs an interval indexed from a to b If no data is given the Interval is filled with length b a 1 zeros One must be careful that data points to an array of at least length elements for otherwise that would be a mistake Finally there is a default constructor which for the most part sets everything to zero The default constructor is useful when one wishes to create an array of Intervals as
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