Abstract. 1 Introduction

Transcription

1 Design of a Wavelet Package on Mathentatica Version 3 S. Sakakibara College of Science and Engineering, Iwaki Meisei University, Iwaki-shi. Fukushima-ken, Japan 970 Abstract Based on the SplineWavelet package, we have developed an updated version of the general analysis package called Wavelet. The new version incorporates many new capabilities that become available due both to the recent development of theory, and the new features of Mathematica Version 3. We discuss changes in the function definitions and their implications, and present the details of some of the most important new features. The new Compile function can speed up the computation by a factor of six to seven. Palettes can serve as an easy-to-use GUI for analysis. 1 Introduction Wavelet [1][2] is a subject of recent extensive study in a variety of fields including mathematics, signal processing, statistics, and engineering. Due to its ability of providing time-frequency representation of a signal, is used in de-noising, signal compression, and other data analysis and signal processing. Wavelets also form bases which are used to solve differential or integral equations numerically. In this work, we present a Mathematica package for a general analysis, called Wavelet. It is basically an updated and expanded version of the SplineWavelet package presented by the author at the IMS'95 [3]. The SplineWavelet package was written for author's book [4], and is contained in a CD-ROM that comes with the book. At present, there are several packages available for the leading computational software,

2 428 Innovation In Mathematics including S-PLUS [5], MATLAB [6], and Mathematica [7], as well as stand alone application softwares. The SplineWavelet package has been used in various applications [8] [9]. Due both to the recent developments in theory and to the new features of Mathematica Version 3, there are a number of important improvements in the Wavelet package over the previous version. The advantage of employing Mathematica is that it provides an uniform platform combining symbolic algebra, numerical computation, arid graphical visualization of data, which are all inevitable ingredients in analysis. On the other hand, Mathematica is not the fastest in dealing with large number of numerical data, arid such a shortcoming must be minimized by a careful design of the package. In Section 2, theory of discrete transform is briefly reviewed. In Section 3, we discuss the changes of function definitions made in the new package, and discuss their implications. In Section 4, we show by an example how Compile can speed up computations. In Section 5, we explain how to make palettes for a graphical user interface of analysis. Section 6 is devoted to our conclusions. 2 Wavelet Theory The construction of a mother starts with a choice of a low-path filter consisting of filter coefficients pk-, k G Z. There is a high-pass conjugate filter {qk} for each set of the filter {pk}- The corresponding scaling function <j>(x) and mother tfi(x) satisfy the dilation equation <^(z) = ^] pt <K2z - &), ^(z) = ]>]<% <X2z - &) (1) &ez kez For x n G Z, the dilation equation for </)(n) becomes an eigenvalue problem, from which, together with the normalization the values of <j>(n) may be determined. A repeated application of the dilation equation (1) to these values, one can obtain (f>(x) and z/>(x) at the dyadic points x n/y. For each pair of conjugate filters {pk} and {%&} associated a pair of dual filters {p^} and {#&}. The dual filters are the same as the primary filters for orthogonal s. A large part of theory concerns with the determination of thefilter,but here we assume that they are known for some commonly used s. Also, an algorithm is defined in the Wavelet package to compute </>(n), n G Z, from thefiltercoefficients.

3 Innovation In Mathematics 429 We define the linear combinations of the scaling function and, respectively, at the resolution level j E Z, as #' ^(2^ - A;) (2) In analysis of a signal {s^ez, the set of coefficients initialized such that the function fo(x) interpolates the data, Joh In many cases, however, one simply assumes that Using Mallat's algorithm, Jo) f-2t one obtains the coefficients at a lower level. Its inverse operation is the reconstruction They are combinations of discrete convolution and up/down sampling, as schematically summarized as \ 21 (3-D " n (5) By the repeated application of Mallat's algorithm, the signal may decomposed into successive lower levels as be \ In this way, the signal fo(x) is decomposed in the sum of components fo(x) = g.i(x) + f-i(x)

4 430 Innovation In Mathematics 3 Functions defined in the Packages We represent a data set { so, -Si,..., Sn} as {0, {SQ, Si,..., s^}}. The function Seal ingsample computes from { } a list of pairs {Ar/2-*, By the special choice c^ ^,o, or {0,{!}} in Mathematica, the values (t>(k/2,i) are obtained, which is used to plot the scaling function. The function Wavelet Sample works in the same way for the tj)(x). Decompose ToScaling and DecomposeToWavelet implement Mallat's algorithm (3). The summary is given in Table 1. Table 1. Some of the functions defined in SplineWavelet package Function ListConvolve[/i Iter, data] UpSample[data] DownSample [data] Seal ingsample [data] WaveletSample[data] DecomposeToScaling[data] DecomposeToWavelet [data] Reconstruct [data./, data2] Findlnterpolat ion [data] PSequence [] QSequence [] GSequence [wave let] HSequence [] Seal ingat Knots [] TimeScalePlot [data] Option SampleLevel DataStyle SampleLevel DataStyle Option Values Pair, Plain Pair, Plain wouew In the old SplineWavelet package, they all use ListConvolve and UpSample or DownSample in combinations. While this corresponds closely to the view (5), this is not fastest in practice. In particular, half of the result is thrown away in the DownSample [ListConvolve [... ]] combination. We therefore define Convolve, UpsampleConvolve, and ConvolveDownsample separately. We also renew the names offilters.the primary (conjugate)filteris represented as PFilter (CFilter), and their duals will be distinguished by the name of dual s. The correspondence between the new and old

5 names is summarized in Table 2. Innovation In Mathematics 431 Table 2. The new and old functions In SplineWavelet package In Wavelet package Convolve[/ilter, data] ListConvolve[filter, data] UpsampleConvolve[filter, data] ListConvolve[fiIter, UpSample[data]] ConvolveDownsample[/ilter, data] DownSample[ListConvolve[filter, data]] PFilter[wave let] PSequence[wave let] CFilter [wav elet] PFilter [dual ] QSequence[] GSequence[wat/elet] CFilter [dual ] HSequence[] 4 Convolution using Compile Most operations in discrete transform are convolution of data with filter coefficients. As it is easy to implement, Mathematica does not have a built-in function for convolution. It is also convenient to define a function for convolution that is appropriate for our purposes. In particular, we have to deal with several types of boundary conditions imposed on the data when the convolution is actually carried out. Furthermore, the new Compile capability of Mathematica Version 3 is very effective in speeding up the operation. When a discrete signal {s^} is filtered, the output signal {s[} is given by the convolution k*n-k (7) where {/%&} is the set of filter coefficients. It is convenient to consider the convolution in terms of multiplication of polynomials. Define the z- transform of the sequences {/&&}, {s^}, and {sj.}, respectively. #(z) = ^2/ttz\ k Then (7) is equivalent to 5%z) = yi-st A k S'(z) = H(z)S(z). (8) We can define the function corresponding to this operation as In[l]:= convolve?[h_list, s_list] := CoefficientList[ Expand [h. (z~range[0, Length [h]-1] ) * s. (z~range[0, Length [s]-1] )],z] which returns the list of coefficients {s^}. Defining the sequence {hk} as a Mathematica list hi, and similarly {sk} as si, 5

6 432 Innovation In Mathematics In[3]:= convolve?[hi, si] // TableForm Out[3]//TableForm= The operation (7) may be implemented in Mathematica basically as Map [(Reverse [hi].#)&, Part it ion [si, Length [hi], 1] ], but we must pad O's which would then produce terms like h^sq. Thus, we define In [4] := convolvel [h_list, s_list] : = Module [{pred, postd}, pred=table [0, {Length [h] -1}] ; postd=table[0, {Length [h]-!}] ; Map [ (Reverse [h].#)&, Partition [Join [pred, s, postd], Length [h], 1]] The input convolvel [hi, si] // TableForm reproduces the same result as in Out [3]. This list manipulation version is much faster. Let hr be a list of 2^ 32 floating numbers, and sr of 2^ = 4096 floating numbers, generated by Random function. We have the following results. In [6] := convolve? [hr, sr] ; // Timing Out [6] = { Second, Null} In [7] := convolvel [hr, sr] ; // Timing Out [7] = { Second, Null} The Compile function of Mathematica 3.0 can deal with lists. When this is used in our case, the reduction of CPU time is remarkable. In [8]:= conv= Compile [{{h,.real, 1}, {s,.real, 1}}, Map [(Reverse [h].#)&, Partition[s, Length [h], 1]]] Out [8] = CompiledFunct ion [{h,s}, (Reverse [h].#!&)/ Part it ion [s, Length [h],1], -CompiledCode-] In [9] := convolvec [h.list, s.list] := Module [{pred, postd}, pred=table [0, {Length [h] -1}] ;

7 postd=table[0, {Length [h]-1}]; conv[h,join[pred, s, postd]] Innovation In Mathematics 433 In[10]:= convolvec[hr, sr]; // Timing Out[10]= {0.25 Second, Null} This is 6.53 times faster than the uncompiled counterpart. In fact, the function Convolve of the Wavelet package has an option Compiled, which is set True by default. While Convolve normally uses the compiled code, it switches to the convolvel version when Compiled is set to False. This is convenient when dealing with exact numbers, such as the filter coefficients of D4, the Dabuechiesfilterwith 2 vanishing moments. Although the polynomial version convolve? is not fast, it is useful when we try to define and check the functionality of various types of boundary conditions, such as periodic or reflected [10]. 5 Wavelet Analysis Palettes The functions defined in the Wavelet package are relatively low-level, and one may want to define higher level functions for easy use. For example, a plot of the Daubechies scaling function with two vanishing moments is generated by ListPlot with ScalingSample in its argument. Thus, we may define a function for this purpose. In[11]:= ScalingFunctionPlot[name_] := ListPlot[ScalingSample[{0, {!}}, SampleLevel->4, ->name], PlotJoined->True] Then the input In [12]:= ScalingFunctionPlot[Daubechies2]; produces the plot shown in Fig. 1 (left). Fig. 1 The Daubechies scaling function and It will be more convenient to make a palette with the buttons contain templates of such functions. An example is shown in Fig. 2. It is even possible

8 434 Innovation In Mathematics to give the definition as the ButtonData in the palette itself. With the palette at hand, analysis of data may be carried out without much typing. The plot in Fig. 1 (right) may be generated with the second button. ScalingFunctiQnPlotj>3 aveletplot[«] :Deco»po3eToScaling[a] :Decompo3eTo avelet[n] Fig. 2 An example of a simple Wavelet Palette 6 Conclusion and Further Developments Other features of Wavelet package not discussed here include packets, waveshrink, lifting, as well as analysis of images. They are still under development, but the new capabilities of Mathematica Version 3 is of vital importance. References [1] Chui, C. K., An Introduction to Wavelets, Academic Press, New York, [2] Daubechies, I., Ten lectures on Wavelets, SIAM, Philadelphia, [3] Sakakibara, S., Denoising with Compactly supported B-spline Wavelets, in Mathematics with Vision, Proceedings of the First International Mathematica Symposium, V. Keranen and P. Mitic, ed., Computational Mechanics Publications, [4] Sakakibara, S., Wavelet Beginner's Guide, Tokyo Denki University Press, 1995, in Japanese. [5] Bruce, A., and H.-Y. Gao, Applied Wavelet Analysis with S-PLUS, Springer, New York, [6] Misiti, M., Y. Misiti, G. Oppenheim, and J-M. Poggi, MATLAB Wavelet Toolbox, MathWorks, Inc., [7] He, Y., Mathematica Wavelet Explorer, Wolfram Research, Inc., [8] Sakakibara, S., A Practice of Data Smoothing by B-spline Wavelets, in Wavelets: Theory, Algorithms, and Applications, ed. by C. K. Chui et al, Academic Press, New York, [9] Fujiwara, Y., and J. Soda, Wavelet Analysis of One-dimensional Cosmological Density Fluctuations, Yukawa Institute Preprint, [10] Strang, G., and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996.