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WaveletsEdward Aboufadel Grand Valley State University, USA aboufade@gvsu.edu \251 2001 Edward Aboufadel

This worksheet demonstrates the use of Maple for exploring the properties of the Daubechies scaling function D[4], along with the family of wavelets associated with this function. The compact support, averaging, orthogonality, and regularity properties of D[4] are explored in this worksheet, along with graphs of the mother, daughter, and son wavelets.

Technical sources for this worksheet are [AS1], [AS2], and [D].There is a growing enthusiasm about the subject of wavelets. Disciplines such as radiology, geology, computer science, music, and engineering provide a wide range of real applications of wavelets, including signal and image processing, denoising of data, and compression and retrieval of data. These applications involve creating a set of basis functions called wavelets that efficiently capture the important information of a signal or function. Generating a family of wavelets begins with a scaling function (also called the father wavelet), and representing functions or data using wavelets gives alternate and often useful ways to analyze the information.A family of wavelets form a basis of a function space and can be used to describe other functions or data. In this way, wavelets are like the sine and cosine waves used in Fourier analysis. Because sines and cosines (the basis functions of Fourier analysis) are periodic, Fourier methods provide excellent information about the big picture of functions, but not about the details. In contrast, wavelets can simultaneously supply both global and local information. This is possible because a family of wavelets is generated by both dyadic scaling and translating parent functions, as opposed to the basic scaling of sines and cosines that occur in Fourier analysis.Ingrid Daubechies proved that it was possible to create a family of wavelets where the scaling function NiMtJSRwaGlHNiMlInRH had the following properties:1. Compact support of 0 < t < 3. (That is, NiMvLSUkcGhpRzYjJSJ0RyIiIQ== if t < 0 or t > 3.)2. Averaging: The integral of NiMtJSRwaGlHNiMlInRH on the interval 0 < t < 3 is 1.3. Orthogonality: Translates NiMtJSRwaGlHNiMsJiUidEciIiImJSJtRzYjRighIiI= and NiMtJSRwaGlHNiMsJiUidEciIiImJSJtRzYjIiIjISIi are orthogonal (NiMmJSJtRzYjIiIi and NiMmJSJtRzYjIiIj are integers).4. Regularity: Constant and linear functions can be reproduced by NiMlJHBoaUc= and its translates.The scaling function is created, based on these conditions. Every scaling function satisfies a dilation equation of the formNiMvLSUkcGhpRzYjJSJ0Ry0lJFN1bUc2JComJiUiY0c2IyUia0ciIiItRiU2IywmKiYiIiNGMEYnRjBGMEYvISIiRjAvRi87LCQlKWluZmluaXR5R0Y2Rjo=where NiMmJSJjRzYjJSJrRw== are constants called refinement coefficients. It can be shown that the four conditions above lead to the following values for the refinement coefficients:NiMvJiUiY0c2IyIiISomLCYiIiJGKi0lJXNxcnRHNiMiIiRGKkYqIiIlISIi NiMvJiUiY0c2IyIiIiomLCYiIiRGJy0lJXNxcnRHNiNGKkYnRiciIiUhIiI= NiMvJiUiY0c2IyIiIyomLCYiIiQiIiItJSVzcXJ0RzYjRiohIiJGKyIiJUYv NiMvJiUiY0c2IyIiJComLCYiIiJGKi0lJXNxcnRHRiYhIiJGKiIiJUYtand the rest of the refinement coefficients are zero. Since only four of the refinement coefficients are non-zero, the scaling function is called NiMmJSJERzYjIiIl. Here is a graph of this function:restart;Load the scaling function NiMmJSJERzYjIiIl. (Note: The files wavelet.ind and wavelet.lib need to be in the Maple library folder. NiMmJSJERzYjIiIl is defined in Maple by a look-up table so that it is constant on intervals of the form [i/512, (i + 1)/512] for integers i. This leads to some numerical inaccuracies below.)with(D4wavelets); with(student);NiMmJSJERzYjIiIl is now defined in Maple as a function of one variable. plot(D4(t), t=-5..5, axes=boxed);D4(1.6); D4(2); D4(-6.2); D4(Pi/2); #Some sample values.

The compact support condition is clear from the graph. In order to investigate the other properties, we need to define the inner product of two functions:ip:= (f,g) -> evalf(simpson(f*g, t=-2..10, 3072));(Note that due to the non-symbolic nature of NiMmJSJERzYjIiIl, a numerical integration method such as Simpson's Rule is preferred. Consequently, the calculated value of the inner product will not be exact.)

Here, we compute the integral of NiMmJSJERzYjIiIl on the compact support, which should equal 1.ip(D4(t), 1);

Here, we compute the inner product of various translates to see if they are zero. Graphs of the product of the two functions are also included.ip(D4(t), D4(t-1));plot(D4(t)*D4(t-1), t=-5..5, axes=boxed);ip(D4(t+2), D4(t-2));plot(D4(t+2)*D4(t-2), t=-5..5, axes=boxed);ip(D4(t+1), D4(t+2));plot(D4(t+1)*D4(t+2), t=-5..5, axes=boxed);

The idea here is that for any function g(t) =mt + b, we can find coefficients NiMmJSJhRzYjJSJrRw== such thatNiMvLSUiZ0c2IyUidEctJSRTdW1HNiQqJiYlImFHNiMlImtHIiIiLSYlIkRHNiMiIiU2IywmRidGMEYvISIiRjAvRi87LCQlKWluZmluaXR5R0Y4Rjw=.In fact, NiMvJiUiYUc2IyUia0csKCUjbWtHIiIiJSJiR0YqKigsJiIiJEYqLSUlc3FydEc2I0YuISIiRiolIm1HRioiIiNGMkYq.As an example, consider g(t)= -2t + 5.g:= t -> -2*t+5;m:=-2; b:=5;a:= k -> m*k + b + (3-sqrt(3))*m/2;We will consider various finite sums.g_rep:= (init, fin) -> Sum(a(k)*D4(t-k), k=init..fin);g_rep(0, 2);In the following, we can see how the finite sum reproduces g(t). As the range on k expands, so does the interval in which the sum equals the linear function.plot([g_rep(0,2),g(t)], t=-5..5, color=[red, blue], axes=boxed);plot([g_rep(-1,2),g(t)], t=-5..5, color=[red, blue], axes=boxed);plot([g_rep(-1,3),g(t)], t=-5..5, color=[red, blue], axes=boxed);plot([g_rep(-2,3),g(t)], t=-5..5, color=[red, blue], axes=boxed);plot([g_rep(-3,3),g(t)], t=-5..5, color=[red, blue], axes=boxed);plot([g_rep(-4,3),g(t)], t=-5..5, color=[red, blue], axes=boxed);

Once the scaling function, or father wavelet, is defined, we can define the mother wavelet, sons, and daughters. For the NiMmJSJERzYjIiIl family, the mother wavelet NiMlJHBzaUc= is defined in terms of the scaling function byNiMsKiooLCYiIiJGJi0lJXNxcnRHNiMiIiRGJkYmLSYlIkRHNiMiIiU2IywmKiYiIiNGJiUidEdGJkYmRiYhIiJGJkYvRjVGNSooLCZGKkYmRidGJkYmLUYsNiNGMkYmRi9GNUYmKigsJkYqRiZGJ0Y1RiYtRiw2IywmRjJGJkYmRiZGJkYvRjVGNSooLCZGJkYmRidGNUYmLUYsNiMsJkYyRiZGM0YmRiZGL0Y1RiY=psiD4:= t -> -(1+sqrt(3))/4*D4(2*t-1)+(3+sqrt(3))/4*D4(2*t)-(3-sqrt(3))/4*D4(2*t+1)+ (1-sqrt(3))/4*D4(2*t+2):plot(psiD4(t), t=-2..3, axes=boxed);Note that the support of the mother wavelet is [-1, 2].The son wavelets are created by dyadic scaling and translations of the father wavelet, while the daughter wavelets are created by dyadic scaling and translations of the mother wavelet. For example:plot(D4(2*t-3), t=-1..4, title=`a first generation son`, axes=boxed);plot(psiD4(2^2*t-1), t=-1..4, title=`a second generation daughter`, axes=boxed);plot(psiD4(2^(-3)*t-1), t=0..30, title=`a great grandmother?`, axes=boxed);

The set of functions {NiMtJSRwc2lHNiMsJiomKSIiIyUibkciIiIlInRHRitGKyUia0chIiI=| n, k are integers} form a basis for NiMqJCUiTEciIiM=(R). Here is an example to demonstrate how to write a function in NiMqJCUiTEciIiM=(R) as a linear combination of wavelets. We will use a function with compact support:f:= t -> piecewise (0<t and t<=1, sin(Pi*t));plot(f(t), t=-1..2, axes=boxed);Our goal is to determine constants NiMmJSJwRzYkJSJuRyUia0c= so thatNiMvLSUiZkc2IyUidEctJSRTdW1HNiQtRik2JComJiUicEc2JCUibkclImtHIiIiLSUkcHNpRzYjLCYqJikiIiNGMUYzRidGM0YzRjIhIiJGMy9GMjssJCUpaW5maW5pdHlHRjtGPy9GMUY9These constants can be found using the Orthogonal Decomposition Theorem. p:= (n,k) -> ip(f(t), psiD4(2^n*t-k))/ip(psiD4(2^n*t-k), psiD4(2^n*t-k));We will compute several of these constants, which will take a bit of time.for n1 from -3 to 1 do for k1 from -1 to 1 dop1[n1,k1]:=p(n1,k1); print(n1, k1, p1[n1, k1]); od; od;f_rep:= t-> sum(sum(p1[n,k]*psiD4(2^n*t-k),k = -1 .. 1),n = -3..1):plot([f_rep(t),f(t)], t=-2..4, color=[red, blue], axes=boxed, thickness=2);The linear combination is beginning to capture the shape of f(t). With many more terms, we can generate a better approximation.

The D4wavelet package was written so that students could explore its properties without having to worry about creating the scaling function through a fixed point method. With this Maple package, it is easy to graph members of this family of wavelets, and to investigate various properties of the family.

[AS1] Aboufadel, E., and Schlicker, S., Discovering Wavelets, 1999, Wiley.[AS2] Aboufadel, E., and Schlicker, S., "Wavelets, Introduction", in Encyclopedia of Physical Science and Technology, 3rd edition (Robert A. Meyers, editor), 2001, Academic Press.[D] Daubechies, I., Ten Lectures on Wavelets, 1992, SIAM.

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