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1 Why Are Wavelets Useful? CS448: Topics in Computer Graphics Lecture 1 Mathematical Mo dels for Computer Graphics Stanford University Thursday, 2 Octob er 1997 Intro duction to Wavelets Lecture 1: Tuesday, 30 Septemb er 1997 Lecturer: Denis Zorin Scrib e: John Owens 1 Why are wavelets useful? Wavelets have a few interesting applications, some of which are mentioned b elow. How- ever, the applications of wavelets by themselves are limited. The ideas b ehind wavelets, whichwe will b e covering in this lecture and future lectures, are more imp ortant. The most common use of wavelets is in signal pro cessing applications. For example: Compression applications. If we can create a suitable representation of a signal, we can discard the \least signi cant" pieces of that representation and thus keep the original signal largely intact. This requires a transformation which separates the \imp ortant" parts of the signal from less imp ortant parts. In the simplest case, we can decomp ose a signal into two parts: a low frequency part, which is some sort of average of the original signal, and a high frequency part, which is what remains after the low frequency part is subtracted from the original signal. If we are interested in the low frequency part and hence discard the high frequency part, what remains is a smo other representation of the original signal with its low frequency comp onents intact. Alternatively,ifwe are most interested in the high frequency part, wemay b e able to discard the low frequency part instead. This approach, that of decomp osing a signal into two parts, is common for all wavelets. Also fundamental to wavelet analysis is a heirarchical decomp osition, in whichwemay apply further transforms to an already decomp osed signal. Edge detection. With this application it is most imp ortant to identify the areas in which the input image changes quickly.We can discard the smo oth low frequency parts. The simplest wavelet basis, the Haar basis to b e discussed later is suitable for this application. Along this vein, the b o ok by Strang and Nguyen describ es a widely used application of wavelets, ngerprint compression, in which edge detection gures prominently. Graphics. Two prominent uses of wavelets in graphics include 2 CS448: Lecture 1 1. Curve and surface representations; and 2. Wavelet radiosity. These two re ect two quite di erent uses of wavelets. Numerical analysis. Wavelets are used in the solution of partial di erential equa- tions and integral equations. 2 History of Wavelets The rst use of wavelets was by Haar in 1909. He was interested in nding a basis on a functional space similar to Fourier's basis in frequency space. In physics, wavelets were used in the characterization of Brownian motion. This work led to some of the ideas used to construct wavelet bases. Wavelets were also used for analysis of coherent states of a particular quantum system. Finally, in the signal pro cessing eld, S. Mallat discovered that lter banks have imp ortant connections with wavelet basis functions. 3 Filters and Filter Banks 3.1 Linearity and Time Invariance Consider a discrete input signal xn, a lter H , and an output y n. We express the op eration of H on the input signal x as y = Hx. We call the lter H \linear" if scaling the input scales the output, and we call H \time invariant" if shifting the input in time corresp ondingly shifts the output. In these notes all lters will b e assumed to b e linear and time invariant. 3.2 Filter op eration If H is linear and time invariant, the we can express its op eration as follows: X hk xn k : y n= k This op eration is called a convolution. The individual co ecients hi are the \im- pulse resp onses" of the system. The equation y = Hx can also b e written as an in nite matrix, with y and x column vectors and H in the following form: 2 3 6 7 h1 h0 h1 6 7 6 7 4 5 h1 h0 h1 CS448: Lecture 1 3 Note that the entries of the lter are in reverse indexed order. If the input stream is nite with n elements, H is a nite n n matrix. 3.3 What is a basis? For a given space of functions, a frame is a collection of functions such that any function in the space is a weighted sum of the functions in the frame. In other words, if the P functions in the frame are f :::f :::, then any function g can b e written as g = a f . 0 n i i i A basis is also a collection of functions. Any basis is a frame, but a basis also has the prop erty of linear indep endence. With a basis, the co ecients a of the expansion of the i P function g written as a weighted sum of the basis functions g = a f are uniquely i i n determined. Another way to state this prop erty is that no function f in the basis can i b e written as a weighted sum of the other functions of the basis. For a function space we can de ne an inner pro duct. One example of an inner pro duct is a dot pro duct, used in a vector space. In a function space we de ne the inner pro duct R <f;g> of two functions f and g as f tg tdt. One desirable prop erty of a basis is orthogonality. With an orthogonal basis, the inner pro duct of two basis functions f and f is equal to zero if i 6= j: i j A second desirable prop erty is orthonormality, which implies that taking the inner pro duct of a basis function f with itself equals 1. For our functional basis we see that i R 2 f tdt =1. In summary, an orthogonal, orthonormal basis implies that 1 i = j <f ;f >= i j 0 i 6= j So why is orthonormality a desired prop erty? Let us de ne a function x as the P weighted sum of the basis functions, x = a f .Wewould like to nd the co ecients i i a , and orthonormality makes this simple. We only need to take the inner pro duct of the i function x with a 's asso ciated basis function to nd the asso ciated co ecient a . i i X <x;f > = a <f;f > j i i j i = a <f ;f > j j j = a j 3.4 Filters and wavelets, Haar lters Eachwavelet basis has two lters asso ciated with it. In general those lters are expressed P in the form y n= hnxn. The Haar basis is particularly simple. The two lters are H and H and are de ned 0 1 as follows: 1 1 xn+ xn 1 H : y n= 0 2 2 4 CS448: Lecture 1 1 1 xn xn 1 H : y n= 1 2 2 For H , all h co ecients are zero except for h 0=1=2 and h 1=1=2. Similarly,in 0 0 0 H , h 0=1=2 and h 1 = 1=2. 1 1 1 H computes a moving average of its input, resulting in a sequence which is smo other 0 than the initial sequence. Hence it is a low pass lter. H computes a moving di erence 1 and serves as a high pass lter. 3.5 Scaling functions and the dilation equation We de ne a function as a \scaling function". Scaling functions ob ey the dilation equation, X t= 2 h k 2t k : 1 0 Note that the lter co ecients used the h 's are from the rst of the pair of lters. We 0 will use the second lter later. If this equation has an appropriate solution t, then we can construct a frame using anyintegral values of j or k in the following generating relation: j 2 t k : 2 Next weintro duce the wavelet equation 3, which uses the co ecients from the second pair of lters: X w t=2 h k 2t k : 3 1 We generate a wavelet basis with a similar generating relation j w 2 t k : Note that scaling functions don't form a basis; wavelets do. But wavelets are con- structed from scaling functions, so wehave to nd a scaling function rst. As can b e seen from the wavelet equation, wavelets are de ned as linear combinations between scaling functions one level b elow, where a \level" is de ned as the set of all scaling functions generated with a given j . For instance, the Haar scaling function satis es the dilation equation with the solution t= 2t+2t 1: And the Haar wavelet satis es the wavelet equation with the solution w t= 2t 2t 1: The Haar basis spans the space of all functions with integrable square. Haar wavelets are orthogonal. CS448: Lecture 1 5 4 Filters and Filter Banks 4.1 De nition of several op erators Here we will de ne four op erators and describ e their op eration Figure 1. H0 2 H1 2 Figure 1: Our 4 op erators The rst op erator, H = H , is the Haar moving average lter. It outputs the average 0 of its current input and its previous input. The second op erator, H , is the Haar moving di erence lter. It outputs half the 1 di erence b etween its current input and its previous input.
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