Haar Decomposition and Reconstruction Algorithms

Transcription

1 Jim Lambers MAT 773 Fa Semester Lecture 15 and 16 Notes These notes correspond to Sections 4.3 and 4.4 in the text. Haar Decomposition and Reconstruction Agorithms Decomposition Suppose we approximate a given function f(x) by a step function f j V j. By Theorem 9, f j has the decomposition f j = f 0 + w 0 + w w j 1, w W. The graph of each w consists of spies of width (+1). For sufficienty arge, such spies can be considered noise, so noise can be removed from f by setting these components to zero. To obtain the above decomposition, we first obtain the representation f j (x) = a φ( j x ) the {a } are the vaues of f j on each subinterva x = [ j, j ( + 1)]. From this representation, we can obtain the desired decomposition. From the graphs of φ(x) and ψ(x), it can easiy be seen that φ(x) = 1 [φ(x) + ψ(x)], φ(x 1) = 1 [φ(x) ψ(x)]. Repacing x with j 1 x yieds the foowing resut. Lemma 10 For x R, Exampe 11 Consider φ( j x) = 1 [φ(j 1 x) + ψ( j 1 x)], φ( j x 1) = 1 [φ(j 1 x) ψ( j 1 x)]. f(x) = φ(4x) + φ(4x 1) + φ(4x ) φ(4x 3). As f V, we can decompose f(x) into components from V 0, W 0 and W 1. Using Lemma 10, we proceed as foows: f(x) = φ(4x) + φ(4x 1) + φ(4x ) φ(4x 3) = [φ(x) + ψ(x)] + [φ(x) ψ(x)] + 1 [φ((x 1/)) + ψ((x 1/))] 1 [φ((x 1/)) ψ((x 1/))] = φ(x) + ψ(x 1) = φ(x) + ψ(x) + ψ(x 1). 1

2 From the preceding exampe, we now derive a genera procedure for decomposing f j. First, we decompose f into sums of even- and odd-numbered terms: f j (x) = Then, from Lemma 10, we obtain a φ( j x ) + a +1 φ( j x 1). φ( j x ) = 1 [φ(j 1 x ) + ψ( j 1 x )], φ( j x 1) = 1 [φ(j 1 x ) ψ( j 1 x )]. It foows that f j (x) = ( ) a + a +1 φ( j 1 x ) + This eads to the foowing theorem. ( a a +1 ) ψ( j 1 x ) = f j 1 + w j 1. Theorem 1 (Haar Decomposition) Let f j (x) = a j φ(j x ) V j. Then f j has a unique decomposition of the form f j = w j 1 + f j 1 w j 1 = f j 1 = ψ( j 1 x ) W j 1, a j 1 φ( j 1 x ) V j 1 with = aj aj +1, a j 1 = aj + aj 1. Once Theorem 1 is appied to f j, then it can be appied to f j 1 to obtain w j and f j, and so on unti we obtain the decomposition f j = w j 1 + w j + + w 1 + w 0 + f 0. Exampe 13 Consider a signa f(x) defined on [0, 1]. We discretize over 8 (equay spaced) nodes, with a 8 = f(/8 ), based on the observation that a mesh size of 1/ 8 seems sufficienty sma to resove the essentia features of the signa. We then have the approximation f 8 (x) = 8 1 =0 f(/ 8 )φ( 8 x ). For such a signa, the W 7 component tends to be sma, except f(x) has sharp spies of approximate width 1/ 8.

3 Reconstruction Once the signa f has been decomposed into a sum of signas from V 0 and W, 0 j, we can either denoise or compress it. To denoise, we can zero out each W component that corresponds to unwanted frequencies. To compress, we can zero out each W component that is considered sufficienty sma. In either case, it remains to reconstruct the modified signa from the remaining components, so that it has the form f(x) = a j φ(j x ). That is, the reconstructed f(x) is a step function with step height a j on the interva [/j, (+1)/ j ]. The reconstruction proceeds as foows. We begin with f(x) in the form f(x) = f 0 (x) + w 0 (x) + w 1 (x) + + w j 1 (x), w W, f 0 (x) = a 0 φ(x ) V 0, w (x) = b ψ( x ) W, = 0, 1,..., j 1. We then use the reations φ(x) = φ(x) + φ(x 1), ψ(x) = φ(x) φ(x 1), which can be seen from graphs. More generay, we have φ( j 1 x) = φ( j x) + φ( j x 1), ψ( j 1 x) = φ( j x) φ( j x 1). We then have f 0 (x) = a 1 φ(x ), a 1 = a0, a1 +1 = a0. That is, the coefficient a 0 that is the vaue of f 0 on [, + 1] is copied to the coefficients a 1, a1 =1 that are the vaues of f 0 on the subintervas [, + 1/] and [ + 1/, + 1]. Simiary, we can rewrite w 0 (x) = b0 ψ(x ) as w 0 (x) = b 1 φ(x ), b 1 = b0, b0 +1 = b0. Combining our expressions for f 0 (x) and w 0 (x) yieds f 0 (x) + w 0 (x) = a 1 φ(x ), 3

4 a 1 = a0 + b0, a1 +1 = a0 b0. Next, given w 1 (x) = b1 ψ(x ), we obtain, in a simiar fashion, f 0 (x) + w 0 (x) + w 1 (x) = a φ( x ), a = a1 + b1, a +1 = a1 b1. Continuing this process yieds the foowing theorem. Theorem 14 (Haar Reconstruction) If f = f 0 + w 0 + w w j 1, with f 0 (x) = a 0 φ(x ) V 0, w (x) = b φ( x ) W, = 0, 1,..., j 1, then f(x) = a j φ(j x ) V j, a = a 1 a +1 = a 1 + b 1, b 1. Exampe 15 Let f(x) be a signa defined on [0, 1]. Its decomposition has ony one term in V 0, because the support of φ(x) is [0, 1]. We can compress this signa by zeroing the smaest 80% of the coefficients b j, and then reconstructing the signa using the procedure outined in Theorem 14. With 80% compression, the reative L error is about 9%, as with 90% compression, the reative L error is about 18%. Fiters and Diagrams The decomposition and reconstruction agorithms can be described in terms of inear operators specificay, fiters. For decomposition, we define the sequences h and by h 1 = 1/, h 0 = 1/, 1 = 0 = 1/, with a other terms of h and equa to zero. We then define the operators H and L by H(x) = h x, L(x) = x. It foows that [H(x)] = 1 x 1 x +1, [L(x)] = 1 x + 1 x +1. We then define the downsamping operator D on by D({x } = ) = {x } =. 4

5 That is, D(x) is the sequence obtained by discarding the odd-numbered terms in x. Then, the coefficients of the decomposition can be expressed as foows: = [DH(a j )], a j 1 = [DL(a j )]. The reconstruction agorithm can be described in an anaogous manner. We define the sequences h and by h 0 = 1, h1 = 1, 0 = 1 = 1, with a other terms of h and equa to zero. As before, we define the operators H and L by It foows that H = h x, L(x) = x. [ H(x)] = x x 1, [ L(x)] = x + x 1. By anaogy with the decomposition agorithm, we define the upsamping operator U as foows: given x, { x =, [U(x)] = 0 = + 1. That is, U(x) is the sequence obtained by aternating the terms of x with zeros, in such a way that the odd-numbered terms are zero. Finay, the coefficients of the reconstruction can be expressed as foows: a j = LUa j 1 + HU. Summary 1. Sampe: We sampe f(x) at J equay spaced points to obtain the coefficients a J = f(/j ), = 0, 1,,..., J 1 and approximation f J (x) = a J φ(j x ). The parameter J shoud be chosen so that J exceeds the Nyquist rate for the signa.. Decomposition: The decomposition of f J proceeds in stages that have the form f J = w J w j 1 + f j 1, j = J, J 1,..., 1 w j 1 = f j 1 = ψ( j 1 x ), a j 1 φ( j 1 x ) with coefficients a j 1 = [DL(a j )], = [DH(a j )], the operators D, H and L are as defined previousy. The resut is the decomposition f J = w J w 1 + w 0 + f 0. 5

6 3. Processing: In terms of coefficients, scaing functions and waveets, the decomposed f J can be written as f J (x) = = J 1 w j + f 0 j=0 ( ) J 1 b j ψ(j x ) + j=0 a 0 φ(x ). Then, we can zero seect coefficients b j depending on whether we wish to denoise or compress the signa. 4. Reconstruction: Finay, the reconstructed signa has the form f J (x) = a J φ(j x ) the coefficients are obtained through the iteration a j = LUa j 1 + HU, j = 1,..., J. Exercises Chapter 4: Exercises 6, 7 6