Haar transform
0.1. Haar wavelets on R.
Definition 0.1. Let j, k Z. Define the interval I to be ∈ j,k −j −j Ij,k = [2 k, 2 (k + 1)).
We say that (j, k) =(j′, k′) if either j = j′ or k = k′. 6 6 6 Lemma 0.2. Let j, j′,k,k′ Z be such that (j, k) = (j′, k′). Then, one of the ∈ 6 following is true:
(i) I I ′ ′ = , or j,k ∩ j ,k ∅ (ii) I I ′ ′ , or j,k ⊂ j ,k (iii) I ′ ′ I . j ,k ⊂ j,k Given the integers j, k Z and the interval I , we define Il (resp., Ir ) to be the ∈ j,k j,k j,k left half of Ij,k (resp., the right half of Ij,k). Thus,
l −j −j −j−1 Ij,k = [2 k, 2 k + 2 ),
r −j −j−1 −j Ij,k = [2 k + 2 , 2 (k + 1)). It is not difficult to see that:
l r Ij,k = Ij+1,2k and Ij,k = Ij+1,2k+1.
With this observation in mind, we have the following lemma.
′ ′ ′ ′ Lemma 0.3. Let j, j ,k,k Z be such that (j, k) =(j , k ). If I I ′ ′ then ∈ 6 j,k ⊂ j ,k l (i) either I I ′ ′ j,k ⊂ j ,k r (ii) or I I ′ ′ . j,k ⊂ j ,k Definition 0.4. The system of Haar scaling functions is defined as follows:
Let χ[0,1) be the characteristic function of interval [0, 1), i.e.,
1 x [0, 1), χ (x)= ∈ [0,1) 0 otherwise. 1 Let p(x)= χ (x), let j, k Z, and let [0,1) ∈ j/2 j p (x) = 2 p(2 x k)= D j T (p)(x), j,k − 2 k 2 where Da is the L -dilation by a: Da(f) = √af(ax) (this means that for all finite energy functions f, f = D (f) ), and T is a translation by b (T also has this k k2 k a k2 b b property: f = T (f) ). k k2 k b k2 Lemma 0.5. For all j, k Z, ∈
pj,k = χIj,k .
It is not difficult to observe that for all j, k Z: ∈ −j/2 pj,k(x) dx = 2 R Z and
2 pj,k(x) dx = 1. R | | Z Definition 0.6. The Haar wavelet system on R is defined as follows: Let h(x)= χ (x) χ (x), let j, k Z, and let [0,1/2) − [1/2,1) ∈ j/2 j h (x) = 2 h(2 x k)= D j T (h)(x). j,k − 2 k For each fixed J Z, we refer to p ,h : k Z, j J as the Haar wavelet ∈ { J,k j,k ∈ ≥ } system of scale J.
Clearly, we have
j/2 r hj,k = 2 χIl χ . j,k − j,k Moreover, for each j, k Z, we have ∈
hj,k(x) dx = 0 R Z and
2 hj,k(x) dx = 1. R | | Z 2 l r Also, because of our observations about Ij,k and Ij,k, the Haar wavelets and scaling functions satisfy the following relations:
1 pl,k = (pl+1,2k + pl+1,2k+1) √2 and 1 hl,k = (pl+1,2k pl+1,2k+1). √2 −
Theorem 0.7. The Haar wavelet system h : j, k Z on R is an orthonormal { j,k ∈ } basis for L2(R).
We shall only look at the proof of orthonormality of the Haar system on R, leaving the completeness part for Advanced Calculus classes. ′ First, fix j Z and suppose that k = k . We have that h (x)h ′ (x) = 0 for all ∈ 6 j,k j,k x R, since the function h assumes non-zero values only in the interval I , and ∈ j,k j,k for k = k′, 6 I I ′ = . j,k ∩ j,k ∅ If k = k′, then
2
l Case I I ′ ′ implies that whenever h is non-zero, h ′ ′ is constantly equal to j,k ⊂ j ,k j,k j ,k 1. Thus,
Theorem 0.8. The Haar wavelet systems of scale J, p ,h : k Z, j J , on R { J,k j,k ∈ ≥ } is an orthonormal basis for l2(R).
(The proof of this result is similar to the previous one.)
0.2. Haar wavelets on [0, 1]. Fix an integer J 0. The Haar wavelet system of ≥ scale J on [0, 1] is defined as
p : 0 k 2J 1 h : j J, 0 k 2j 1 . { J,k ≤ ≤ − } ∪ { j,k ≥ ≤ ≤ − } When J = 0 we refer to this system simply as the Haar wavelet system on [0, 1]. Here, the choice of k’s and the assumption about J 0 are necessary so that the ≥ system we have created is a collection of functions which are non-zero only in the interval [0, 1]. Remark. Note that each and every Haar system on [0, 1] consists of both Haar wavelet functions and Haar scaling functions. This is to compensate the fact that we have restricted the set of possible parameters j, k.
Theorem 0.9. The Haar wavelet system of scale J on [0, 1] is an orthonormal basis on [0, 1].
0.3. Discrete Haar transform. Fix N > 0 and let c0(k) =< f,pN,k >, k = 0,..., 2N 1. This is our starting finite sequence of length 2N . One may think − of this sequence as of a finite approximation to a given signal f of length 1. We fix J > 0 and for each 1 j J we define the coefficients: ≤ ≤
cj(k)=< f,pN−j,k > 4 and
dj(k)=< f,hN−j,k > .
Because of the relationship that holds together Haar wavelets and Haar scaling func- tions (stated in these notes before Theorem 0.7), the following formulas hold for the coefficients: 1 cj(k)= (cj−1(2k)+ cj−1(2k + 1)) √2 and 1 dj(k)= (cj−1(2k) cj−1(2k + 1)). √2 − These equations can be rewritten equivalently in the matrix-vector form:
cj(k) 1 1 1 cj−1(2k) = . dj(k)! √2 1 1! cj−1(2k + 1)! − By taking the inverse of the above matrix (which has a non-zero determinant, and so, is invertible), we can re-write the matrix-vector equation above as follows:
cj−1(2k) 1 1 1 cj(k) = . cj−1(2k + 1)! √2 1 1! dj(k)! − Definition 0.10. Given J d (k) : 1 j J; 0 k 2N−j 1 c (k) : 0 k 2N−j 1 , { j ≤ ≤ ≤ ≤ − } ∪ { J ≤ ≤ − } where we use the following formulas to compute cj’s and dj’s: 1 cj(k)= (cj−1(2k)+ cj−1(2k + 1)) √2 and 1 dj(k)= (cj−1(2k) cj−1(2k + 1)). √2 − 5 Define the following auxiliary l/2 L matrices: × 1 1 00 . . . 0 1 0 0 110 . . . H = √2 . . . 0 ...... 0 1 1 and 1 1 0 0 . . . 0 − 1 0 0 1 1 0 . . . G = − √2 . . . 0 ...... 0 1 1 − In the matrix-vector notation, the DHTJ produces the vector (d1, d2, ... dJ , cJ ) where each d is a vector of length 2N−j, 1 j J, and c is a vector of length 2N−J , j ≤ ≤ J and where cj H = cj−1. dj! G! Here, H and G are 2N−j−2 2N−j−1 matrices, and × In the extreme case when J = N 1, d and c are both vectors of length 2. − J J The inverse of DHTJ is computed by applying the formula: ∗ ∗ cj−1 = H (cj)+ G (dj), up to j = 1. This is related to the fact that the matrix 1 1 1 √2 1 1! − is its own inverse; hence, the inverse of H G! is (H∗,G∗). 6