A Study on Preconditioning Multiwavelet Systems for Image Compression

We present a study on applications of multiwavelet analysis to image compression, where filter coefficients form matrices. As a multiwavelet filter bank has multiple channels of inputs, we investigate the data initialization problem by considering prefilters and postfilters that may give more efficient representations of the decomposed data. The interpolation postfilter and prefilter are formulated, which are capable to provide a better approximate image at each coarser resolution level. A design process is given to obtain both filters having compact supports, if exist. Image compression performances of some multiwavelet systems are studied in comparison to those of single wavelet systems. 1 Nonorthogonal Multiwavelet Subspaces Let us define a multiresolution analysis of L(R) generated by several scaling functions, with an increasing sequence of function subspaces {Vj}j∈Z in L(R): {0} ⊂ . . . ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ . . . ⊂ L(R). (1) Subspaces Vj are generated by a set of scaling functions φ, φ, . . . , φ (namely, multiscaling functions) such that Vj := closL2(R) < φj,k : 1 ≤ m ≤ r, k ∈ Z >, ∀ j ∈ Z, (2) i.e., Vj is the closure of the linear span of {φj,k}1≤m≤r, k∈Z in L(R), where φj,k(x) := 2 j/2φm(2jx− k), ∀x ∈ R. (3) Then we have a sequence of multiresolution subspaces {Vj} generated by a set of multiscaling functions, where the resolution gets finer and finer as j increases. Let us define inter-spaces Wj ⊂ L(R) such that Vj+1 := Vj +̇Wj , ∀ j ∈ Z, where the plus sign with a dot (+̇) denotes a nonorthogonal direct sum. Wj is the complement to Vj in Vj+1, and thus Wj and Wl with j = l are disjoint but may not be orthogonal to each other. If Wj ⊥ Wl, ∀ j = l, we call them Y. Y. Tang et al. (Eds.): WAA 2001, LNCS 2251, pp. 22–36, 2001. c © Springer-Verlag Berlin Heidelberg 2001 A Study on Preconditioning Multiwavelet Systems for Image Compression 23 semi-orthogonal wavelet spaces [1]. By the nature of construction, subspaces Wj can be generated by r base functions, ψ, ψ, . . . , ψ that are multiwavelets. The subspace Wj is the closure of the linear span of {ψm j,k}1≤m≤r, k∈Z: Wj := closL2(R) < ψ j,k : 1 ≤ m ≤ r, k ∈ Z >, ∀ j ∈ Z, (4) where ψ j,k(x) := 2 j/2ψm(2jx− k), ∀x ∈ R. (5) We may express multiscaling functions and multiwavelets as vector functions: φ(x) :=   φ(x) .. φ(x)   , ψ(x) :=   ψ(x) .. ψ(x)   , ∀x ∈ R. (6) Also, in vector form, let us define φj,k(x) := 2 j/2φ(2jx− k) and ψj,k(x) := 2j/2ψ(2jx− k), ∀x ∈ R. (7) Since the multiscaling functions φ ∈ V0 and the multiwavelets ψ ∈ W0 are all in V1, and since V1 is generated by {φ1,k(x) = 21/2φm(2x−k)}1≤m≤r, k∈Z, there exist two 2 matrix sequences {Hn}n∈Z and {Gn}n∈Z such that we have a two-scale relation for the multiscaling function φ(x): φ(x) = 2 ∑ n∈Z Hnφ(2x− n), x ∈ R, (8) which is also called as a two-scale matrix refinement equation (MRE), and for multiwavelet ψ(x): ψ(x) = 2 ∑ n∈Z Gnφ(2x− n), x ∈ R, (9) where Hn and Gn are r×r square matrices. We are interested in finite sequences of Hn and Gn, namely, FIR (Finite Impulse Response) filter pairs. Using the fractal interpolation, Geronimo, Hardin, and Massopust successfully constructed a very important multiwavelet system [2,3,4] which has two orthogonal multiscaling functions and two orthogonal multiwavelets. Their four matrix coefficients Hn satisfy the MRE for a multiscaling function φ(x): H0 = " 3 10 4 √ 2 10 − √ 2 40 − 3 20 # , H1 = 3 10 0 9 √ 2 40 1 2 , H2 = 0 0 9 √ 2 40 − 3 20 , H3 = 0 0 − √ 2 40 0 , (10) and other four matrix coefficients Gn generate a multiwavelet ψ(x): G0 = " − √ 2 40 − 3 20 − 1 20 − 3 √ 2 20 # , G1 = 9 √ 2 40 − 1 2 9 20 0 , G2 = " 9 √ 2 40 − 3 20 − 9 20 3 √ 2 20 # , G3 = − √ 2 40 0 1 20 0 (11) 24 Wonkoo Kim and Ching-Chung Li 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 GHM multiscaling function 1 (a) φ -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 GHM multiscaling function 2 (b) φ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 GHM multiwavelet 1 (c) ψ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 GHM multiwavelet 2 (d) ψ Fig. 1. Geronimo-Hardin-Massopust orthogonal multiscaling functions and multiwavelets