Application of symmetric orthogonal multiwavelets and prefilter technique for image compression

Journal of Computer Science & Technology, April, 2003 by Jiazhong Chen, Jingli Zhou, Shengsheng Yu, Qian Xiao, Jun Xu

ABSTRACT

Multiwavelets are new addition to the body of wavelet theory. There are many types of symmetric multiwavelets such as Geronimo-Hardin-Massopust (GHM) and Chui-Lian (CL) multiwavelets. However, the matrix filter generating the GHM system multiwavelets does not satisfy the symmetric property. For this reason, this paper presents a new method to construct the symmetric orthogonal matrix filter, which leads to the symmetric orthogonal multiwavelets (SOM). Moreover, we analyze the prefilter technique, corresponding to the symmetric orthogonal matrix filter, to get a good combining frequency response. To prove the good property of SOM in image compression application, we compared the compression effect with other writers' work, which was in published literature.

Key words: Image compression, Multiwavelets, Prefilter technique.

1. INTRODUCTION

Multiwavelets have several advantages in comparison with scalar wavelets. The features such as compact support, orthogonality, symmetry, and high order vanish moments are known to be important in signal processing. A scalar wavelet can not possess all these properties at the same time but multiwavelets can.

The study of multiwavelets was initiated by Goodman, Lee and Tang in [1]. Then Goodman and Lee in [2] discovered the characterization of scaling functions wavelets. In [3], Jia constructed a class of continuous orthogonal double wavelets with symmetry, short support, and orthogonality. The special case of [3] with multiplicity 2 and support [0, 2], was studied by Chui and Lian [4]. In [5], Hong and Wu constructed a class of multiwavelets with multiplicity 4 and support [0, 2]. Generally, after the presentation of prefilter technique, multiwavelets with multiplicity 2 can be applied in image compression application successfully [6][7][8].

The matrix filter generating the GHM, is not symmetric. So GHM can not solve the edge problem accurately in image coding unless the matrix filter used to do transform is symmetric. Though the matrix filter generating CL is symmetric, the construction of matrix filter is lack of universality. So we present a general method to construct a lowpass matrix filter at first. For scalar wavelets, the highpass filter is determined by an automatic way from the lowpass filter but it often fails here. The reason is that the lowpass fliter are matrices and they can not commute [9]. A new construction procedure is needed for multiwavelets. So we give some simple formulas to construct the highpass filter. For the complexity of construction the lowpass and highpass fiters, we focus only on the orthogonality but omit the good frequency response. So a prefilter technique should be presented to get a good combining frequency property of matrix filter together with prefilter.

In order to evaluate the performance of multiwavelets for image coding at low bit rate, efficient SPIHT coding of multiwavelet coefficients has been realized, accomplished with a suitable scanning strategy across scales and inside each detail subband.

2. SYMMETRIC MATRIX FILTER

Let [[phi].sub.1], [[phi].sub.2],..., [[phi].sub.r] be a multiwavelet system and compactly support [[??].sup.2]([??]) functions, and [[??].sub.0] = span{[[phi].sub.i](. - j): i=1,2,..., r, j [member of] Z}, is a close space. Then [[??].sub.0] is called a finitely generated shift in variant space. Let [([[??].sub.p]).sub.p[member of]Z] be given by [[??].sub.p] = {[phi]([2.sup.p].): [phi] [member of] [??]}. The sequence ([[??].sub.p]) is called a MRA generated by [[phi].sub.1], [[phi].sub.2], ..., [[phi].sub.r] if (a) the spaces are nested ...[[??].sub.-1] [subset] [[??].sub.0] [subset] [[??].sub.1] ..., and (b) the generator [[phi].sub.1], [[phi].sub.2], ..., [[phi].sub.r] and their integer translates form a Riesz basis for [[??].sub.0] [10]. Because of (a) and (b), we can write [[??].sub.j 1], = [[??].sub.j] [direct sum] [[??].sub.j], The space [[??].sub.0] is called multiwavelet space, and if [[psi].sub.1], [[psi].sub.2], ..., [[psi].sub.r] generate a shift-invariant basis for [[??].sub.0], then these functions are called wavelet function.

Then (a) and (b) imply that [??] = {[phi].sub.1], [[phi].sub.2], ..., [[phi].sub.r]}.sup.T] satisfies the dilation equation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

The multiwavelets [??] = [{[[psi].sub.1], [[psi].sub.2], ..., [[psi].sub.r]}.sub.T] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where [[??].sub.k] are a r x r matrix lowpass filter sequences and [[??].sub.G] are a r x r matrix highpass filter sequences. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then Eq.(1) and Eq.(2) can be formulated in the Fourier domain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

For scalar wavelets, [[??].sub.k] are determined by an automatic way from [[??].sub.k] but it often fails here. The orthogonality of [??] and [??] no longer follows identities like [[??].sub.0][[??].sub.1] = [[??].sub.1] [[??].sub.0], because [[??].sub.k] are matrices and they can not commute. Therefore a new construction procedure is needed for multiwavelets.