New biorthogonal multiwavelets for image compression☆
Introduction
The study of multiwavelets as an extension from scalar wavelets has received considerable attention from the wavelets research communities both in theory [4], [8] as well as in applications such as signal compression and denoising [16], [18], [22]. It was shown in [1], [4] that symmetry, orthogonality, compact support and approximation order k>1 can be simultaneously achieved for multiwavelets although this is not possible for scalar wavelets. In our preceding paper [13], we introduced a class of symmetric-antisymmetric orthonormal multiwavelet filters (SAOMFs) which can be constructed directly in an easy manner from orthonormal scalar wavelets. Such a direct construction is made possible due to the concept of an “equivalent system of scalar filters” and the idea of “good multifilter properties” (GMPs), which we introduced earlier in [18], [19]. The first concept formulates a relationship in which any multiwavelet filter with multiplicity r can be sufficiently represented in terms of identical input-output relationships by an equivalent set of r scalar wavelet filters. The idea of GMPs provides a set of design criteria that can be imposed on the equivalent scalar filters, which in turn can be translated directly as eigenvector properties of the designed multiwavelet filters. The class of SAOMFs that possess GMPs have been shown to be useful for image compression. In addition, such a direct construction is attractive as the derived multiwavelet filters can possess linear phase which is lacking in the orthonormal scalar wavelets. By integrating the designed SAOMFs into a general framework for multiwavelet decomposition and reconstruction, we could achieve improved image compression performance with lower computational requirements.
In this paper, we will extend the study of symmetric-antisymmetric multiwavelet filters that possess GMPs to the biorthogonal case. Unlike orthonormal scalar wavelet filters, it is already possible to design biorthogonal scalar wavelets that have linear phase property. Nevertheless, the motivation of our extension work is twofold. First, we show that the proposed biorthogonal multiwavelet filters (BMFs) can provide a new decomposition and reconstruction framework for the application of even-length biorthogonal scalar wavelets. BMFs derived from existing even-length scalar wavelets can in fact produce improvement in image compression over the direct application of scalar wavelets. Second, the availability of matrix coefficients, rather than scalar coefficients, provides us with more design parameters (degrees of freedom) for designing multiwavelet filters than for designing scalar wavelet filters of the same filter lengths. For example, the length-4 scalar orthonormal wavelet D4 has a maximum vanishing moment of 2, while that for length-4 orthonormal multiwavelet of multiplicity 2 is 3 [1]. In this paper, we take advantage of these extra degrees of freedom to construct better multiwavelet filters that possess GMPs.
The rest of the paper is organised as follows. In Section 2, we begin with some relevant basic theory of BMFs with multiplicity r. In Section 3, we review the definition of GMPs in the context for constructing BMFs. In Section 4, we propose two methods for the construction of symmetric-antisymmetric BMFs (SABMFs) possessing GMPs. The first method shows that any even-length, linear-phase biorthogonal scalar wavelet filter can generate an SABMF. This also means that we can alternatively implement even-length biorthogonal scalar wavelet filters using the proposed multiwavelet framework. The second method shows that a pair of non-symmetric scalar sequences satisfying some given conditions can also generate an SABMF. Examples of SABMFs of different lengths constructed using both methods are given. In Section 5, we will describe a general framework for the application of multiwavelet filters, and address the problem of multiwavelet initialization or pre-filtering. An extensive performance analysis of the proposed families of SABMFs is carried out in Section 6. Comparisons of objective and subjective image compression performances as well as the computational complexity of various multiwavelet filters and scalar filters are made. Finally some suggestions for future research and the conclusions are drawn in Section 7.
Notation Bold-faced characters are used to denote vectors and matrices. The matrices and denote respectively the transpose and the inverse of the matrix . In addition, denotes the similarity transformation of using a transition matrix ; i.e., . Symbols and 0 denote the identity and zero matrices, respectively. For a given function will denote its Fourier transform. For brevity, we will express the eigenvector of an operator corresponding to an eigenvalue λ as the λ-eigenvector of . Throughout the paper, j will denote .
Section snippets
Preliminaries of biorthogonal multiwavelet theory
We present here only a subset of the basic theory for biorthogonal multiwavelets necessary for our exposition. For a more complete and rigorous presentation, interested readers can refer to [3], [8].
A biorthogonal multiwavelet system (BMWS) consists of two multiscaling function vectors and where r>1 is an integer. The component functions here are referred to as the multiscaling functions. In this paper, we will restrict ourself to consider only multiscaling
Design criteria for biorthogonal multiwavelet filters
For simplicity of exposition, we will focus on designing BMFs with multiplicity r=2 in this paper. Extensions to a higher multiplicity can also be carried out. In [18], we proposed a new design criterion called “good multifilter properties” (GMPs) as a useful tool for analyzing and constructing orthogonal multiwavelets. In this section, we will extend the concept of GMPs to the biorthogonal setting with the main aim of constructing some BMFs which can perform better than both popular orthogonal
Construction of SABMFs
We begin with some basic properties of SABMFs. Two methods for the construction of SABMFs will then be presented. Several examples of SABMFs with different lengths are also given.
For our following discussion, we need to refer to two special matrices:
For SABMFs, it is easy to see that is parallel to [1,0]T, and thus one can use this to determine (see GMP definition) the required orthogonal matrix for transforming to . For our purpose, we fix it as
Application of multiwavelets to image compression
This section briefly explains how the designed SABMFs can be applied for multiscale signal decomposition and reconstruction. It should be noted that Mallat's multiresolution algorithm [10] for scalar wavelets cannot be used directly for multiwavelet filters, each of which requires a vectorized input signal. The problem of obtaining the vector input streams from a given signal is known as multiwavelet initialization or pre-filtering. In [18], we proposed a framework for multiwavelet
Performance analysis of wavelet filters
In this section, we investigate the performance of various scalar and multiwavelet filters for image compression. For the purpose of fair and consistent comparisons, we have chosen one of the best wavelet-based image codecs called “set partitioning in hierarchical trees” (SPIHT), as proposed by Said and Pearlman [12]. We found that other codecs such as [17] have also demonstrated comparable relative performance improvements. Since the proposed SABMFs are symmetric or antisymmetric, it was shown
Conclusions and future research directions
We have successfully extended our earlier work on the construction of symmetric-antisymmetric orthonormal multiwavelet filters that possess good multifilter properties (GMPs) to the biorthogonal case. Two methods for the design of new families of symmetric-antisymmetric biorthogonal multiwavelet filters (SABMFs) were proposed. The first method allows direct construction from any even-length, linear-phase biorthogonal scalar wavelets. The second method gives a step-by-step procedure to construct
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This work is supported by the Wavelets Strategic Research Programme (WSRP) funded by the National Science and Technology Board and the Ministry of Education under Grant RP960 601/A.