Design of Interpolating Biorthogonal Multiwavelet Systems with Compact Support

Abstract

luozq@mcmaster.ca, and wongkm@mcmaster.ca Communicated by P. P. VaidyanathanReceived May 18, 2000; revised May 16, 2001 In this paper we characterize all totally interpolating biorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provide a design framework for corresponding compactly supported multiwavelet systems with high approximation order. In these systems, each component of the analysis and synthesis portions possesses the interpolating property. The design framework is based on scalar filter banks, and examples with approximation order two and three are provided. We show that our multiwavelet systems preserve almost all of the desirable properties of the generalized interpolating scalar wavelet systems, including the dyadic-rational nature of the filter coefficients, equality of the flatness degree of the low-pass filters and the approximation order of the corresponding functions, and equality between the uniform samples of a signal and its projection coefficients for a given scale. This last property allows us to avoid the cumbersome prefiltering associated with standard multiwavelet systems. We also show that there are no symmetric totally interpolating biorthogonal multifilter banks of multiplicity two. Finally, we point out that our design framework incorporates a simple relationship between the multiscaling functions and multiwavelets that substantially simplifies the implementation of the system. 

Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

“…In Zhang et al [35] focus on discussing interpolating biorthogonal multiwavelets of the special case s = 1, m = −1. From the above theorem and discussion to follow in the next section, we can easily find that for various values of s and m, different forms of the transferring armlet order will bring a variety of multifilter banks, which will generate biorthogonal multiwavelets with different properties.…”

“…The corresponding multiwavelet Ψ can be likewise an interpolating function, which has been discussed in [25,35,36] and [16]. Differing from the traditional wavelets, a discrete multiwavelet transform employs a multiple-input, multiple-output (MIMO) filter bank.…”

“…3 shows the corresponding different smoothness of Ψ and Ψ for different values of r and r 1 , in which stands for the smoothness of Ψ and Ψ in Example 3. This case only has different phases with the one in [35]. But different values of s and m will bring different forms of transferring armlet order, which will affect the construction of multifilter banks.…”

Approximation and Balancing Orders for Totally Interpolating Biorthogonal Multiwavelet Systems

“…If each function in a biorthogonal multiwavelet system {Φ,Φ, Ψ,Ψ} has interpolating property, then this system is said to be totally interpolating [21].…”

“…Our result is based on a different approach and can be regarded as a biorthogonal analogue. Consequently, as is mentioned in [1,11,18,19,21], a prefiltering can be avoided if a totally interpolating biorthogonal multiwavelet system satisfies suitable approximation order conditions. This paper is organized as follows: basic notations and elementary facts on biorthogonal multiwavelet systems with the interpolating property are introduced in Section 2.…”