Biorthogonal Multiwavelets with Sampling Property and Application in Image Compression

Abstract

This paper discusses biorthogonal multiwavelets with sampling property. In such systems, vector-valued refinable functions act as the sinc function in the Shannon sampling theorem, and their corresponding matrix-valued masks possess a special structure. In particular, for the multiplicity r = 2, a biorthogonal multifilter bank can be reduced to two scalar-valued filters. Moreover, if the vector-valued scaling functions are interpolating, three different concepts: balancing order, approximation order and analysis-ready order, will be shown to be equivalent. Based on this result, we introduce the transferring armlet order for constructing biorthogonal balanced multiwavelets with sampling property. Also, some balanced biorthogonal multiwavelets will be obtained. Finally, application of biorthogonal interpolating multiwavelets in image compression is discussed. Experiments show that for the same length, the biorthogonal multifilter bank is superior to the orthogonal case. Moreover, certain biorthogonal interpolating multiwavelets are also better than the classical Daubechies wavelets.

A Brief Survey of the Graph Wavelet Frame

“…e related theory of compactly supported functions with good vanishing moments can be found in references [9,51]. Moreover, some scholars have also studied the construction of multiwavelet and multiwavelet frames [52][53][54][55][56]. Based on two-direction refinable functions, Li and Yang [52] studied the construction of dual multiwavelet frames with symmetry and discussed the vanishing moment of the constructed multiwavelet frames.…”

“…For OEP-based tight framelets, Han [54] also extended to high dimensional case and constructed compactly supported tight M-wavelet frames from compactly supported M-refinable functions for any d × d dilation matrix M. Based on the approach of polyphase matrix extension of multiscaling vectors, Cen et al [55] presented a novel approach for the construction of symmetric compactly supported bi-orthogonal multiwavelets with multiplicity of 2. Li and Peng [56] analyzed the sampling property of biorthogonal multiwavelets and discussed the application of bi-orthogonal multiwavelets in image compression filter bank. In addition, some researchers investigated the construction and application of multivariate wavelet frames.…”