High-order balanced multiwavelets: theory, factorization, and design

Abstract

Abstract-This paper deals with multiwavelets and the different properties of approximation and smoothness associated with them. In particular, we focus on the important issue of the preservation of discrete-time polynomial signals by multifilterbanks. We introduce and detail the property of balancing for higher degree discrete-time polynomial signals and link it to a very natural factorization of the refinement mask of the lowpass synthesis multifilter. This factorization turns out to be the counterpart for multiwavelets of the well-known zeros at condition in the usual (scalar) wavelet framework. The property of balancing also proves to be central to the different issues of the preservation of smooth signals by multifilterbanks, the approximation power of finitely generated multiresolution analyses, and the smoothness of the multiscaling functions and multiwavelets. Using these new results, we describe the construction of a family of orthogonal multiwavelets with symmetries and compact support that is indexed by increasing order of balancing. In addition, we also detail, for any given balancing order, the orthogonal multiwavelets with minimum-length multifilters.

Symbolic Computation for Moments and Filter Coefficients of Scaling Functions

“…Here we calculate infinitely many solutions, which are due to the additional degree of freedom imposed by neglecting a vanishing moment condition. Applications of Gröbner bases to the design of wavelets and digital filters are for example described in Chyzak et al [8], Lebrun & Selesnick [18], Lebrun & Vetterli [19] and Selesnick & Burrus [25]. Gröbner bases were introduced by Buchberger in [4] and [5].…”

High-order balanced M-band orthogonal multiwavelet: Construction and application

“…Accordingly, the Gröbner base algorithms extend Gaussian elimination to multivariate polynomial systems, and offer an efficient way to obtain solutions, since the equation system is always too complicated to be solved directly. Lebrun ( , 2001 and Selesnick (1998Selesnick ( , 1999Selesnick ( , 2000 used Singular software to carry out the Gröbner base computation in designing wavelets and multiwavelet systems, and presented the detailed procedures and complete programs in their homepages. We analyzed the procedures and improved these programs according to the particular properties of M-band multiwavelet system, and finally constructed three families of multiwavelets in the similar way.…”

“…Meanwhile, researchers have designed strategies for the problem, such as to intricately preprocess discrete-time data, or to particularly transform multiwavelet basis, but they always destroy certain pretty properties that the multiwavelet basis originally owns, such as symmetry ororthogonality, for example (Lebrun and Vetterli, 2001). Besides these, directly constructing specialized balanced multiwavelets is another way for the problem.…”

“…In this study, three families of high-order balanced M-band multiwavelets were constructed by using the Wang et al 1885 Gröbner base technique, inspired by the construction scheme of Selesnick (1998Selesnick ( , 1999Selesnick ( , 2000 and Lebrun ( , 2001) used in their two-band counterpart construction. The article was organized as follows; the fundamental theories on the key properties of M-band multiwavelets were first studied, such as orthogonality, symmetry, flipping and balancing as well.…”

High-order balanced M-band multiwavelet packet transform-based remote sensing image denoising

“…The balanced order ρ of a multiwavelet system corresponds to its ability to represent images sparsely [17,18]. Recent studies show that the 2-band MWTs based on balanced (ρ = 1) multiwavelets (i.e., balanced 2-band MWT, as shown in Fig.…”