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Biorthogonal system

About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.


Papers
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Journal ArticleDOI
TL;DR: In this article, the authors show that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.
Abstract: The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel–Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V1, V2 in two different variables, these kernels may be expressed in terms of finite dimensional “windows” spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V1, V2. The vectors formed by such subsequences satisfy “dual pairs” of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V1 or V2 and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V1 and V2. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.

140 citations

Journal ArticleDOI
TL;DR: These simulations verified that both orthogonal and biorthogonal multiwavelets that possess GMPs and employ the proposed initialization technique can perform better than the popular scalar wavelets such as Daubechies'D8 wavelet and the D(9/7) wavelet, and some of these multi wavelets achieved this with lower computational complexity.
Abstract: This paper proposes a general paradigm for the analysis and application of discrete multiwavelet transforms, particularly to image compression. First, we establish the concept of an equivalent scalar (wavelet) filter bank system in which we present an equivalent and sufficient representation of a multiwavelet system of multiplicity r in terms of a set of r equivalent scalar filter banks. This relationship motivates a new measure called the good multifilter properties (GMPs), which define the desirable filter characteristics of the equivalent scalar filters. We then relate the notion of GMPs directly to the matrix filters as necessary eigenvector properties for the refinement masks of a given multiwavelet system. Second, we propose a generalized, efficient, and nonredundant framework for multiwavelet initialization by designing appropriate preanalysis and post-synthesis multirate filtering techniques. Finally, our simulations verified that both orthogonal and biorthogonal multiwavelets that possess GMPs and employ the proposed initialization technique can perform better than the popular scalar wavelets such as Daubechies'D8 wavelet and the D(9/7) wavelet, and some of these multiwavelets achieved this with lower computational complexity.

137 citations

Journal ArticleDOI
TL;DR: In this paper, a discrete-time chain associated with the generalized eigenvalue problem for two Jacobi matrices is derived, and a class of rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed.
Abstract: A discrete-time chain, associated with the generalized eigenvalue problem for two Jacobi matrices, is derived. Various discrete and continuous symmetries of this integrable equation are revealed. A class of its rational, elementary and elliptic functions solutions, appearing from a similarity reduction, are constructed. The latter lead to large families of biorthogonal rational functions based upon the very-well-poised balanced hypergeometric series of three types: the standard hypergeometric series 9 F 8, basic series 10ϕ9 and its elliptic analogue 10 E 9. For an important subclass of the elliptic biorthogonal rational functions the weight function and normalization constants are determined explicitly.

134 citations

Journal ArticleDOI
TL;DR: In this article, the authors give a formula for compactly supported biorthogonal multi-wavelets, which makes construction of compactly supportable biorghogonal uniwavelets easy like in the construction of BOWs.

116 citations

Proceedings ArticleDOI
31 Mar 1996
TL;DR: The results extend previous results since no bit allocation procedures have been proposed for biorthogonal wavelet coders which minimize the output mean square error (MSE).
Abstract: This paper shows how to derive bit allocations for biorthogonal wavelet coders which result in minimum reconstruction error. It is shown that the effect of using biorthogonal filters is to weight the amount of quantization error which appears in the reconstructed output. The amount of weighting can be determined from the biorthogonal filters used and is a measure of the closeness of the biorthogonal filters to the class of orthogonal filters. Once the weights are computed, they can be used to derive bit allocations which minimize the output mean square error (MSE). These results extend previous results since no bit allocation procedures have been proposed for biorthogonal wavelet coders which minimize the MSE.

115 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202329
2022105
202155
202058
201960