Multiwavelet prefilters. 1. Orthogonal prefilters preserving approximation order p≤2

Hardin, Douglas P.; Roach, Dennis P.

doi:10.1109/82.718820

Cited by 79 publications

(41 citation statements)

References 3 publications

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“…The situation is very much the same as in the scalar case, when the generating function is noninterpolating (see Section III-D). Given the equidistant samples (or measurements) of a signal , the expansion coefficients are usually obtained through an appropriate digital prefiltering procedure (analysis filterbank) [54], [140], [146]. The initialization step-or prefiltering-can be avoided for the class of so-called balanced multiwavelets [71], [103].…”

Section: Finite Elements and Multiwaveletsmentioning

confidence: 99%

Sampling-50 years after Shannon

2000

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Section: Finite Elements and Multiwaveletsmentioning

confidence: 99%

Sampling-50 years after Shannon

2000

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“…Consequently, we introduced the concept of balanced multiwavelets, which is now also further investigated by several other authors [17], [27], [28]. One of the goals of this concept is to avoid the intricate steps of pre/post filtering [11], [37] that are required with systems based on multiwavelets that do not satisfy the interpolation/approximation properties of balancing. Inspired by some of the results from [4], [23], and [24], we will clarify the relations between balancing order (discrete-time property) and approximation power (continuous-time property) and prove that the notion of balancing order is truly central to the whole issue of regularity for multiwavelets.…”

Section: Introductionmentioning

confidence: 99%

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

Lebrun

Vetterli

2001

IEEE Trans. Signal Process.

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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.

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“…The indirect approach is to apply certain appropriate prefiltering to the input data sequence {x k } as well as to the low-pass output of each wavelet decomposition level to be used as input for the next level of wavelet decomposition (see [1,7,19,20]). On the other hand, the direct approach is to design Φ and Ψ so that the decomposition algorithm (1.1) ensures polynomial output {y L k } of degree K −1 (or order K) and zero output {y H k }, when the polynomial data sequences {x k } = {v s,k,m }, k ∈ Z, for 0 ≤ s ≤ r − 1 and 0 ≤ m ≤ K − 1, are used as input sequences in (1.1), where {P k }/{Q k } are the refinement (or two-scale) sequences corresponding to the orthonormal Φ and Ψ.…”

Section: Introductionmentioning

confidence: 99%

Balanced multi-wavelets in $\mathbb R^s$

Chui

Jiang

2004

Math. Comp.

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Abstract. The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K − 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K ≥ 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the Kbalancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K ≥ 1.

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Multiwavelet prefilters. 1. Orthogonal prefilters preserving approximation order p≤2

Cited by 79 publications

References 3 publications

Sampling-50 years after Shannon

Sampling-50 years after Shannon

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

Balanced multi-wavelets in $\mathbb R^s$

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