Symmetric wavelet tight frames with two generators

Selesnick, Ivan; Abdelnour, A. Farras

doi:10.1016/j.acha.2004.05.003

Cited by 97 publications

(45 citation statements)

References 14 publications

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“…Symmetric tight framelet filter banks with two high-pass filters have been studied in [13,16,17]. Tight framelet filter banks with or without symmetry have been constructed in [1]- [21] and references therein. This paper is largely motivated by several results in [11,13,17] to further study tight wavelet frames and their associated filter banks with symmetry and two high-pass filters.…”

mentioning

confidence: 99%

“…Tight wavelet frames and tight framelet filter banks without symmetry have been extensively studied in a lot of papers, to mention only a few here, see [2,3,4,5,8,12,18,19,20] and many papers therein. Interesting examples of real-valued tight framelet filter banks with symmetry have been obtained in [3,5,6,13,14,15,16,17,18,21].…”

Section: Introductionmentioning

confidence: 99%

“…For discussion on tight wavelet frames, see [1]- [21] and many references therein. It is interesting to point out that there is indeed a one-to-one correspondence between filter banks and frequency-based framelets in the distribution space, recently established in [10,Theorem 2].…”

Section: Introductionmentioning

confidence: 99%

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Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

Han¹

2013

Applied and Computational Harmonic Analysis

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Abstract. The oblique extension principle introduced in [3,5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two high-pass filters have been studied in [13,16,17]. Tight framelet filter banks with or without symmetry have been constructed in [1]- [21] and references therein. This paper is largely motivated by several results in [11,13,17] to further study tight wavelet frames and their associated filter banks with symmetry and two high-pass filters. Our study not only leads to a simpler algorithm for the construction of tight framelet filter banks with symmetry and two high-pass filters, but also allows us to obtain a wider class of tight wavelet frames with symmetry which are not available in the current literature. The key ingredient in our investigation is a complete characterization of splitting positive semi-definite 2 × 2 matrices of Laurent polynomials with symmetry. Several examples are provided to illustrate the results and algorithms in this paper.

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confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

Han¹

2013

Applied and Computational Harmonic Analysis

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“…However, for any given low-pass filter a and a moment correcting filter Θ, the necessary and sufficient condition in [7] is still too restrictive. As a matter of fact, there are only a handful examples of symmetric tight framelet filter banks {a; b 1 , b 2 } Θ with two high-pass filters known in the literature ( [2,3,7,9,12,13,14,15] and references therein).…”

Section: Introduction and Motivationsmentioning

confidence: 99%

Symmetric tight framelet filter banks with three high-pass filters

Han

2014

Applied and Computational Harmonic Analysis

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Abstract. In this paper we continue our investigation of symmetric tight framelet filter banks (STFFBs) with a minimum number of generators in [7]. In particular, we shall systematically study STFFBs with three high-pass filters which are derived from the oblique extension principle. To our best knowledge, except the papers [1,11], there are no other papers in the literature so far systematically studying this problem. In this paper we show that there are two different types, types I and II, of STFFBs with three high-pass filters. Then we provide a detailed analysis and a complete algorithm to obtain all type I STFFBs with three high-pass filters. Our results not only significantly generalize the results in [1,11], but also help us answer several unresolved problems on STFFBs. Based on [7], we also propose an algorithm to construct all type II STFFBs with three high-pass filters and with the shortest possible filter supports. Several examples are given to illustrate the results and algorithms in this paper.

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“…Earlier papers discussing tight frame design can be found in [11,12] where 3-channel tight frame minimum length filterbanks are obtained. Three channel symmetric TF filterbanks are discussed in [13,14,15]. Four channel symmetric TF filterbanks have been discussed in [13,16,17,18,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Symmetric tight frame wavelets with dilation factor M=4

Abdelnour

2011

Signal Processing

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In this paper we discuss a new set of symmetric tight frame wavelets with the associated filterbank outputs downsampled by four at each stage. The frames consist of seven generators obtained from the lowpass filter using spectral factorization, with the lowpass filter obtained via a simple method using Taylor polynomials. The filters are simple to construct, and offer smooth scaling functions and wavelets. Additionally, the filterbanks presented in this paper have limited redundancy while maintaining the smoothness of underlying limit functions. The filters are linear phase (symmetric), FIR, and the resulting wavelets possess vanishing moments.

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Symmetric wavelet tight frames with two generators

Cited by 97 publications

References 14 publications

Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

Symmetric tight framelet filter banks with three high-pass filters

Symmetric tight frame wavelets with dilation factor M=4

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