Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions

Diao, Chenzhe; Han, Bin

doi:10.1016/j.acha.2018.12.001

Cited by 14 publications

(11 citation statements)

References 39 publications

(80 reference statements)

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“…In this paper, we are interested in quasi-tight framelets with two generators having the symmetry property. Our investigation is inspired by the recent development of quasi-tight framelets [7,8,[18][19][20] and the importance of (anti-)symmetric wavelet frames in many applications. Using Theorem 1.1, we can obtain a necessary and sufficient condition for the existence of quasi-tight framelets with symmetry.…”

Section: Introduction Motivations and Main Resultsmentioning

confidence: 99%

“…[18, Example 3.2.2] motivates all later research on quasi-tight framelets in [7,8,19,20] but all such constructed quasi-tight framelets there lack the symmetry property. As an application of our main results Theorems 1.1 and 1.3 in this paper, we now provide a few more examples here.…”

Section: Illustrative Examplesmentioning

confidence: 99%

“…This motivates us to consider quasi-tight framelets, which behave almost identical to tight framelets, but have much more flexibility and are much less difficult to construct. As shown in [7,8,19,20], quasi-tight framelets can be constructed from arbitrary compactly supported refinable (vector) functions, with the desired features (e.g., symmetry, a high order of vanishing moments, and high balancing order) being achieved. This makes quasi-tight framelets of interest in both theory and applications.…”

Section: Introduction To Quasimentioning

confidence: 99%

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Generalized Matrix Spectral Factorization with Symmetry and Applications to Symmetric Quasi-Tight Framelets

Diao¹,

Han²,

Lu³

2021

Preprint

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Factorization of matrices of Laurent polynomials plays an important role in mathematics and engineering such as wavelet frame construction and filter bank design. Wavelet frames (a.k.a. framelets) are useful in applications such as signal and image processing. Motivated by the recent development of quasi-tight framelets, we study and characterize generalized spectral factorizations with symmetry for 2 × 2 matrices of Laurent polynomials. Applying our result on generalized matrix spectral factorization, we establish a necessary and sufficient condition for the existence of symmetric quasi-tight framelets with two generators. The proofs of all our main results are constructive and therefore, one can use them as construction algorithms. We provide several examples to illustrate our theoretical results on generalized matrix spectral factorization and quasi-tight framelets with symmetry.

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Section: Introduction Motivations and Main Resultsmentioning

confidence: 99%

Section: Illustrative Examplesmentioning

confidence: 99%

Section: Introduction To Quasimentioning

confidence: 99%

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Generalized Matrix Spectral Factorization with Symmetry and Applications to Symmetric Quasi-Tight Framelets

Diao¹,

Han²,

Lu³

2021

Preprint

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“…Similar types of directional tight wavelet filter banks under a more general setting, including the case of all lowpass filters with nonnegative coefficients, are constructed in Ref. 6. Construction of various complex tight wavelet filter banks exhibiting directionality is also available in the literature.…”

Section: Introductionmentioning

confidence: 99%

Tight Wavelet Filter Banks with Prescribed Directions

Hur¹

2022

Preprint

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Constructing tight wavelet filter banks with prescribed directions is challenging. This paper presents a systematic method for designing a tight wavelet filter bank, given any prescribed directions. There are two types of wavelet filters in our tight wavelet filter bank. One type is entirely determined by the prescribed information about the directionality and makes the wavelet filter bank directional. The other type helps the wavelet filter bank to be tight. In addition to the flexibility in choosing the directions, our construction method has other useful properties. It works for any multi-dimension, and it allows the user to have any prescribed number of vanishing moments along the chosen directions. Furthermore, our tight wavelet filter banks have fast algorithms for analysis and synthesis. Concrete examples are given to illustrate our construction method and properties of resulting tight wavelet filter banks.

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“…In this paper, motivated by the recent development of directional Haar framelet systems on R d [13,22,32,35] as well as wavelet-like systems for graph signal processing [1,8,33,50,51], we focus on the development of directional multiscale representation systems for signals defined on digraphs. We are going to investigate the following two main problems: 1) How to construct directional Haar tight framelets on bounded domains with adaptivity?…”

Section: Introductionmentioning

confidence: 99%

Adaptive Directional Haar Tight Framelets on Bounded Domains for Digraph Signal Representations

Xiao

Zhuang

2021

J Fourier Anal Appl

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Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set K ⊆ R d . In particular, on the unit block [0, 1] d , such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.

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Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions

Cited by 14 publications

References 39 publications

Generalized Matrix Spectral Factorization with Symmetry and Applications to Symmetric Quasi-Tight Framelets

Generalized Matrix Spectral Factorization with Symmetry and Applications to Symmetric Quasi-Tight Framelets

Tight Wavelet Filter Banks with Prescribed Directions

Adaptive Directional Haar Tight Framelets on Bounded Domains for Digraph Signal Representations

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