Wavelet bi-frames with few generators from multivariate refinable functions

Ehler, Martin; Han, Bin

doi:10.1016/j.acha.2008.04.003

Cited by 69 publications

(25 citation statements)

References 20 publications

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“…For dimension 2 only a few examples of bivariate Hermite interpolatory masks are known, for example see [39,. In fact, even the seemingly simpler case of constructing multivariate multiwavelet frames derived from a single refinable function with a number of vanishing moments is a rather complicated task, see [23,58]. These two papers show how complicated the construction of wavelet masks from pairs of refinable multivariate functions.…”

Section: Remarkmentioning

confidence: 99%

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Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Atreas

Papadakis

Stavropoulos

2015

J Fourier Anal Appl

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In this work we prove that any pair of homogeneous dual multiwavelet frames of L 2 (R s ) constructed from a pair of refinable function vectors gives rise to a pair of nonhomogeneous dual multiwavelet frames and vice versa. We also prove that the Mixed Oblique Extension Principle characterizes dual multiwavelet frames. Our results extend recent characterizations of affine dual frames derived from scalar refinable functions obtained in [3].

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Section: Remarkmentioning

confidence: 99%

“…They are mainly used for the construction of affine dual multiwavelet systems (framelets) arising from a pair of refinable functions (e.g. see [8,15,16,18,20,22,23,32,36,55,56]). Extension Principles provide great flexibility allowing the designed framelets to often combine several desirable properties.…”

Section: Introductionmentioning

confidence: 99%

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Atreas

Papadakis

Stavropoulos

2015

J Fourier Anal Appl

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“…Recently, researchers are very interested in some types of frames, such as tight wavelet frames, dual wavelet frames and so on [2,8,4,10,6,5]. In [20], Weber proposed orthogonal wavelet frames, which are useful in multiple access communication systems and characterizations of superframes.…”

Section: Introductionmentioning

confidence: 99%

“…As a redundant wavelet system, a wavelet frame may have many desirable properties and is of interest in application such as signal processing and numerical analysis [2,8,17,18,4,10,11,6,5]. Recently, researchers are very interested in some types of frames, such as tight wavelet frames, dual wavelet frames and so on [2,8,4,10,6,5].…”

Section: Introductionmentioning

confidence: 99%

Explicit construction of symmetric orthogonal wavelet frames in L2(Rs)

Yang

2010

Journal of Approximation Theory

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Recently, some researchers propose the concept of orthogonal wavelet frames, which are useful for multiple access communication systems. In this article, we first give two explicit algorithms for constructing paraunitary symmetric matrices (p.s.m. for short), whose entries are symmetric or antisymmetric Laurent polynomials. We also give two algorithms for constructing orthogonal wavelet frames from existing tight or dual wavelet frames in L 2 (R s ). The constructed orthogonal wavelet frames are also tight or dual ones. Furthermore, based on the constructed p.s.m. and the existing symmetric tight (dual) wavelet frames, we can obtain symmetric orthogonal (s.o. for short) tight (dual) wavelet frames in L 2 (R s ). From the constructed s.o. wavelet frames in L 2 (R s ), we can obtain s.o. wavelet frames in L 2 (R m ) by the projection method, where m ≤ s. To illustrate our results, we construct s.o. wavelet frames in L 2 (R) and L 2 (R 2 ) from the quadratic B-spline B 3 (x). Especially, in Example 2, we obtain nonseparable tight 2I 2 -wavelet frames in L 2 (R 2 ) from a separable tight 2I 2 -wavelet frame constructed by tensor product.

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“…Though many particular constructions of various dual framelet filter banks with or without symmetry appeared in the literature (see [2,4,5,7,8,10,16,17,21,22] and references therein), to our best knowledge, so far there is no systematic algorithm available in the literature to construct all possible dual framelet filter banks ({ã;b 1 ,b 2 }, {a; b 1 , b 2 }) Θ with symmetry and with the shortest possible filter supports derived from any given filters a,ã, Θ with symmetry. The only method that we know so far is [17,Appendix] where a system of nonlinear equations has to be solved in order to obtain some nontrivial examples of dual framelet filter banks other than the two particular constructions in (1.10)-(1.13).…”

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confidence: 99%

Algorithm for constructing symmetric dual framelet filter banks

Han

2014

Math. Comp.

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Abstract. Dual wavelet frames and their associated dual framelet filter banks are often constructed using the oblique extension principle. In comparison with the construction of tight wavelet frames and tight framelet filter banks, it is indeed quite easy to obtain some particular examples of dual framelet filter banks with or without symmetry from any given pair of low-pass filters. However, such constructed dual framelet filter banks are often too particular to have some desirable properties such as balanced filter supports between primal and dual filters. From the point of view of both theory and application, it is important and interesting to have an algorithm which is capable of finding all possible dual framelet filter banks with symmetry and with the shortest possible filter supports from any given pair of low-pass filters with symmetry. However, to our best knowledge, this issue has not been resolved yet in the literature and one often has to solve systems of nonlinear equations to obtain nontrivial dual framelet filter banks. Given the fact that the construction of dual framelet filter banks is widely believed to be very flexible, the lack of a systematic algorithm for constructing all dual framelet filter banks in the literature is a little bit surprising to us. In this paper, by solving only small systems of linear equations, we shall completely settle this problem by introducing a step-by-step efficient algorithm to construct all possible dual framelet filter banks with or without symmetry and with the shortest possible filter supports. As a byproduct, our algorithm leads to a simple algorithm for constructing all symmetric tight framelet filter banks with two high-pass filters from a given low-pass filter with symmetry. Examples will be provided to illustrate our algorithm. To explain and to understand better our algorithm and dual framelet filter banks, we shall also discuss some properties of our algorithms and dual framelet filter banks in this paper.

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Wavelet bi-frames with few generators from multivariate refinable functions

Cited by 69 publications

References 20 publications

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Extension Principles for Dual Multiwavelet Frames of $$L_2(\mathbb {R}^s)$$ L 2 ( R s ) constructed from Multirefinable Generators

Explicit construction of symmetric orthogonal wavelet frames in L2(Rs)

Algorithm for constructing symmetric dual framelet filter banks

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