On construction of multivariate symmetric MRA-based wavelets

Krivoshein, A.

doi:10.1016/j.acha.2013.04.001

Cited by 10 publications

(5 citation statements)

References 32 publications

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“…. , r, r ≥ m − 1, defined by (20), the V M n property for the corresponding wavelet system is equivalent to the fact that the wavelet masks m ν , ν = 1, . .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…In [18] M. Skopina and co-authors suggest a method for the construction of refinable interpolatory masks and wavelet masks which are point-symmetric (point-antisymmetric) for several dilation matrices. In [20] the point and axial symmetric frame-like wavelet systems were constructed. The aim of this paper is to extend the results in [18] and to generalize the method in [20] on any symmetry group and any appropriate dilation matrix in order to construct the H-symmetric dual wavelet frames generated by interpolatory refinable masks.…”

Section: Introductionmentioning

confidence: 99%

“…In [20] the point and axial symmetric frame-like wavelet systems were constructed. The aim of this paper is to extend the results in [18] and to generalize the method in [20] on any symmetry group and any appropriate dilation matrix in order to construct the H-symmetric dual wavelet frames generated by interpolatory refinable masks.…”

Section: Introductionmentioning

confidence: 99%

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Symmetric interpolatory dual wavelet frames

Krivoshein

2017

St. Petersburg Math. J.

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For any symmetry group H and any appropriate matrix dilation we give an explicit method for the construction of H-symmetric refinable interpolatory refinable masks which satisfy sum rule of arbitrary order n. For each such mask we give an explicit technique for the construction of dual wavelet frames such that the corresponding wavelet masks are mutually symmetric and have the vanishing moments up to the order n. For an abelian symmetry group H we modify the technique such that each constructed wavelet mask is H-symmetric.

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“…. , r, r ≥ m − 1, defined by (20), the V M n property for the corresponding wavelet system is equivalent to the fact that the wavelet masks m ν , ν = 1, . .…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric interpolatory dual wavelet frames

Krivoshein

2017

St. Petersburg Math. J.

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“…With a more general dilation matrix, Skopina [46] (see also [47], [45]) describes an algorithm to construct compactly supported wavelet frames with vanishing moments. In Krivoshein [30], wavelet frame systems providing any desired approximation order are constructed for any matrix dilation.…”

Section: Introductionmentioning

confidence: 99%

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Antolín

Zalik

2015

J Fourier Anal Appl

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Abstract. Let A ∈ R d×d , d ≥ 1 be a dilation matrix with integer entries and | det A| = 2. We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is 2 d + d and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case d = 2 and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values.

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“…Additionally, the canonical form of matrix mask (see, for example, [34]) can be used for the construction of symmetric matrix masks starting from any appropriate refinable mask in the scalar case (see, e.g., [22] for details). For instance, any known interpolating symmetric refinable masks can be used (see, for example, [22,27,[35][36][37] for examples of such masks and for methods of their construction). This provides the initial step in the construction of symmetric multiwavelets.…”

Section: This Is Equivalent Tomentioning

confidence: 99%

Multivariate Symmetric Interpolating Dual Multiwavelet Frames

Krivoshein

2022

Symmetry

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The construction of symmetric multiwavelets in the multivariate case with useful in applications properties is a challenging task, mainly due to the complexity of the matrix extension problem. Nevertheless, for the interpolating case, a general technique can be developed. For an appropriate pair of symmetry group H and matrix dilation M and for a given H-symmetric interpolating refinable matrix mask, a method for the construction of H-symmetric dual refinable matrix masks with a preassigned order of sum rule is suggested. Wavelet matrix masks are constructed using a certain explicit matrix extension algorithm, and their symmetry properties are studied via its polyphase components. The resulting multiwavelet systems form dual multiwavelet frames, where wavelet functions have symmetry properties, preassigned order of vanishing moments and preassigned order of the balancing property. Several examples are presented.

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On construction of multivariate symmetric MRA-based wavelets

Cited by 10 publications

References 32 publications

Symmetric interpolatory dual wavelet frames

Symmetric interpolatory dual wavelet frames

Some Smooth Compactly Supported Tight Wavelet Frames with Vanishing Moments

Multivariate Symmetric Interpolating Dual Multiwavelet Frames

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