Symmetric Multivariate Wavelets

Abstract

For arbitrary matrix dilation M whose determinant is odd or equal to ±2, we describe all symmetric interpolatory masks generating dual compactly supported wavelet systems with vanishing moments up to arbitrary order n. For each such mask, we give explicit formulas for a dual refinable mask and for wavelet masks such that the corresponding wavelet functions are real and symmetric/antisymmetric. We proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0. For matrix dilations M with |det M | = 2, we also give an explicit method for construction of masks (non-interpolatory) m 0 symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n. It is proved that for some matrix dilations (in particular, for the quincunx matrix) such a mask does not have a dual mask. Some of the constructed masks were successfully applied for signal processes.

Multiresolution analysis for compactly supported interpolating tensor product wavelets

“…Krivoshein and Skopina [35] have presented convergence results for frame-like wavelets that are not frames. Karakaz'yan et al [32] have described symmetric interpolatory masks generating dual compactly supported wavelet systems and they have also given formulas for dual refinement masks. Shui et al [48] have shown how to construct M -band wavelets with all the following properties: compact support, orthogonality, linear-phase, regularity, and interpolation.…”

Symmetric interpolatory dual wavelet frames

“…Let us fix p. Then the matrix K ∈ H can be represented as K = E (n) F, where E (n) ∈ Γ <sp,0> , F ∈ H <sp,0> . In this case the map j(p, i, K) simply means i ⊕ n, since KE (i) = E (i⊕n) F, Thus, (18) can be written as…”

On construction of multivariate symmetric MRA-based wavelets

“…The order of vanishing moments is preserved. S. Karakazyan, M. Skopina, M. Tchobanou in [23] described all real interpolatory masks which are symmetric with respect to the origin and generate symmetric/antisymmetric compactly supported biorthonormal bases or dual wavelet systems, generally speaking, with vanishing moments up to arbitrary order n for matrix dilations whose determinant is odd or equal ±2.…”

“…Trigonometric polynomials Π α (ξ) have real Fourier coefficients, then (see, e.g. [23,Lemma 4]) ReΠ α (ξ) is an even function and ImΠ α (ξ) is an odd function. Therefore, it is clear that T (ξ) is an even (odd) trigonometric polynomial of semi-integer degrees associated with σ.…”

“…In recent years, methods for the construction of wavelet systems with different types of symmetry are actively developed (see [1], [3], [4], [6], [9], [10], [15], [16], [18], [19], [20], [21], [22], [23], [26], [27] and the references therein). In this paper, we study the construction of symmetric framelike wavelet systems.…”