Symmetric multivariate wavelets

TLDR: It is proved that an interpolatory mask whose center of symmetry is different from the origin cannot generate wavelets with vanishing moments of order n > 0, and an explicit method is given for construction of masks (non-interpolatory) m0 symmetric with respect to a semi-integer point and providing vanishing moments up to arbitrary order n.

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) m0 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.