TY - JOUR

T1 - Symmetric multivariate wavelets

AU - Karakaz'yan, S.

AU - Skopina, M.

AU - Tchobanou, M.

N1 - Funding Information: This paper is supported by RFBR grant, project N 09-01-00162, by DFG grant, project 436 RUS 113/951 and by a common grant of RFBR and JSPS, project # 06-07-91751-YaF a.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Interpolatory mask

KW - Matrix dilation

KW - Symmetric/antisymmetric wavelet function

KW - Unitary Extension Principle

KW - Wavelet system

UR - http://www.scopus.com/inward/record.url?scp=66349129813&partnerID=8YFLogxK

U2 - 10.1142/S0219691309002921

DO - 10.1142/S0219691309002921

M3 - Article

AN - SCOPUS:66349129813

VL - 7

SP - 313

EP - 340

JO - International Journal of Wavelets, Multiresolution and Information Processing

JF - International Journal of Wavelets, Multiresolution and Information Processing

SN - 0219-6913

IS - 3

ER -