Optimal construction of compactly-supported multidimensional wavelets

The wavelet construction from a multiresolution generated by a finite number of compactly supported scaling functions in any dimension can be reduced to the problem of extending a matrix with Laurent polynomial entries. As the extended matrix is not unique, one can consider the set of all possible extensions which produces a design space or parametrization for wavelet construction. The paper aims to clarify the process of obtaining such a design space and subsequently optimizing the wavelet construction with respect to certain design goals (e.g., frequency response, regularity, linear phase, etc.). The method relies on Grobner basis computation to solve the algebraic relations produced during the process. A conjecture is proposed regarding the feasibility of paraunitary matrix completion.