Coset Sum: an alternative to the tensor product in wavelet construction

TL;DR

This paper introduces the coset sum, an alternative to tensor product, for constructing multivariate wavelet systems from univariate masks, offering similar properties and potential computational advantages.

Contribution

The paper proposes the coset sum method as a new approach for multivariate wavelet construction, preserving key features of tensor products while enabling faster algorithms in some cases.

Findings

Coset sum preserves biorthogonality and accuracy of univariate masks.

It can lead to faster wavelet algorithms compared to tensor products.

Experimental results demonstrate effectiveness on 2-D images.

Abstract

A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in Multiresolution Analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: (1) it preserves the biorthogonality of univariate refinement masks, (2) it preserves the accuracy number of the univariate refinement mask, and (3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.