Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks with Fast Algorithms

TL;DR

This paper introduces prime coset sum, a systematic method for constructing multi-dimensional non-separable wavelet filter banks from 1-D filters, applicable to any dimension and prime dilation, with guaranteed vanishing moments and faster algorithms.

Contribution

It presents a new prime coset sum method for multi-D wavelet construction that is systematic, general, and yields faster algorithms than existing tensor product approaches.

Findings

Applicable to any spatial dimension and prime dilation

Guarantees vanishing moments based on initial filters

Produces faster algorithms than tensor product methods

Abstract

As constructing multi-D wavelets remains a challenging problem, we propose a new method called prime coset sum to construct multi-D wavelets. Our method provides a systematic way to construct multi-D non-separable wavelet filter banks from two 1-D lowpass filters, with one of whom being interpolatory. Our method has many important features including the following: 1) it works for any spatial dimension, and any prime scalar dilation, 2) the vanishing moments of the multi-D wavelet filter banks are guaranteed by certain properties of the initial 1-D lowpass filters, and furthermore, 3) the resulting multi-D wavelet filter banks are associated with fast algorithms that are faster than the existing fast tensor product algorithms.