Preconditioning filter bank decompositions using structured normalized tight frames

TL;DR

This paper introduces a method to convert arbitrary filter banks into unitary, perfect-reconstruction filter banks with equal-norm filters using structured normalized tight frames, preserving essential features and enabling efficient computation.

Contribution

It presents a novel approach to precondition filter bank decompositions into tight frames with equal norms, maintaining fast algorithms and perfect reconstruction.

Findings

Achieves perfect reconstruction with normalized tight frames.

Ensures equal-norm filters for uniform energy distribution.

Preserves fast matrix-vector multiplication schemes.

Abstract

We turn a given filter bank into a filtering scheme that provides perfect reconstruction, synthesis is the adjoint of the analysis part (so-called unitary filter banks), all filters have equal norm, and the essential features of the original filter bank are preserved. Unitary filter banks providing perfect reconstruction are induced by tight generalized frames, which enable signal decomposition using a set of linear operators. If, in addition, frame elements have equal norm, then the signal energy is spread through the various filter bank channels in some uniform fashion, which is often more suitable for further signal processing. We start with a given generalized frame whose elements allow for fast matrix vector multiplication, as for instance, convolution operators, and compute a normalized tight frame, for which signal analysis and synthesis still preserve those fast algorithmic schemes.