Composite Wavelet Matrix-Based Transforms and Applications

TL;DR

This paper introduces composite wavelet-like transforms that enhance energy concentration and sparsity over traditional wavelets, leading to improved denoising performance in signals and images.

Contribution

It develops new invertible, stable transforms from combinations of orthogonal wavelet matrices, surpassing classical wavelet filterbanks in sparsity and denoising effectiveness.

Findings

Composite transforms induce stronger signal energy concentration.

They achieve lower mean-squared error in denoising tasks.

Applications show better structure preservation and noise suppression.

Abstract

Orthogonal wavelet transforms are a cornerstone of modern signal and image denoising because they combine multiscale representation, energy preservation, and perfect reconstruction. In this paper, we show that these advantages can be retained and substantially enhanced by moving beyond classical single-basis wavelet filterbanks to a broader class of composite wavelet-like matrices. By combining orthogonal wavelet matrices through products, Kronecker products, and block-diagonal constructions, we obtain new unitary transforms that generally fall outside the strict wavelet filterbank class, yet remain fully invertible and numerically stable. The central finding is that such composite transforms induce stronger concentration of signal energy into fewer coefficients than conventional wavelets. This increased sparsity, quantified using Lorenz curve diagnostics, directly translates into improved denoising under identical thresholding rules. Extensive simulations on Donoho-Johnstone benchmark signals, complex-valued unitary examples, and adaptive block constructions demonstrate consistent reductions in mean-squared error relative to single-basis transforms. Applications to atmospheric turbulence measurements and image denoising of the Barbara benchmark further confirm that composite transforms better preserve salient structures while suppressing noise.