Composite Wavelet Matrix-Based Transforms and Applications

2026-03-02

AI Summary

This work addresses the limited sparsity of traditional orthogonal wavelet transforms—despite their multiscale representation and energy conservation properties—which constrains their denoising performance. Departing from the classical wavelet filter bank framework, the paper proposes a composite construction method based on orthogonal wavelet matrices, leveraging matrix products, Kronecker products, and block-diagonal structures to design novel invertible, numerically stable, unitary-like transforms. These transforms preserve perfect reconstruction while substantially enhancing coefficient sparsity, as quantified by the Lorenz curve. Experimental results demonstrate consistently lower mean squared error across diverse datasets, including Donoho–Johnstone test signals, Barbara images, and atmospheric turbulence measurements, indicating superior noise suppression with improved preservation of intrinsic signal structure.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• wavelet transforms
• signal denoising
• sparsity
• energy concentration
• composite transforms

Innovation

• composite wavelet transforms
• sparsity enhancement
• unitary matrix constructions
• signal denoising
• Kronecker product