M-RWTL: Learning Signal-Matched Rational Wavelet Transform in Lifting Framework

TL;DR

This paper introduces M-RWTL, a novel method for learning signal-matched rational wavelet transforms within the lifting framework, enabling adaptive, invertible transforms from limited data, improving compressed sensing reconstruction.

Contribution

It extends the lifting framework to rational wavelets and enables learning from a single signal without large training datasets, enhancing wavelet adaptivity and application scope.

Findings

M-RWTL outperforms standard wavelets in compressed sensing tasks.

The method ensures invertibility and modularity of the learned wavelet transforms.

It successfully incorporates nonlinear filters, broadening application potential.

Abstract

Transform learning is being extensively applied in several applications because of its ability to adapt to a class of signals of interest. Often, a transform is learned using a large amount of training data, while only limited data may be available in many applications. Motivated with this, we propose wavelet transform learning in the lifting framework for a given signal. Significant contributions of this work are: 1) the existing theory of lifting framework of the dyadic wavelet is extended to more generic rational wavelet design, where dyadic is a special case and 2) the proposed work allows to learn rational wavelet transform from a given signal and does not require large training data. Since it is a signal-matched design, the proposed methodology is called Signal-Matched Rational Wavelet Transform Learning in the Lifting Framework (M-RWTL). The proposed M-RWTL method inherits all the advantages of lifting, i.e., the learned rational wavelet transform is always invertible, method is modular, and the corresponding M-RWTL system can also incorporate nonlinear filters, if required. This may enhance the use of RWT in applications which is so far restricted. M-RWTL is observed to perform better compared to standard wavelet transforms in the applications of compressed sensing based signal reconstruction.