Wavelet Design in a Learning Framework

TL;DR

This paper introduces a novel learning-based framework for wavelet design using filterbank autoencoders trained on Gaussian data, enabling the creation of data-independent wavelets with desired properties.

Contribution

It presents a new approach that leverages autoencoder training to design wavelets, recovering known families and discovering new wavelets without relying on specific datasets.

Findings

Successfully recovers Daubechies and Cohen-Daubechies-Feauveau wavelets

Learns wavelets with properties like orthogonality and symmetry through architecture design

Demonstrates wavelet design without dataset dependence

Abstract

Wavelets have proven to be highly successful in several signal and image processing applications. Wavelet design has been an active field of research for over two decades, with the problem often being approached from an analytical perspective. In this paper, we introduce a learning based approach to wavelet design. We draw a parallel between convolutional autoencoders and wavelet multiresolution approximation, and show how the learning angle provides a coherent computational framework for addressing the design problem. We aim at designing data-independent wavelets by training filterbank autoencoders, which precludes the need for customized datasets. In fact, we use high-dimensional Gaussian vectors for training filterbank autoencoders, and show that a near-zero training loss implies that the learnt filters satisfy the perfect reconstruction property with very high probability. Properties of a wavelet such as orthogonality, compact support, smoothness, symmetry, and vanishing moments can be incorporated by designing the autoencoder architecture appropriately and with a suitable regularization term added to the mean-squared error cost used in the learning process. Our approach not only recovers the well known Daubechies family of orthogonal wavelets and the Cohen-Daubechies-Feauveau family of symmetric biorthogonal wavelets, but also learns wavelets outside these families.