Wavelet digital filtering - Fundamentals

The strong interest in the world of wavelets and their applications dates back almost three decades, and has led to the publication of an impressive number of books and scientific articles, theoretical foundations and syntheses. Our modest contribution here is to introduce wavelet-based filtering via multi-resolution analysis, with a view to providing skills and know-how to as many users as possible in the academic world (students, junior researchers or non-specialists in the field) and the socio-economic world (technicians and engineers).

The current boom in wavelet and wavelet packet transforms is mainly due to two specific properties resulting from decompositions on orthogonal wavelet bases: representation parsimony and the tendency to transform a stationary random process into decorrelated Gaussian sequences.

In the context of noise reduction, more commonly known as "denoising", the success of wavelet-based multiresolution analysis is precisely ensured by its decorrelation capability (separation of noise and useful signal) and by the notion of parsimony in its representation.

This parsimony is materialized by a small number of high-amplitude wavelet coefficients (or, more precisely, wavelet transform coefficients) representing the useful signal, which is assumed to be regular, or piecewise regular. As for the noise, often assumed to be white and stationary, it will tend to spread over all the wavelet components or coefficients.

Relying on these two properties (sparsity and decorrelation), appropriate filtering in the wavelet domain and calculation of the corresponding inverse wavelet transform will yield the denoised signal.

The performance of this filtering will be analyzed, both in terms of mean square error and SNR (signal-to-noise ratio) or PSNR (peak signal-to-noise ratio), and in terms of visual quality in the case of images.

This article aims to provide a simple formulation and a new perspective, to make a priori abstruse knowledge and information accessible, to merge, marry or complement multi-resolution analysis with filtering and feature extraction. It aims to provide the keys to deciphering and operationalizing concepts that seem abstract. The most advanced theoretical aspects will therefore be bypassed and referred to a rich specialized bibliography.