Wavelets and their use

TL;DR

This paper surveys discrete wavelet theory and its practical deployment for analyzing nonstationary signals via multiresolution analysis (MRA) and the fast wavelet transform. It covers foundational blocks (Haar and Daubechies wavelets, scaling spaces $V_j$, detail spaces $W_j$), multidimensional extensions, and the relationship to Fourier analysis, as well as operator representations and nonstandard matrix forms. The authors emphasize regularity, vanishing moments, and biorthogonal/setups (coiflets, wavelet packets) to optimize compression, reconstruction, and stability, and illustrate broad applications—from solid-state physics and high-energy multiparticle processes to medicine, biology, data compression, and microscopy. The review also develops advanced tools such as two-microlocal analysis and fractal multifractal frameworks to quantify local regularity and scaling, underscoring wavelets as both a mathematical microscope and a practical engine for pattern recognition and numerical analysis. Overall, the work highlights the versatility and impact of wavelet methods across science and engineering, while acknowledging the need for problem-specific wavelet selection and careful numerical treatment.

Abstract

This review paper is intended to give a useful guide for those who want to apply discrete wavelets in their practice. The notion of wavelets and their use in practical computing and various applications are briefly described, but rigorous proofs of mathematical statements are omitted, and the reader is just referred to corresponding literature. The multiresolution analysis and fast wavelet transform became a standard procedure for dealing with discrete wavelets. The proper choice of a wavelet and use of nonstandard matrix multiplication are often crucial for achievement of a goal. Analysis of various functions with the help of wavelets allows to reveal fractal structures, singularities etc. Wavelet transform of operator expressions helps solve some equations. In practical applications one deals often with the discretized functions, and the problem of stability of wavelet transform and corresponding numerical algorithms becomes important. After discussing all these topics we turn to practical applications of the wavelet machinery. They are so numerous that we have to limit ourselves by some examples only. The authors would be grateful for any comments which improve this review paper and move us closer to the goal proclaimed in the first phrase of the abstract.