Irregularities and Scaling in Signal and Image Processing: Multifractal Analysis

TL;DR

This paper reviews the concepts of scale invariance, self-similarity, and fractal dimensions, relating them to multifractal analysis, reformulating the tools in a wavelet framework, and illustrating their applications across diverse fields.

Contribution

It unifies key notions of fractal geometry, develops a wavelet-based reformulation of multifractal analysis, and demonstrates its practical applications in real-world data.

Findings

Unified theoretical framework for scale invariance and fractal concepts

Wavelet reformulation enhances analysis and implementation

Successful application examples across various domains

Abstract

B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is to review and relate together these key notions, explore their interplay and show that they are different facets of a same intuition. Second, it will explain how these notions lead to the derivation of the mathematical tools underlying multifractal analysis. Third, it will reformulate these theoretical tools into a wavelet framework, hence enabling their better theoretical understanding as well as their efficient practical implementation. B. Mandelbrot used his concept of fractal geometry to analyze real-world applications of very different natures. As a tribute to his work, applications of various origins, and where multifractal analysis proved fruitful, are revisited to illustrate the theoretical developments proposed here.