Fractal and multifractal analysis: A review R. Lopes a,b , N. Betrouni a, * a Inserm, U703, Pavillon Vancostenobel, CHRU Lille, Lille Cedex 59037, France b Laboratoire d’automatique LAGIS, CNRS UMR 8146, USTL, Bâtiment P2, Villeneuve d’Ascq 59655, France article info Article history: Received 8 November 2007 Received in revised form 1 April 2009 Accepted 15 May 2009 Available online 27 May 2009 Keywords: Fractal analysis Fractal dimension Multifractal analysis Multifractal spectrum Texture Characterization abstract Over the last years, fractal and multifractal geometries were applied extensively in many medical signal (1D, 2D or 3D) analysis applications like pattern recognition, texture analysis and segmentation. Applica- tion of this geometry relies heavily on the estimation of the fractal features. Various methods were pro- posed to estimate the fractal dimension or multifractal spectral of a signal. This article presents an overview of these algorithms, the way they work, their benefits and their limits. The aim of this review is to explain and to categorize the various algorithms into groups and their application in the field of medical signal analysis. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The idea of describing natural phenomena by studying statisti- cal scaling laws is not recent. Indeed, many studies were carried out on this topic (Bachelier, 1900; Frish, 1995; Kolmogorov, 1941; Mandelbrot, 1963). However, there has been a recent resur- gence of interest in this approach. A great number of physical sys- tems tend to present similar behaviours on different scales of observation. In the 1960s, the mathematician Benoît Mandelbrot used the adjective ‘‘fractal” to indicate objects whose complex geometry cannot be characterized by an integral dimension. The main attraction of fractal geometry stems from its ability to describe the irregular or fragmented shape of natural features as well as other complex objects that traditional Euclidean geometry fails to analyse. This phenomenon is often expressed by spatial or time-domain statistical scaling laws and is mainly characterized by the power-law behaviour of real-world physical systems. This con- cept enables a simple, geometrical interpretation and is frequently encountered in a variety of fields, such as geophysics, biology or fluid mechanics. To this end, Mandelbrot introduced the notion of fractal sets (Mandelbrot, 1977), which enables to take into ac- count the degree of regularity of the organizational structure re- lated to the physical system’s behaviour. Fractal geometry is widely used in image analysis problems in general and especially in the medical field. It is applied in different ways with different results. However, there has been no review pa- per to digest these different methods and their application. The purpose of this paper is to provide a survey of these methods and to discuss the principal results. This research may provide assistance to researchers aiming to use this geometry in medical imaging applications. It is organised as follow: in the next section, we introduce more formally the fractals; Section 3 discusses the relevance of fractals in image analysis. Section 4 gives the survey of the methods, their principles and limitations. Sections 5 and 6 are respectively reserved to multifractal analysis and the associ- ated algorithms. Section 7 discusses the main applications of frac- tals/multifractals in the medical image analysis procedures and the methods used. 2. Fractals and dimensions A definition that can illustrate the notion of fractal can be as fol- lows: consider an object. One has to take an element of this object. One has to surround it with a sphere of a given radius R and count the amount of object elements R inside the sphere. The measure of R can be arbitrary. Here, of importance is only the dependence of R on the sphere radius after averaging over the element put in its ori- gin. This definition takes into account the fact that the relevant dimension of an object depends on the spatial scale. A fundamental characteristic of fractal objects is that their mea- sured metric properties, such as length or area, are a function of the scale of measurement. A classical example to illustrate this prop- erty is the ‘‘length” of a coastline (Mandelbrot, 1967). When mea- sured at a given spatial scale d, the total length of a crooked coastline L(d) is estimated as a set of N straightline segments of 1361-8415/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.media.2009.05.003 * Corresponding author. Tel.: +33 320 446 7 22; fax: +33 320 446 715. E-mail address: n-betrouni@chru-lille.fr (N. Betrouni). Medical Image Analysis 13 (2009) 634–649 Contents lists available at ScienceDirect Medical Image Analysis journal homepage: www.elsevier.com/locate/media