Medical Image Analysis 13 (2009) 634–649
Contents lists available at ScienceDirect
Medical Image Analysis
journal homepage: www.elsevier.com/locate/media
Fractal and multifractal analysis: A review
R. Lopes a,b, N. Betrouni a,*
Inserm, U703, Pavillon Vancostenobel, CHRU Lille, Lille Cedex 59037, France
Laboratoire d’automatique LAGIS, CNRS UMR 8146, USTL, Bâtiment P2,Villeneuve d’Ascq 59655, France
Received 8 November 2007
Received in revised form 1 April 2009
Accepted 15 May 2009
Available online 27 May 2009
Over the last years, fractal and multifractal geometries were applied extensively inmany medical signal
(1D, 2D or 3D) analysis applications like pattern recognition, texture analysis and segmentation. Application of this geometry relies heavily on the estimation of the fractal features. Various methods were proposed 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 andtheir 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.
The idea of describing natural phenomena by studying statistical 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 resurgence of interest in this approach. A great number of physical systems 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 integraldimension.
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 concept 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 account the degree of regularity of the organizational structure related to the physical system’s behaviour.
Fractal geometry iswidely 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* Corresponding author. Tel.: +33 320 446 7 22; fax: +33 320 446 715.
E-mail address: firstname.lastname@example.org (N. Betrouni).
1361-8415/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.media.2009.05.003
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 3discusses 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 associated algorithms. Section 7 discusses the main applications of fractals/multifractals in the medical image analysis procedures and the
2. Fractals and dimensions
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