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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
2. MULTIFRACTAL ANALYSIS BASICS 
The multifractal formalism was created to describe the 
properties of very turbulent systems with change in scale. It is 
the case, in particular, in the study of fluids turbulences in 
physics, (Grassberger and Procaccia, 1983) or (Frisch and 
Parisi, 1985). It is used to describe the local behaviour and 
nature of the singularities of irregular functions in a geometrical 
or statistical way. A more complete theoretical description of 
the multifractal analysis can be found in (Abry et al, 2001; 
Lévy-Vehel et al, 2001). 
2.1 Regularity and Hôlder exponent 
The concept of specific regularity, in a point xp , was created to 
quantize, using a positive real number « , the "roughness" of the 
graph of a function in this point. The Holder regularity is a 
generalization of the concepts of derivability and continuity of a 
function. It is defined as following: 
Let œ be a positive real number and x; € R; a function 
g:R > is said to be C^ (xg) if it exists a polynomial P 
with maximum degree [a] such that: 
g(x) — Px — xo ) S C|x = xo” (1) 
  
where [a] -q-l if a isan integer. 
Therefore, the regularity of a function g is computed using an 
estimation of the maximum difference of g with respect to a 
polynomial P of degree lesser or equal to [a]. 
From the definition of the regularity we can define the Hólder 
exponent a, of a function g as: 
Go(Xo)7 supe: Ug is C" (xg) (2) 
Thus, from its definition, it is obvious that the Hólder exponent 
can characterize the regularity of a function g in each point. 
2.2 Thesingularity spectrum 
The multifractal analysis is generally used to study signal 
without any a priori knowledge on its nature. It is a very useful 
tool to describe and analyze the variations of the local 
regularity of an unspecified signal. Most of the time it is used to 
characterize the regularity of highly irregular signals whose 
singularity spectrum is not a single point. The singularity 
spectrum of such signals is function of time, contrary to 
(mono)fractals signals which are entirely characterized by 
single exponent. These signals are called multifractal because 
they are characterized by infinity of fractal sets. Those sets have 
to be studied in order to deduce the signal singularity spectrum. 
This spectrum is a global description of the singularities 
distribution. It exist various singularity spectrum types 
according to whether one uses a statistical or geometrical 
approach to estimate it. They all try, as well as possible, to 
approach the theoretical singularity spectrum by using different 
methods (Berroir, 1994). One of them is called the large 
deviation spectrum f; and its definition is given below. The 
proposed algorithm is based on this spectrum. 
To obtain the large deviation spectrum, we first have to 
compute the Holder exponent for each point of the signal. From 
the image of Holder exponents, we extract the fractal 
component sets F, which are formed by the points having the 
same Hôlder exponents: F, = d: "A63 = aj. In practice, this 
e 
spectrum can be calculated by using the widened spectral 
components F7 based on a quantization of a, noted a, : 
Lu 
— 
FE =fl:a-e<a(f)<a+el, 
because the number of different exponents can be large. The 
spectrum is then obtained by the following formula: 
log N, (@,) 
1 
log— 
; 
Jo(a) = lim limsup (4) 
£20 / 40 
where N,(@,) is the number of balls C of size r which 
contain a Holder exponent a, , so: 
N.(a,) zt :a ela - 2a e]. (5) 
. This is equivalent to the computation of fractal dimension of 
33 
each fractal component F7 . Thus, the large deviation spectrum 
can be approximated by using the box dimension of the fractal 
component sets, which is the slope of the points whose 
coordinates are (logr,-log N,(a,)). 
The spectrum fg is a statistical approach of the multifractal 
spectrum and can be interpreted as the probability of finding a 
Holder exponent of order @ in a ball of radius r centred in 
xg. This is equivalent to: 
P(a, (x9) ~ a) ~ pool (6) 
The aim of the multifractal formalism is to establish a strong 
relation between the different multifractal spectra. If one makes 
the assumption, not proved in our case, that the multifractal 
formalism holds, then f = /ç, but in a more general way: 
f £ fg, where f is the theoretical spectrum. 
2.3 The wavelet transform and the Holder exponent 
The Fourier transform is a powerful tool for studying the 
singularities of a signal, but it is only possible to perform a