Self-similarity properties of natural images 
ANTONIO TURIEL,* GERMAN MATO, l NSTOR PARGA I 
Departamento de Fsica Tedfica . Universidad Autdnoma de Madrid 
Cantoblanco, 2809 Madrid, Spain 
and JEAN-PIERRE NADAL  
Laboratoire de Physique Statistique de I'E.N.S.  Ecole Normale Sup4rieure 
2, rue Lhomond, F-75231 Paris Cedex 05, France 
Abstract 
Scale invariance is a fundamental property of ensembles of nat- 
ural images [1]. Their non Gaussian properties [15, 16] are less 
well understood, but they indicate the existence of a rich statis- 
tical structure. In this work we present a detailed study of the 
marginal statistics of a variable related to the edges in the images. 
A numerical analysis shows that it exhibits extended self-similarity 
[3, 4, 5]. This is a scaling property stronger than self-similarity: 
all its moments can be expressed as a power of any given moment. 
More interesting, all the exponents can be predicted in terms of 
a multiplicative log-Poisson process. This is the very same model 
that was used very recently to predict the correct exponents of 
the structure functions of turbulent flows [6]. These results allow 
us to study the underlying multifractal singularities. In particular 
we find that the most singular structures are one-dimensional: the 
most singular manifold consists of sharp edges. 
Category: Visual Processing. 
I Introduction 
An important motivation for studying the statistics of natural images is its relevance 
for the modeling of the visual system. In particular, the epigenetic development 
* e-mail: amturiel@delta.ft.uam.es 
te-maih matog@cab.cnea.edu.ar 
*To whom correspondence should be addressed. e-mail: parga@delta.ft.uam.es 
 e-mail: nadallps.ens.fr 
Laboratoire associ au C.N.R.S. (U.R.A. 1306), k FENS, et aux Universit,s Paris VI 
et Paris VII. 
Self-similarity Properties of Natural Images 837 
could lead to the adaptation of visual processing to the statistical regularities in the 
visual scenes [8, 9, 10, 11, 12, 13]. Most of these predictions on the development of 
receptive fields have been obtained using a gaussian description of the environment 
contrast statistics. However non Gaussian properties like the ones found by [15, 16] 
could be important. To gain further insight into non Gaussian aspects of natural 
scenes we investigate the self similarity properties of an edge type variable [14]. 
Scale invariance in natural images is a well-established property. In particular it 
appears as a power law behaviour of the power spectrum of luminosity contrast: 
i (the parameter / depends on the particular images that has been 
S ( f ) oc ]-yy 
included in the dataset). A more detailed analysis of the scaling properties of the 
luminosity contrast was done by [15, 16]. These authors noted the possible analogy 
between the statistics of natural images and turbulent flows. There is however no 
model to explain the scaling behaviour that they observed. 
On the other hand, a large amount of effort has been put to understand the statistics 
of turbulent flows and to develop predictable models (see e.g. [17]). Qualitative 
and quantitative theories of fully developed turbulence elaborate on the original 
argument of Kolmogorov [2]. The cascade of energy from one scale to another is 
described in terms of local energy dissipation per unit mass within a box of linear 
size r. This quantity, er, is given by: 
ix dx' E[Oivj(x') + Ojvi(x')] 2 (1) 
e(x) oc -x'l<r 
where vi (x) is the ith component of the velocity at point x. This variable has Self- 
Similarity (SS) properties that is, there is a range of scales r (called the inertial 
range) where: 
< > rrp, (2) 
here < %P > denotes the pth moment of the energy dissipation marginal distribution. 
A more general scaling relation, called Extended Self-Similarity (ESS) has been 
found to be valid in a much larger scale domain. This relation reads 
< > < (3) 
where p(p, q) is the ESS exponent of the pth moment with respect to the qth mo- 
ment. Let us notice that if SS holds then -p = -qp(p, q). In the following we will 
2 
refer all the moments to < % >. 
2 The Local Edge Variance 
For images the basic field is the contrast c(x), that we define as the difference 
between the luminosity and its average. By analogy with the definition in eq. (1) we 
will consider a variable that accumulates the value of the variation of the contrast. 
We choose to study two variables, defined at position x and at scale r. The variable 
eh,r (x) takes contributions from edges transverse to a horizontal segment of size r: 
- dy (4) 
= r 
A vertical variable ev,r (x) is defined similarly inteating along the verticM direction. 
We will refer to the vMue of the derivative of the controt Mong a given direction 
 an edge transverse to that direction. This is justified in the sense that in the 
presence of borders this derivative will te a great value, d it will almost vish 
838 A. Tuffel, G. Mato, N. Parga and J-P Nadal 
if evaluated inside an almost-uniformly illuminated surface. Sharp edges will be the 
maxima of this derivative. According to its definition, el,r (x) ( l = h, v ) is the local 
linear edge variance along the direction I at scale r. Let us remark that edges are 
well known to be important in characterizing images. A recent numerical analysis 
suggests that natural images are composed of statistically independent edges [18]. 
We have analyzed the scaling properties of the local linear edge variances in a set 
of 45 images taken into a forest, of 256 x 256 pixels each (the images have been 
provided to us by D. Ruderman; see [16] for technical details concerning them). An 
analysis of the image resolution and of finite size effects indicates the existence of 
upper and lower cut-offs. These are approximately r = 64 and r - 8, respectively. 
First we show that SS holds in a range of scales r with exponents -h,p and 'rv,p. 
This is illustrated in Fig. (1) where the logarithm of two moments of horizontal 
and vertical local edge variances are plotted as a function of In r; we see that SS 
holds, but not in the whole range. 
ESS holds in the whole considered range; two representative graphs are shown in 
Fig. (2). The linear dependence of In  P  vs In  e 2 
t,r t,r  is observed in 
both the horizontal (l = h) and the vertical (l = v) directions. This is similar to 
what is found in turbulence, where this property has been used to obtain a more 
accurate estimation of the exponents of the structure functions (see e.g. [17] and 
references therein). The exponents Ph(p, 2) and Pv (P, 2), estimated with a least 
squares regression, are shown in Fig. (3) as a function of p. The error bars refer to 
the statistical dispersion. From figs. (1-3) one sees that the horizontal and vertical 
directions have similar statistical properties. The SS exponents differ, as can be 
seen in Fig(l); but, surprisingly, ESS not only holds in both directions, but it does 
it with the same ESS exponents, i.e. pn(p, 2) - pv (p, 2). 
3 ESS and multiplicative processes 
Let us now consider scaling models to predict the p-dependence of the ESS expo- 
nents pt (p, 2). (Since ESS holds, the SS exponents -t,p can be obtained from the 
pt(p, 2)s by measuring -1,2). The simplest scaling hypothesis is that, for a random 
variable er(x) observed at the scale r (such as el,'.(x)), its probability distribution 
15,. (e'. (x) = e) can be obtained from any other scale L by 
1 
= .(r.L) PL .(;.L) (5) 
<> /P (for any p(p, 2) c p; 
From this one derives easily that a(r,L) - [<->j p) and if 
SS holds, -p c p: for turbulent flows this corresponds to the Kolmogorov prediction 
for the SS exponents [2]. Fig (3) shows that this naive scaling is violated. 
This discrepancy becomes more dramatic if eq. (5) is expressed in terms of a 
normalized variable. Taking e = limpc  erP+  /  erP  ( that can be shown 
to be the maximum value of e'., which in fact is finite ) the new variable is defined 
as f'. = e'./e; 0  f'. ( 1. If P'.(f) is the distribution of f'., the scaling relation 
eq.(5) reads P'.(f) = P(f); this identity does not hold as can be seen in Fig. (4). 
A way to generalize this scaling hypothesis is to say that ct is no longer a constant 
as in eq. (5), but an stochastic variable. Thus, one has for P'.(f): 
P'.(f) = f G'.r(lna)Pr dlna (6) 
This scaling relation has been first introduced in the context of turbulent flows 
[6, 19, 7]. Eq. (6) is an integral representation of ESS with general (not necessarily 
Self-similarity Properties of Natural Images 839 
linear) exponents: once the kernel GrL is chosen, the p(p, 2)'s can be predicted. 
It can also be phrased in terms of multiplicative processes [20, 21]: now fr = 
where the factor a itself becomes a stochastic variable determined by the kernel 
Gr(ln a). Since the scale L is arbitrary (scale r can be reached from any other 
scale L') the kernel must obey a composition law, Gr, G, - GrL. Consequently 
fr can be obtained through a cascade of infinitesimal processes G =_ 
Specific choices of G define different models of ESS. The She-Leveque (SL) [6] 
model corresponds to a simple process such that a is I with probability 1 - s and 
1 in(< > 
is a constant/3 with probability s. One can see that s = 1--:- ) and that 
this stochastic process yields a log-Poisson distribution for a [22]. It also gives ESS 
with exponents p(p, q) that is expressed in terms of the parameter/3 as follows [6]: 
p(p, q) = I -/3P - (1 - 
I -/3q - (1 - 
(7) 
We can now test this models with the ESS exponents obtained with the image data 
set. The resulting fit for the SL model is shown in Fig. (3). Both the vertical and 
horizontal ESS exponents can be fitted with/3 - 0.50 4. 0.03. 
The integral representation of ESS can also be directly tested on the probability 
distributions evaluated from the data. In Fig. (4) we show the prediction for Pr (f) 
obtained from P(f) using eq. (6), compared with the actual Pt(f). 
The parameter/3 allows us to predict all the ESS exponents p(p, 2). To obtain the 
SS exponents -p we need another parameter. This can be chosen e.g. as -2 or as the 
asymptotic exponent A, given by e ocr -A , r >> 1; we prefer A. As -p = -2 p(p, 2), 
then from the definition of e one can see that A = - rs A least square fit of 
was used to determine A, obtaining An = 0.4 4. 0.2 for the horizontal variable and 
Av = 0.5 4- 0.2. for the vertical one. 
4 Multifractal analysis 
Let us now partition the image in sets of pixels with the same singularity exponent 
h of the local edge variance: er ocr h. This defines a multifractal with dimensions 
D(h) given by the Legendre transform of -p (see e.g. [17]): D(h) -- infp(ph+d--p), 
where d = 2 is the dimension of the images. We are interested in the most singular 
of these manifolds; let us call D its dimension and hmin its singularity exponent. 
Since e is the maximum value of the variable er, the most singular manifold 
is given by the set of points where er = e, SO hmin = --A. Using again that 
-p = -A (1 -/3) p(p, 2) with p(p, 2) given by the SL model, one has D = d- 1:' 
From our data we obtain Do,h = 1.3 4- 0.3 and D,v = 1.1 4- 0.3. As a result 
we can say that D,a " Dc,v " 1: the most singular structures are almost one- 
dimensional. This reflects the fact that the most singular manifold consists of sharp 
edges. 
5 Conclusions 
We insist on the main result of this work, which is the existence of non trivial 
scaling properties for the local edge variances. This property appears very similar 
to the one observed in turbulence for the local energy dissipation. In fact, we have 
seen that the SL model predicts all the relevant exponents and that, in particular, 
it describes the scaling behaviour of the sharpest edges in the image ensemble. It 
would also be interesting to have a simple generative model of images which - apart 
840 A. Tuffel, G. Mato, N. Parga and JoP Nadal 
from having the correct power spectrum as in [23] - would reproduce the self-similar 
properties found in this work. 
Acknowledgements 
We are grateful to Dan Ruderman for giving us his image data base. We warmly 
thank Bernard Castaing for very stimulating discussions and Zhen-Su She for a 
discussion on the link between the scaling exponents and the dimension of the most 
singular structure. We thank Roland Baddeley and Patrick Tabeling for fruitful 
discussions. We also acknowledge Nicolas Brunel for his collaboration during the 
early stages of this work. This work has been partly supported by the French- 
Spanish program "Picasso" and an E.U. grant CHRX-CT92-0063. 
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Self-similarity Properties of Natural Images 841 
3 
In < e r > a 
5 
In < e r > 
3 35 4 
In r 
Figure 1: Test of SS. We plot In < e p 
t,r > vs. ln r for p = 3 and 5; r from 8 to 64 
pixels. a) horizontal direction, I = h. b) vertical direction, I = v. 
3 
In < e r > a b 
5 
In < e r > 
2 2 
In < e r > In < e r > 
Figure 2: Test of ESS. We plot In < e p > vs. In < e 2 
t,r t,r > for p=3, 5; r from 8 to 
r = 64 pixels. a) horizontal direction, I = h. b) vertical direction, I = v. 
842 A. Tuffel, G. Mato, N. Parga and J-P Nadal 
p(p, 2) 
2 
a 
p 
b 
1 2 3 4 5 6 7 6 9 10 
Figure 3: ESS exponents p(p, 2), for the vertical and horizontal variables. a) hor- 
izontal direction, ph(p, 2). b) vertical direction, pv(p, 2). The solid line represents 
the fit with the SL model. The best fit is obtained with v - h - 0.50. 
P 
16 
14 
12 
10 
4- 
+ 
0 
0 0.05 
+4-+ 
+ ++44- 
+ 
+44-++ 
+++ 
I I 
0.1 0.15 0.2 
Figure 4: Verification of the validity of the integral representation of ESS, eq.(6) 
with a log-Poisson kernel, for horizontal local edge variance. The largest scale is 
L = 64. Starting from the histogram P;(f) (denoted with crosses), and using a 
log-Poisson distribution with parameter  = 0.50 for the kernel GrL, eq.(6) gives 
a prediction for the distribution at the scale r = 16 (squares). This has to be 
compared with the direct evaluation of Pr(f) (diamonds). Similar results hold for 
other pairs of scales. Although not shown in the figure, the test for vertical case is 
as good as for horizontal variable. 
