Figure 1: High-resolution Spot image (20 x 20 m per pixel)
acquired in the Near Infrared channel. In this scene, we
can recognize some fields, roads and also poorly delimited
areas which correspond to cities.
e Second, an universal propagation kernel is used to
partially reconstruct typical object shapes from the
values of the gradient of the original image over the
previous subset.
The paper is organized as follows. In section 2, we review
the multifractal approach for extracting the meaningful en-
tities in the images. In section 3, we introduce the method
for reconstructing the images. We show that this method
provides good results on Spot images with spatial resolu-
tion of 20 x 20 m? (see Fig. 1) and that it could be applied
to any kind of data. We also consider an alternative ver-
sion of this method. As a conclusion, we will discuss the
advantages of our methodology.
2 THE MULTIFRACTAL DECOMPOSITION
Multifractality is a property of turbulent-like systems which
is generally reported on intensive, scalar variables of chaotic
structure (Frisch, 1995). However, methods based on this
approach have also shown to produce meaningful results in
the context of real world images where irregular graylevel
transitions are analyzed (Turiel and Parga, 2000). Natu-
ral images can be characterized by their singularities, i.e.
the set of pixels over which the most drastic changes in
graylevel value occur. In (Turiel and Parga, 2000), Turiel
and Parga introduced a novel technique based on multifrac-
tal formalism to analyze these singularities. We develop in
the following the basis of this technique.
In this approach, the points in a given image are hierarchi-
cally classified according to the strength of the transtion of
the image around them. The first step concerns the defi-
nition of an appropriate multifractal measure. Let us de-
note / an image and VI its spatial derivative. We then de-
fine a measure j: through its density du (x) = dx |VI|(x),
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
1126
so that the measure of a ball B,.(x) of radius r centered
around the point x is given by:
HB = fd MIO. —
J B(x
Here |VI| denotes the modulus of the gradient VI. This
measure gives an idea of the local variability of the graylevel
values around the point x. It was observed that for a large
class of natural images, such a measure is multifractal (Turiel
and Parga, 2000), i.e. the following relation holds:
u(B,(x)) + oa(x) r?*^09. (2)
All the scale dependance in eq. (2) is provided by the term
p2+h(<) where the exponent I(x) depends on the partic-
ular point considered. This exponent is called singularity
exponent and quantifies the multifractal behaviour of the
mesure zi. The second step of the approach regards the es-
timation of these local exponents. This is done through a
wavelet projection 44 (Daubechies, 1992) of the measure,
for which the same kind of relation as eq. (2) holds in a
continuous framework, so that the exponents (x) can be
easily retrieved. The reader is refered to (Turiel and Parga,
2000) for a full description of the method. By this way,
each pixel x of the image can be assigned a feature which
measures the local degree of regularity of the signal VI
at this point. The singularity exponents computed on the
Spot image of the Fig. | are represented in the Fig. 2. We
see that the lowest values of singularity are mainly con-
centrated near the boundaries of the objects of the image
(large culture fields in the middle part, roads in the lower
left part). Some of them are also visible inside the culture
fields, as they are not completely flatly illuminated, and in
city areas (upper right part) that are rather heterogeneous.
computed on the image of the Fig. 1 in the range [0, 255]
of graylevel values; the brighter the pixel, the greater the
singularity exponent a this point. Right: corresponding
distribution of the singularity exponent; the range of values
obtained is [—0.80, 0.81].
The image can then be hierarchically decomposed in dif-
ferent subsets 77, gathering the pixels of similar features:
F,zíix|h(x)mh). (3)
We know, from the multifractal theory, that each one of
these subsets is fractal, i.e. it exhibits the same geomet-
rical structure at different scales. In particular, the aris-
ing decomposition allows isolating a meaningful fractal
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