a la physique de l'information - Lisa - Université d'Angers

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Fig. 9 Two gray-level images with size 512 × 512 pixels and Q = 256 levels, coded as RGB
color images with three identical color components R = G = B; and corresponding number
N(a) of covering boxes with size a to cover their three-dimensional histogram.
reported in their spatial organization [23,22,14]. In this respect, current application
of fractal concepts to image compression is more related to the spatial organization
of images [15,10,7], while here fractality in the colorimetric organization constitutes a
novel direction for exploration. Also, similar fractal approaches as addressed here for
the three-dimensional histogram of color images, could be extended to multispectral
images, to characterize the complex structure of their multidimensional histograms,
contribute to assess their intrinsic dimensionality, and suggest efficient representations
in reduced coordinate systems [17].
For color images as considered here, the fractal dimensions resulting from such
analyses of their three-dimensional histograms, could be taken as simple characteristic
parameters, which can be helpful to various purposes such as image classification or
indexing, or contribute to metrics for realistic synthesis of images [11,1,5,18,25]. Also,
a fractal organization indicates clusters of occupied colorimetric cells exhibiting many
sizes, over many scales, in the three-dimensional histograms. This is to be contrasted
with simpler structures composed of a few clusters, with a few definite sizes, for the
occupied colorimetric cells of the histograms. This may bear relevance to segmentation
of color images based on pixel clustering from the color histogram. The presence of
a fractal organization suggests that there is no such thing as a small number of well
defined clusters that would emerge in the three-dimensional histograms, but on the
contrary, many clusters and sub-clusters co-existing over many scales in a self-similar
way.
Uncovering fractal structures also provides clues useful to developing models for
natural images [26,12]. This is an important task for many areas of image processing
and computer vision. The fractal organization observable in the three-dimensional color
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