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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
component, the so-called MSM (Most Singular Compo- 
nent, denoted F,,) associated with the strongest singulari- 
ties (i.e. the lowest - or most negative - exponent, denoted 
ho). In Fig. 3, the MSM was computed on the original 
image at a rather coarse resolution: the pixels for which 
hoo = —0.42 + 0.3 were included in the MSM. From in- 
formation theory, the MSM can be interpreted as the most 
relevant, informative set in the image (Turiel and del Pozo, 
2002). In (Grazzini et al., 2002), Grazzini et al. have evi- 
dencied that this subset, computed on meteorological satel- 
lite data, is related with the maxima of some well-defined 
local entropy. From multifractal theory, the MSM can be 
regarded as a reconstructing manifold. In (Turiel and del 
Pozo, 2002), Turiel and del Pozo have shown that using 
only the information contained by the MSM and the gradi- 
ent on it, it is possible to predict the value of the intensity 
field at every point. We describe the algorithm for recon- 
structing images in the next section and we show that it 
can be used as a technique for edge-preserving smoothing 
of images. 
3 THE RECONSTRUCTED IMAGES 
In (Turiel and del Pozo, 2002), Turiel and del Pozo pro- 
posed an algorithm supposed to produce a perfect recon- 
struction FRI (Fully Reconstructed Image) from the MSM. 
The authors introduced a rather simple vectorial kernel ÿ 
capable to reconstruct the signal I from the value of its 
spatial gradient VI over the MSM. Namely, let us define 
the density function d(x) of the subset F, which equals 
lifx € F, and 0 if x ¢ F.. Let also define the gradient 
restricted to the same set Vx = VI 0.5, Le. the field which 
equals the gradient of the original image over the MSM 
and is null elsewhere: this will be the only data required 
for the reconstruction. The reconstruction formula is then 
expressed as: 
I(x) = § * Uoo(X) (4) 
where x denotes the convolution. The reconstruction ker- 
nel j is easily represented in Fourier space by: 
g(r) = if/|f (5) 
where ^ stands for the Fourier transform, f denotes the fre- 
quency and i the imaginary unit. The principle is that of a 
propagation of the values of the signal VI over the MSM 
to the whole image. The full algorithm for reconstruction 
is given in (Turiel and del Pozo, 2002). 
The multifractal model described above also allows to con- 
sider a related concept, that of RMI (Reduced Multifractal 
Image), as it was introduced in (Grazzini et al., 2002). It 
consists in propagating, through the same reconstruction 
formula (4), instead of 7, another field vo so that a more 
uniform distribution of the luminance in the image is ob- 
tained. Namely, the field v is simply built by assigning to 
every point in the MSM an unitary vector, perpendicular 
to the MSM and with the same orientation as the original 
gradient VI. Thus, by introducing in eq. (4) this rather 
naive vector field, we obtain the RMI. Consequently, the 
RMI has the same multifractal exponents (and so, the same 
1127 
Figure 3: Top: MSM (dark pixels) extracted at hoo = 
—0.42 + 0.3 from the image of Fig. 1; quantitatively, the 
MSM gathers 22.29% of the pixels of the image. Middle: 
FRI computed with eq. (4) on the field vc (PSNR = 
24.93dB). Bottom: RMI computed with the field vg 
(PSNIR 2). 34d B).