International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
global analysis. Thus, it is not adapted to analyze the spatial
distribution of discontinuities. However, the wavelet transform
(Daubechies, 1992) enables to locally analyze a signal in time
and frequency, and therefore to compute the local regularity of
a signal.
In (Abry et al, 2001), it is shown that the continuous wavelet
transform is an efficient tool for the computation of the Hólder
exponent of a signal. For an accurate estimate of the Hólder
exponent of a function, it is better to use only the maxima lines
of the wavelet coefficients (the Wavelet Transform Modulus
Maxima method) as introduced in (Mallat and Hwang, 1992;
Muzy et al, 1993).
However, for the segmentation of an image, we need to
estimate this exponent in each point of the image; therefore this
method is not appropriate. Moreover, it is not the accuracy of
the estimation which is important in our case, but rather the
discriminating power of the resulting multifractal spectrum.
The choice of the wavelet basis is important. It should have a
sufficiently large number of vanishing moments to eliminate the
polynomial trends present in the signal. These polynomial
trends are sometimes so important that it is impossible to study
the local singularities of the signal. The more the wavelet basis
has vanishing moments, the larger the degree of the polynomial
we can get rid off. The wavelet basis must also have a relatively
small support size, in order to preserve the local aspect of the
analysis. It is desirable that the value of the wavelet transform
of a function in a point depends only on the values of this
function in its vicinity. A more complete study on the choice of
the wavelet basis is made in (Turiel, 1998).
It was proved that for a multifractal signal g, its wavelet
o. : J S
coefficients d. are such that at the location ¥ and for the scale
r , we have:
2 9 ;
Ea, Gr) - r7" when r0, (7)
2: :
where Ea, .r) is the mean of the squared wavelet
coefficients.
Thus, the wavelet coefficients follow a power law and the
Hólder exponent can be estimated as the slope of the regression
line of the points (logr,log(E|d, (Er) - Therefore,
according to the definition of the Hôlder regularity, it is the
decrease of the amplitude of the wavelet coefficients through
the scales which characterizes the local regularity of a signal.
3. THE PROPOSED SEGMENTATION ALGORITHM
3.1 Principle
The proposed algorithm is based on the idea that an image can
be divided into sets of points having a similar singularity
spectrum. We make the assumption that a texture is a particular
combination of Holder exponents. In other words, a texture
consists of singularities and the nature and the spatial
distribution of these singularities are enough to entirely
characterize this texture.
34
The proposed method is based on the estimation of the
spectrum fi; in each point of the image. The formulas given in
the previous sections are used. The spectrum is not computed
on the whole image, but on a sliding window of fixed. size.
During the computation of the spectrum we also introduce a
weighting filter. It represents the distance of the point studied
compared to the nearest singularity. Its coefficients are
computed as follows:
1
weight = ———p—p=— 8
; dist(x X) + e
where E. and E are the coordinate vectors of respectively, the
pixel for which we compute the spectrum and the nearest
à jp L
singularity location. dist(a,b) corresponds to the Euclidean
; p Lb
distance between vectors à and 5.
This weighting filter was introduced in order to prevent all the
points in the neighbourhood of a singularity to have the same
spectrum. To some extent, this factor takes into account the
neighbourhood of the singularity by affecting a decreasing
weight according to the distance.
Obviously, the local analysis of a signal is very important,
because an image is made of several thematic classes, each one
having its own texture. If a particular discontinuity is rather rare
in a class, it can be much more frequent in another. Locally, the
histogram of the singularities of a texture can be very different
according to the class that is studied. Therefore, to achieve a
good segmentation result, it is necessary to take account of
both, the strength and the spatial distribution of the
singularities.
An interesting advantage of the multifractal analysis is that it is
completely independent of the grey level values of the image.
Therefore, it is not affected by the low frequency variations of
the grey levels within a class. As this method is based on the
local spectrum multifractal, we name it the LMS method in the
remainder of this article. In (Arduini et a/, 1991; Kam and
Blanc-Talon, 1999), methods of image classification based on
multifractal tools were proposed and give interesting results
based on different ideas.
The method that we propose is unsupervised and is based on the
k-means clustering algorithm, therefore only the number of
classes in the image is required as input to the method.
3.2 The method parameters
The first step of this algorithm is the computation of the wavelet
transform of the image. A two-dimensional "Mexican hat"
wavelet is used, but other tests with the Morlet wavelet lead to
similar results. We chose to use the continuous wavelet
transform rather than the discrete one because it better fits the
needs for the analysis, which are a high degree of accuracy and
invariance under translation. The redundant information present
in the continuous wavelet transform enables to compute more
robust estimators, and thus to have a better stability of the
estimate. Tests showed that 5 to 8 levels of decomposition were
sufficient to obtain good segmentation results. The Hôlder
exponents were computed for each point of the image by linear
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