

Multiscale analysis of SAR Polarimetric Correlations in the Eastern Pyrenees
Jose M. Redondo^{(1)}^{(1)} Univ. Politecnica de Catalunya, B5 Campus nord UPC, Barcelona 08034, Spain
Abstract
We compare diferent SAR images of the Eastern Pyrenees, evaluating the changes in structure as a function of average height. The multiple correlations between HH HV VV polarizations and the images are used to
calculate the fractal dimension with the BoxCounting method. The distribution of the boxes is accomplished systematically for each SAR intensity level, the intersection of these boxes with the images gives N(i,e) boxes with a non void intersection ( Being i the local SAR intensity, e the box size and D(i) = log N(i,e) / log e the fractal dimension (Redondo and Linden 1996)
With this methodology an additional unique value is obtained that characterizes the overall spatial fractal dimension of the system. The steps are described as follows (Grau 2005) make an image segmentation to obtain the interest region.(m and n are the xy discrete coordinates). Compute the FT (Fourier Transform) to obtain the frequency spectrum representation. (Iuv, u and v are the frequency discrete coordinates). Compute the square of the signal intensity or energy Suv. Obtain the radial representation, as the radial distribution of Suv and finally find the exponent b from Sr = r^(b). Using the radius as an isotropic length scale.With a linear fit from a loglog representation of Sr we may obtain the spatial spectral value of the set of all SAR image intensities and using as the Euclidean surface dimension E = 2 and the fractal dimension relationship we relate F(D(i), b ) as a function of height.
And thus we also have a global, indirect measure of the average fractal dimension < D(i)> from the local radial spectral energy <b>. ImaCalc [Grau, 2005] performs most of the multifractal box counting methods as well as the spectral ones.
Several polarizations of the SAR are compared exhibiting their different structure functions (Mahjoub et al.2001) up to 6th order.
The flatness or Kurthosis is a statistic which indicates the grade of planitud of the pdfs of the SAR intensity, and is defined from the structure functions as F = S4/S2*S2. The flatness seems to be a very good indicator of the degree of existing structure; when flatness changes with scale following a potencial law, intermittency is present.
Both D(i)and F are found to be useful to measure intermittency and even more so, when it is applied to the correlations between the different SAR polarizations. Comparisons with the standard multifractal formalism (Dathe et al 2006) also reveal the importance of anisotopy.
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Grau, J. 2005. Analysis of the Meteosat images sequences using the digital processing method. PhD Thesis UPC, Barcelona
Mahjoub O. B., Granata T. and Redondo J. M.: Scaling laws in geophysical flows, Phys. Chem. Earth (B), 26, 281285, 2001.
Dathe, A., Perrier, E., Tarquis, A., (2006). Multifractal análisis of twodimensional soil porous structure on natural images. Geoderma 134, 318–326.
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