Urban Texture Analysis (3/3)

 

PARAMETERS NORMALIZATION

We now have to normalize the parameters with respect to the different directions. Let the step of the lattice be the unity distance. Thus, if one considers the direction (E/W),  s is at a distance one from each of its neighbours. On the other hand, if one considers the direction (NWw/SEe), s is at a distance sqrt(5)from each of its neighbours. These different distances introduce a bias. That is why we are going to normalize the considered parameters in order to correct the anisotropy of the lattice.
We choose the 1D reference model that will represent the lattice with a reference sample step. Then we calculate the parameters of others models (which are defined on the lattices given by the data)  in function of the parameters of the reference model . With this aim we use the renormalization method by decimation. This method consists in integrating the reference model on the set of sites that we want to decimate in order to get the model that is defined on the lattice given by the data. This is equivalent to calculate the marginal law of the model at a lower resolution corresponding to the resolution of the considered direction. Notice that the model on the decimated lattice remains Markovian.
We want to calculate the parameters of the models whose lattice steps are 1, sqrt(2) or sqrt(5) with the above method. Using the renormalization technique, if we consider a reference lattice of resolution 1 we can compute the parameters for integer resolutions. We have  to approximate these non necessary rational numbers by rational fractions so as to  integrate on a finite number of sites.

We shall keep the rational fractions that give the best compromise between a precise approximation of  sqrt(2) and sqrt(5) and a small number of variables on which we have to integrate.





 

For pratical considerations we do not look for the equivalent parameters in relation to the reference lattice. In fact we calculate the equivalent parameters in relation to the lattice of step 1. We do this both for lattice of step sqrt(2) and for lattice of step sqrt(5). Thus we keep the conditional variances estimated in the directions N/S and E/W, and we normalize the estimated conditional variances in all the others directions. We use the reference lattice only for sake of easier calculus.

 

So that we could point up the interest of the normalization we extract a quasi-isotropic subimage from an urban area in the original image. Indeed the distributions of the eight parameters should be the same for anfor an isotropic image after the normalization. So we draw the histograms of the conditional variances before normalization and after normalization. We calculate the Kolmogorov-Smirnov distance dKS between each distribution and the distribution of the conditional variance estimated in the direction NE. In fact the directions E and N are privileged directions. Both directions correspond to the orientation of the streets and of the buildings. That is why we only compare the distributions corresponding to the 6 others directions. If the image were exactly isotropic dKS between the distributions relating to the directions NE and NW (lattices of same step sqrt(2)) should be zero before and after normalization.
In our test this value is 0.12 . This value is thus used as a reference value to compare the others dKS . The dKS values decrease after the normalization and are closer to the reference value 0.12. This points up the interest of such a normalization.
 

Kolmogorov-Smirnov distances w.r.t the distribution of the conditional variance in the direction NE
 
Before normalization
After normalization
NE
0
0
NEe
0.39
0.12
NEn
0.46
0.19
NOn
0.37
0.14
NO
0.12
0.12
NOo
0.36
0.14
 

Last modified: Thu Sep 3 15:36:03 MET DST 1998