Classification

INTRODUCTION

In the previous section we have defined models so as to transform grey-level information into more discriminating information. In this section we classify the image of parameters through a modified Fuzzy Cmeans algorithm. The new algorithm proposed herein includes an entropy term.
In this framework the word fuzzy refers to fuzzy sets. Each set is characterized by a continuous function defined on [0,1]. That is to say each object (for us a pixel) belongs to each set with some degree of membership.
Thus a partition of the image is characterized by a partition matrix U=[uij].
uij represents the degree of membership of the object i in the set j.
We set up a clustering method to estimate this matrix. In our framework the objects are the pixels and the sets are the different classes.

MODIFIED FUZZY CMEANS

The classification consists in minimizing the following algorithm:

Under the following constraint:

Where:

    m is the degree of fuzzyness
    C is the number of clusters
    N is the number of pixels
    c_i is the centroide of cluster i.
    d²(x_j,c_i) is the euclidian distance.
    is the a priori probability of cluster i.

ALGORITHM

We minimize J with respect to c_i and then we minimize with respect to u_ij. We remove classes whose probability is under a fixed threshold.

Parameters are updated as follows (m=2):

RESULTS

These results point up the great interest of the modified Fuzzy Cmeans algorithm that does not need to know any a priori number ofclasses. Thus, for our application of urban areas extraction, we do not need to a priori know whether there is an urban area or not in the image.
left column: an urban area
right column: no urban area
Images c and d were obtained with the classical algorithm and a fixed number of 2 clusters. An urban area is detected for both cases.
Images e and f were obtained with modified algorithm. So the urban area is well detected on the left image and no one is detected on the right image.

Last modified: Tue May 5 17:08:07 MET DST 1998