Next: . About wavelets Up: Supervised classification for textured Previous: . Introduction

# . Classification

Partition, level set approach

The image is considered as a function (where is an open subset of )

We denote . The collection of open sets forms a partition of if and only if

, and if      Ø

We denote the boundary of (except points belonging also to ), and the interface between and (see figure 1).

In order to get a functional formulation rather than a set formulation, we suppose that for each there exists a lipschitz function such that:

is thus completely determined by .

Regularization

In our equations, there will appear some Dirac and Heaviside distributions and . In order that all the expressions we write have a mathematical meaning, we use the classical regular approximations of these distributions (see figure 2):

Figure 3 shows how the regions are defined by these distributions and the level sets.

Functional

Our functional will have three terms:

1) A partition term:

 (2.1)

2) A regularization term:

 (2.2)

In practice, we seek to minimize:

 (2.3)

3) A data term:

 (2.4)

The functional we want to minimize is the sum of the three previous terms:

 (2.5)

Next: . About wavelets Up: Supervised classification for textured Previous: . Introduction
Jean-Francois Aujol 2002-12-03