**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:

**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) |