We compute a packet wavelet decomposition of the image (up to the second order in practice): we get channels ( in practice).

We call the energy at pixel the vector , where is the square of the wavelet coefficient in the sub-band at pixel .

**Hypotheses**:

(H
**1) **We assume that, for each texture
, in each channel
, the square of the wavelet coefficients follows a law of the type (4.4).

(H
**2) **We consider that the different channels are independent.

**Data term**

We want to maximize , where is the assumed class (Maximum Likelihood Estimator).

Our fitting term to the data is:

And 5.1 is given by:

(5.2) |

**The functional**

We are now able to completely write the functional which models the classification problem for textured images:

**Euler-Lagrange Equations**

We get the following system composed of -coupled PDE's: we have for

**Dynamical scheme**

To solve the PDE's system (5.4), we embed it in the following dynamical scheme (
):

with the initial condition is the Euclidean signed distance function to the zero level set .

We discretize this system with finite differences.

**Reinitialization**

The are initialized as Euclidean signed distance functions. Nevertheless, as in the classical active contour method, the evolution of the with respect to (5.5) does not keep them as Euclidean signed distance functions to their zero level set.

To reinitialize into the Euclidean signed distance function, we use the PDE: