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Introduction to the classification problem

Let $ v$ be the original image defined on an open set $ \Omega$ of $ \mathbb{R}^2$ , and let $ C_i, 1\le i\le n$ , be $ n$ classes (i.e. grey or color levels). We first assume that thesse classes are predefined. The goal of image classification is to find a partition of $ \Omega$ in subsets $ \{\Omega_i\}_{i=1\dots n}$ , such that $ v$ is close to $ C_i$ in $ \Omega_i$ .

A variational approach can be defined: it consists of a cost function measuring the difference between the original image and the classified image:

$\displaystyle J((\Omega_i)_{i=1\dots n}) = \sum_{i=1}^n \int_{\Omega_i} (v(x)-C_i)^2\,dx+\alpha \sum_{i\ne j}\vert\Gamma_{ij}\vert,$ (2.36)

where $ \Gamma_{ij}$ represents the interface $ \Omega_i\cap\Omega_j$ between two subsets.

The main difficulty of this approach is that the unknowns are sets, and not variables. This is why the topological asymptotic analysis seems to be appropriate for solving this problem. The topological gradient and the corresponding numerical results are presented in [24].

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