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Restoration and classification coupling

Another solution consists of coupling classification with restoration, and to adapt the approach introduced in section 2.4. The idea is to first consider an iteration of the topological asymptotic analysis for the image restoration problem in order to smooth the image, and then to classify this smooth image without any regularization. If we remove the regularization term from equation (2.36), which leads to the unregularized classification problem, then the optimal subset $ \Omega_i$ is the set of pixels that are closer to $ C_i$ than to any other $ C_j$ . In other words, each pixel is assigned to the subset corresponding to its closest class.

In the perturbed problem (2.29), instead of setting $ c=0$ (or $ c=\varepsilon$ from the numerical point of view) on the edge set and $ c=c_0$ elsewhere, we set

$\displaystyle c_1 = \left\{\begin{array}{l} \varepsilon \textrm{ on the edge se... ...\ \displaystyle \frac{c_0}{\varepsilon} \textrm{ elsewhere.} \end{array}\right.$ (2.37)

The algorithm is then the following:

$ \bullet$
Application of the restoration algorithm defined in section 2.4, with $ c_1$ defined by (2.37) instead of (2.29).
$ \bullet$
Unregularized classification of the image $ u_1$ , using for example the closest class algorithm (in which each pixel is assigned to the subset corresponding to its closest class).

As previously seen, the complexity of this algorithm is $ \mathcal{O}(n.\log(n))$ , and the various numerical results presented in [24] show the relative efficiency of these approaches. Moreover, it is possible to regularize more or less the image by choosing different values of $ c_1$ , and it allows us to also obtain good results on noisy images.

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