A GAMMA-CONVERGENCE APPLIED TO MULTISPECTRAL IMAGE CLASSIFICATION AND RESTORATION
M.Iddir Zait · 2001
The main objective of this paper is to develop a model which combines in the same process image classification and restoration. Image classification consists of assigning a label to each site of an image to produce a partition into homogeneous labeled areas. The classification problem concerns many applications, like in the field of remote sensing: land use management, monitoring, urban areas. Observed images are often affected by degradations. The purpose of restoration is to find an original image describing a real scene from the observed one. This problem can be identified by inverse problem. In general, it is ill-posed in the sense of Hadamard. The existence and uniqueness of the solution are not guaranteed. It is therefore necessary to introduce an a priori constraint on the solution. This operation is the regularization. We can distinguish two types of regularization: the linear one and the non-linear. In this paper, we develop a model proposed by C.Samson, combining classification and restoration with non linear regularization. It’s based on works developed for phase transitions in fluid mechanics by Van der Walls-Cahn-Hilliard, and uses a Gamma-convergence theory. This model is named variational model, due to the fact that calculus of variations is its main tool. The classificationrestoration is obtained by minimizing a sequence of functionals. The result is a classified and restored image, and corresponds to an image composed of homogeneous classes, separated by minimum length boundaries. The minimization problem is transformed by Euler-Lagrange equations into PDEs (Partial Differential Equations) resolution problem. We have experimented this model on synthetic and satellite images. For real images, we have considered images from SPOT-1 satellite representing the regions of Blida in south-west of Algiers (capital of Algeria). We will discuss at the end of the paper the results we have obtained. RESUME : L'objectif principal de ce papier est developper un modele qui combine dans un meme processus une operation de classification d’image et une operation de restauration. La classification consiste partitionner une image en regions reperees par des etiquettes differentes. Le probleme de classification concerne beaucoup d'applications telles que la gestion de la couverture terrestre en teledetection, le suivi de l’urbanisation etc... Les images observees sont souvent degradees. Le but de la restauration est de retrouver l’image originale a partir de celle observee. Ce probleme est un probleme inverse mal pose au sens d’Hadamard. L'existence et l’unicite de la solution ne sont pas assurees. Il est alors necessaire de regulariser la solution par l’introduction d’un a priori. Nous pouvons distinguer deux types de regularisation: lineaire et non-lineaire. Dans ce papier, nous developpons un modele variationnel, propose par C.Samson, qui combine classification et restauration avec une regularisation non lineaire. Il est base sur les travaux de Van der walls Cahn-Hilliard developpes pour les transitions de phase en mecanique des fluides, et utilise la theorie de la Gamma Convergence. La classification restauration est obtenue en minimisant une sequence de fonctionnelles. Le resultat correspond a une image composee de classes homogenes separees par des interfaces de longueur minimales. Le probleme de minimisation est transforme par les equations d’Euler-Lagrange en un probleme de resolution d’equations aux derivees partielles (EDP). Nous avons teste ce modele sur des images de synthese et sur des images satellitaires de la serie SPOT-1 recouvrant la region de Blida dans sud est d'Alger (capital d'Algerie). Nous presenterons a la fin du papier les resultats obtenus.