A Gamma-convergence Applied to Multispectral Image Classification and Restoration

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. ln 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. ln this paper, we develop a model proposed by C.Samson, combining classification and restoration with non linear regularization. Ifs based on works developed for phase transitions in fluid mechanics by Van der Walls-Cahn-Hilliard, and uses a Gamma-convergence theory. This mode ! is named variational model, due to the fact that calculus of variations is its main too !. The classification-restoration 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 Differentiai 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 ofthe paper the results we have obtained.

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Smara Y.

Zaita I.

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