Luminita A. Vese

November, 1996

This thesis, formed by three parts, is devoted to the study of a class of variational problems arising from image analysis and to some degenerated and elliptic PDE's of second order, related to the reconstruction of noisy and blurred images and to curves evolution.

In the first part, we study a denoising-deblurring problem, by variational methods. The reconstruction model is posed as a minimization problem, and the energy to be minimized depends on the solution $u$ and on its gradient $Du$, by a regularizing term. We study this problem in the framework of functions of bounded variation $BV$ and we give optimality conditions on the solution. We present two methods to approach the solution in continuous variables, together with a regularity result. In the end of this part, we describe two numerical methods to compute the solution.

The second part is devoted to some fully nonlinear elliptic and parabolic PDE's of second order, having viscosity solutions, with two applications. The first, considering quasilinear equations, is related to image analysis. The second is devoted to a method to convexify functions, by technics of curves evolution, like the mean curvature motion. We present an algorithm and numerical results to compute the convex envelope of a function, for curves and surfaces.

Finally, in the third part of this thesis we compare the algorithms and potentials presented in Part I and Part II, on several numerical results of image reconstruction.