ROF MODEL FOR IMAGE DENOISING: analysis and implementation

The purpose of this thesis is presenting the Rudin-Osher-Fatemi filter for image analysis. We will consider mostly the application for denoising, however there are other relevant application, such as restoring damaged inscriptions. This filter involves solving a minimization problem with the objective function built summing an $L^2$-norm squared, which measure the distance from the reference image, and the total variation functional. Therefore, one of the main concerns of this work is to understand the behaviour of this problem under a variational point of view, this involves studying the total variation functional and the main properties of the bounded variation functions, on which it is defined. In the first chapter the analytical preliminaries will be presented, in particular we will focus on the $BV$ function, the Caccioppoli sets and the coarea formula, which links the total variation with the perimeter of the function level sets (which are Caccioppoli sets). Then, the idea of reduced boundary will be presented and studied, in order to provide a better understanding of the perimeter of Caccioppoli sets and a glimpse on the differentiation of $BV$ functions. This preliminary section will end with the definition and some basic properties of $\Gamma$-convergence. The second chapter is mostly dedicated to the study of the study of the Rudin-Osher-Fatemi problem, in particular: the existence and uniqueness of the solution, finding the associated Euler-Lagrange equation and finding an analytical solution for a simple test case. The last chapter will open with an idea for the discretization of the considered problem and the proof that it actually converges to it (in the sense of $\Gamma$-convergence). Then a modified Arrow-Hurwitz algorithm will be presented with the purpose of implementing a solver for this problem. Finally we will comment some of the images produced in this way and compare the analytical solution with the computational one.