Title: Variational methods and partial differential equations for image
analysis and curve evolution
Luminita A. Vese
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.