Dottorato in Matematica Applicata XXVII ciclo Analysis and Approximation of Non-Lambertian Shape-from-Shading Models Advisor: Chiar.mo Prof.

I feel the need to thank many people who have accompanied me during these years. First of all, I want to thank my advisor Prof. Maurizio Falcone who believed in me and supported me during these years, allow me to grow up in the professional field but also in the human one. Thanks goes to Professor Michael Breuß for his kind hospitality and the availability during my stay in Cottbus at the Brandenburg University of Technology (BTU Univer-sity) that produced a fruitful collaboration with him and his research group. Thanks goes to Alessandra Seghini and the whole staff of the Centro di Calcolo, placed in my Department, for computer support and the great availability and competence. Thanks to the Istituto Italiano di Tecnologia (IIT) placed in Genova, where I had the possibility to work and collaborate with valuable people. Thanks to all of my collaborators of these years, people with which I worked and wrote papers and many thanks to all the research group of Numerical Analysis of my Depart-ment, for the friendly environment that allowed me to spend happily and pleasantly also from a social perspective these years of my Ph.D. and for the fruitful discussions that have contributed to my professional growth. A really heartfelt thanks to my family, always by my side, ready to get behind me and give me confidence to believe in me and in my ability whenever I needed it. Many thanks to my boyfriend who endured me and supported me during the last de-manding and difficult period of this important year, always ready to make every effort, move heaven and earth in order to inject happiness and calm into me. Finally, but not the lasts, I want to thank my closest colleagues with whom I shared the office in these years, which have been shown not only colleagues but also true friends. Contents Introduction i 1. A review of some non-Lambertian models 1 1.1. Formulation of the general model.......................