Seminari del Dipartimento di Matematica

Seminario del 2021

3D shape reconstruction via PDEs approach: looking for well-posedness

nell'ambito della serie: TOPICS IN MATHEMATICS 2020/2021

analisi numerica

The three-dimensional reconstruction of an object is an interesting topic with many applications in different fields and has attracted several researchers. The applications range goes from the biomedical 3D reconstruction of human tissues to the approximation of the surface of astronomical objects, from archeology for the digitization of artistic works to the recent development of 3D printing. The first being interested in this problem were some opticians in the Fifties-Sixties. Afterwards, B.K.P. Horn first formulated the Shape-from-Shading (SfS) problem for a single gray-level image of the object. The goal was to get the 3D surface represented in the input image solving a partial differential equation or a variational problem. This problem gave rise to an expansion in the field of mathematics and some researchers tried to prove the well-posedness in the framework of weak solutions. The first works of Lions, Rouy and Tourin in the early 90s inserted the SfS problem in the context of the viscosity solutions frameworks, hence in a much more theoretical area. In this seminar I will start dealing with the orthographic SfS problem with Lambertian reflectance model, the classical and simplest setup for this ill-posed problem that can be modeled by first order Hamilton-Jacobi equations. During the seminar I will briefly introduce some notions of Hamilton-Jacobi equations, viscosity solutions and other ingredients necessary to understand the problem in a general setting. I will continue exploring some non-Lambertian reflectance models and we will see how it is possible to derive a well-posed problem adding information in a natural way. Finally, I will talk about the more recent Shape-from-Polarization problem and the advantages of it with respect to the SfS.