# Lecture SS 18 Advanced Topics in Scientific Computing

## Image and Surface Processing

Where/When: *Tu 8.15-10.00 and Th 10.15-12.00, Room 2.040*

### Topics

This course is dedicated to the analytical and numerical treatment of selected methods in image and surface processing. To this end, imaging fundamental problems of image reconstruction with applications for CT and MRI imaging and a shape calculus for large triangulated surfaces with applications in modelling and animation in computer graphics will be investigated.

In the first part, we will consider thin shell models along with a suitable discretization, e.g. via discrete differential geometry and discuss numerical simulations of thin shell deformations. The computation of smooth deformation paths which obey certain interpolation constraints is a fundamental task in typical computer graphics applications such as keyframe animation. To this end, an adaption of the classical spline model to Riemannian manifolds along with a suitable time discretization is presented. Furthermore, we will discuss efficient numerical schemes, for example representing a triangle mesh in terms of edge lengths, triangle areas, and angles between neighboring triangles as principal degrees of freedom.

In the second part, we will study variational methods in image reconstruction related to the famous Mumford Shah problem. This will lead to problems in the space of functions of bounded variation and a powerful convex optimisation duality calculus for the resulting functionals will be developed. Taking into account appropriate spatial finite element discretizations one obtains very high dimensional, nonlinear optimization problems. To solve these problems a particular focus will be on novel, efficient algorithms based on suitable primal-dual formulations. We will discuss concrete examples for the reconstruction of 3D images in computer tomography and magnetic resonance imaging.

Knowledge in analysis, basic knowledge on finite element methods and functional analysis are required.