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Lecture WS 20/21 Advanced Topics in Numerical Analysis

Analysis and Computation in Shape Spaces

Prof. Martin Rumpf

The lecture course investigates the rich theory of spaces of shapes, discusses the underlying geometric concepts as well as its numerical discretization, and studies a variety of applications in image processing, computer vision and computer graphics.

Instead of processing single images or surfaces we take a more global perspective and study images or surfaces as objects in a space of images or a space of surfaces. Over the last decade, concepts from Riemannian geometry have been applied to design and investigate such nonlinear and frequently infinite-dimensional shape spaces, with applications in shape morphing and modeling, in computational anatomy, as well as shape statistics and data driven approaches in character animation and editing. In particular, we will discuss the space of thin shell surfaces and show how surface morphing, motion extrapolation, or surface editing can be modeled mathematically and computed efficiently. We will also show how a natural blending between images can be understood as a geodesic path in the space of images with a length measurement involving transport and intensity modulation along transport paths. The lecture course will cover the analytical treatment such as the questions of well-posedness of different models. It will present numerical discretizations including a variational time discretization of geodesic curves and the convergence of these discrete models to their continuous counterparts.

As prerequisites some knowledge of functional analysis and numerical analysis including the theory of finite element methods are requested. The lecture course does not require advanced knowledge of differential geometry. The geometric concepts will be introduced when needed and underpinned with a detailed exposition of the theory of surfaces embedded in Euclidean space.

Please register for the course in eCampus, so that we can contact you for the invitation to the online lectures in zoom and for other organisatorial matters.

Zoom Meeting Link, Meeting-ID: 983 0263 8576, Code: 755470