Geometrie computing in shape space

In geometric computing, a shape is typically viewed as a set of points and then represented accordingly, depending on the available data and the application. However, it has been known for a long time that simple shapes may be treated more elegantly by viewing them as points in a higher-dimensional space. Examples include line and sphere geometries, kinematical geometry and Lie groups. Recently, fundamental mathematical properties of spaces of more complicated objects have been studied and applied to certain problems in image processing. In this talk, the speaker will present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes - triangular meshes or more general straight line graphs in Euclidean space - are considered as points in a shape space. We show how to equip shape space with useful semi-Riemannian metrics which aid the user in design and modeling tasks. An important example of such a metric enables us to explore the space of isometric deformations of a given shape. Much of the work relies on an efficient computation of geodesies in shape spaces; for this, we present a multi-resolution framework to solve the boundary value problem as well as the initial value problem. Working in shape space, various problems from geometric modeling and geometry processing can be treated in a consistent and unified way by linking them to geometric concepts such as parallel transport or the exponential map. These applications include shape morphing, deformation transfer, shape exploration and the computation of piecewise developable shapes such as D-forms from their unfolding. This is joint work with Martin Kilian and Niloy Mitra.