Constrained Optimization with Geometric Objects

In Geometry Processing, the constrained optimization with geometric objects is of significant interest due to an ever increasing availability of three-dimensional geometric data in a variety of application areas ranging from imaging science to geosciences. Encouraged by the excellent results obtained in the FWF funded research project `Geometric Optimization with Moving and Deformable Objects` we propose the present follow-up project. In the new project we focus our research on two main topics. The first topic is that we add constraints to the optimization tasks. This allows us to study the constrained positioning or deformation of geometric objects such as curves, curve networks, B-spline surfaces and triangle meshes. The constraints we employ are of a geometric nature. We will study, e.g. the optimization of curve networks in the presence of obstacles, which have to be avoided, the uni-directional approximation of point clouds by B-spline surfaces, the rigid registration of 3D shapes such that they do not penetrate each other, or the computation of energy minimizing curves on surfaces that are constrained to lie in areas of high mean curvature of the surface. The second topic deals with the formulation of the optimization problems. We will replace the commonly used L2 norm (sum of squared error terms) by the L1 norm (sum of error terms). This leads us to non-smooth optimization, due to the fact that the L1 norm is not everywhere differentiable. Although the optimization tasks become more difficult to solve we expect the benefit of increased robustness if outliers are present in the data. The initial results we have obtained so far are very promising and encourage us to pursue this new research direction. Together with our national and international collaborators we are expecting important contributions to fundamental research with potential applications that go beyond Geometry Processing and include brain imaging science and geosciences.

In Geometry Processing, the constrained optimization with geometric objects is of significant interest due to an ever increasing availability of three-dimensional geometric data in a variety of application areas ranging from imaging science to geosciences. Encouraged by the excellent results obtained in the FWF funded research project `Geometric Optimization with Moving and Deformable Objects` we propose the present follow-up project. In the new project we focus our research on two main topics. The first topic is that we add constraints to the optimization tasks. This allows us to study the constrained positioning or deformation of geometric objects such as curves, curve networks, B-spline surfaces and triangle meshes. The constraints we employ are of a geometric nature. We will study, e.g. the optimization of curve networks in the presence of obstacles, which have to be avoided, the uni-directional approximation of point clouds by B-spline surfaces, the rigid registration of 3D shapes such that they do not penetrate each other, or the computation of energy minimizing curves on surfaces that are constrained to lie in areas of high mean curvature of the surface. The second topic deals with the formulation of the optimization problems. We will replace the commonly used L2 norm (sum of squared error terms) by the L1 norm (sum of error terms). This leads us to non-smooth optimization, due to the fact that the L1 norm is not everywhere differentiable. Although the optimization tasks become more difficult to solve we expect the benefit of increased robustness if outliers are present in the data. The initial results we have obtained so far are very promising and encourage us to pursue this new research direction. Together with our national and international collaborators we are expecting important contributions to fundamental research with potential applications that go beyond Geometry Processing and include brain imaging science and geosciences.