Model Reduction for Interactive Geometry Processing

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The research field of geometry processing is concerned with the representation, analysis, modeling, simulation and optimization of geometric data. In this thesis, we introduce novel techniques and efficient algorithms for problems in geometry processing, such as the modeling and simulation of elastic deformable objects, the design of tangential vector fields or the automatic generation of spline curves. The complexity of the geometric data determines the computation time of algorithms within these applications. The high resolution of modern meshes, for example, poses a big challenge when geometric processing tools are expected to perform at interactive rates. To this end the goal of this thesis is to introduce fast approximation techniques for problems in geometry processing. One line of research to achieve this goal will be to introduce novel model order reduction techniques to problems in geometry processing. Model order reduction is a concept to reduce the computational complexity of models in numerical simulations, energy optimizations and modeling problems. New specialized model order reduction approaches are introduced and existing techniques are applied to enhance tools within the field of geometry processing. In addition to introducing model reduction techniques, we make several other contributions to the field. We present novel discrete differential operators and higher order smoothness energies for the modeling of tangential n-vector fields. These are used, to develop novel tools for the modeling of fur, stroke based renderings or anisotropic reflection properties on meshes. We propose a geometric flow for curves in shape space that allows for the processing and creation of animations of elastic deformable objects. A new optimization scheme for sparsity regularized functionals is introduced and used to compute natural, localized deformations of geometrical objects. Lastly, we reformulate the classical problem of spline optimization as a sparsity regularized optimization problem.