Preprint C171/2014
Optimized Quasi-Interpolators for Image Reconstruction and Consistent Volumetric Discretizations Inside Self-Intersecting Surfaces
Leonardo Koller Sacht
Keywords: interpolation | approximation | tetrahedralization

We first provide a review on the sampling and reconstruction problem. This review focuses mainly on theoretical and historic aspects that will be the fundamental to present our new framework. We then obtain new quasi-interpolators for continuous reconstruction of sampled images by minimizing a new objective function that takes into account the approximation error over the full Nyquist interval. To achieve this goal, we optimize with respect to all possible degrees of freedom in the approximation scheme. We consider three study cases offering different trade-offs between quality and computational cost: a linear, a quadratic, and a cubic scheme. Experiments with compounded rotations and translations confirm that our new quasi- interpolators perform better than the state-of-the-art for a similar computational cost.

We then turn our attention to a different problem on the field geometry processing: the one of generating consistent tetrahedral discretizations inside self-intersecting triangle meshes. With the goal of proposing an algorithm for this problem, we first introduce the main concepts of tetrahedral meshes, geometric flows, deformation energies and parametrization.  We then observe that most steps in the geometry processing pipeline for surfaces, like deformation, smoothing, subdivision and decimation, may create self-intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet-meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self-intersections in the input surface. This enables volumetric processing on self-intersecting models. We leverage conformalized mean-curvature flow, which removes self-intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.