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We introduce FreeLence, a lossless single-rate connectivity compression algorithm for triangle surface meshes. Based upon a geometry-driven traversal scheme we present two novel and simple concepts: free-valence connectivity encoding and entropy coding based on geometric context. Together these techniques yield significantly smaller rates for connectivity compression than current state of the art approaches - valence-based algorithms and Angle- Analyzer, with an average of $36\%$ improvement over the former and an average of $18\%$ over the latter on benchmark 3D models, combined with the ability to well adapt to the regularity of meshes. We also prove that our algorithm exhibits a smaller worst case entropy for a class of "'well-behaved"' triangle meshes than valence-driven connectivity encoding approaches.

The use of point sets instead of meshes became more popular during the last years. We present a new method for anisotropic fairing of a point sampled surface using an anisotropic geometric mean curvature flow. The main advantage of our approach is that the evolution removes noise from a point set while it detects and enhances geometric features of the surface such as edges and corners. We derive a shape operator, principal curvature properties of a point set, and an anisotropic Laplacian of the surface. This anisotropic Laplacian reflects curvature properties which can be understood as the point set analogue of Taubin's curvature-tensor for polyhedral surfaces. We combine these discrete tools with techniques from geometric diffusion and image processing. Several applications demonstrate the efficiency and accuracy of our method.

A new method for noise removal of arbitrary surfaces meshes is presented which focuses on the preservation and sharpening of non-linear geometric features such as curved surface regions and feature lines. Our method uses a prescribed mean curvature flow (PMC) for simplicial surfaces which is based on three new contributions: 1. the definition and efficient calculation of a discrete shape operator and principal curvature properties on simplicial surfaces that is fully consistent with the well-known discrete mean curvature formula, 2. an anisotropic discrete mean curvature vector that combines the advantages of the mean curvature normal with the special anisotropic behaviour along feature lines of a surface, and 3. an anisotropic prescribed mean curvature flow which converges to surfaces with an estimated mean curvature distribution and with preserved non-linear features. Additionally, the PMC flow prevents boundary shrinkage at constrained and free boundary segments.

We provide conditions for convergence of polyhedral surfaces and their discrete geometric properties to smooth surfaces embedded in Euclidian $3$-space. The notion of totally normal convergence is shown to be equivalent to the convergence of either one of the following: surface area, intrinsic metric, and Laplace-Beltrami operators. We further s how that totally normal convergence implies convergence results for shortest geodesics, mean curvature, and solutions to the Dirichlet problem. This work provides the justification for a discrete theory of differential geometric operators defined on polyhedral surfaces based on a variational formulation.

We present a new algorithm for fairing of space curves with respect spatial constraints based on a vector valued curvature function. Smoothing with the vector valued curvature function is superior to standard Frenet techniques since the individual scalar components can be modeled similar to curvature-based curve smoothing techniques in 2d. This paper describes a curve smoothing flow that satisfies strict spatial constraints and allows simultaneous control of both curvature functions.

This paper proposes a generalization of the ordinary de Casteljau algorithm to manifold-valued data including an important special case which uses the exponential map of a symmetric space or Riemannian manifold. We investigate some basic properties of the corresponding Bézier curves and present applications to curve design on polyhedra and implicit surfaces as well as motion of rigid body and positive definite matrices. Moreover, we apply our approach to construct canal and developable surfaces.