Fast and Memory-Efficient Voronoi Diagram Construction on Triangle Meshes

© 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. Geodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window-VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window-VTP algorithm runs 3–8 times faster than Liu et al.'s method [LCT11] , 1.2 times faster than its FWP-MMP variant and more importantly uses 10–70 times less memory than both of them

A triangulation-invariant method for anisotropic geodesic map computation on surface meshes

pre-printThis paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner

Layered Fields for Natural Tessellations on Surfaces

Mimicking natural tessellation patterns is a fascinating multi-disciplinary problem. Geometric methods aiming at reproducing such partitions on surface meshes are commonly based on the Voronoi model and its variants, and are often faced with challenging issues such as metric estimation, geometric, topological complications, and most critically parallelization. In this paper, we introduce an alternate model which may be of value for resolving these issues. We drop the assumption that regions need to be separated by lines. Instead, we regard region boundaries as narrow bands and we model the partition as a set of smooth functions layered over the surface. Given an initial set of seeds or regions, the partition emerges as the solution of a time dependent set of partial differential equations describing concurrently evolving fronts on the surface. Our solution does not require geodesic estimation, elaborate numerical solvers, or complicated bookkeeping data structures. The cost per time-iteration is dominated by the multiplication and addition of two sparse matrices. Extension of our approach in a Lloyd's algorithm fashion can be easily achieved and the extraction of the dual mesh can be conveniently preformed in parallel through matrix algebra. As our approach relies mainly on basic linear algebra kernels, it lends itself to efficient implementation on modern graphics hardware.Comment: Natural tessellations, surface fields, Voronoi diagrams, Lloyd's algorith

Combinatorial Mesh Optimization

International audienceA new mesh optimization framework for 3D triangular surface meshes is presented, which formulates the task as an energy minimization problem in the same spirit as in Hoppe et al. (SIGGRAPH’93: Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, 1993). The desired mesh properties are controlled through a global energy function including data attached terms measuring the fidelity to the original mesh, shape potentials favoring high quality triangles, and connectivity as well as budget terms controlling the sampling density. The optimization algorithm modifies mesh connectivity as well as the vertex positions. Solutions for the vertex repositioning step are obtained by a discrete graph cut algorithm examining global combinations of local candidates.Results on various 3D meshes compare favorably to recent state-of-the-art algorithms. Applications consist in optimizing triangular meshes and in simplifying meshes, while maintaining high mesh quality. Targeted areas are the improvement of the accuracy of numerical simulations, the convergence of numerical schemes, improvements of mesh rendering (normal field smoothness) or improvements of the geometric prediction in mesh compression technique

Fast exact and approximate geodesics on meshes

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm. thereby obtaining an exact solution even more quickly.Engineering and Applied Science

Fast and Exact Discrete Geodesic Computation Based on Triangle-Oriented Wavefront Propagation

Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. In this problem, an effective window pruning strategy can significantly affect the actual running time. Due to its importance, we conduct an in-depth study of window pruning operations in this paper, and produce an exhaustive list of scenarios where one window can make another window partially or completely redundant. To identify a maximal number of redundant windows using such pairwise cross checking, we propose a set of procedures to synchronize local window propagation within the same triangle by simultaneously propagating a collection of windows from one triangle edge to its two opposite edges. On the basis of such synchronized window propagation, we design a new geodesic computation algorithm based on a triangle-oriented region growing scheme. Our geodesic algorithm can remove most of the redundant windows at the earliest possible stage, thus significantly reducing computational cost and memory usage at later stages. In addition, by adopting triangles instead of windows as the primitive in propagation management, our algorithm significantly cuts down the data management overhead. As a result, it runs 4-15 times faster than MMP and ICH algorithms, 2-4 times faster than FWP-MMP and FWP-CH algorithms, and also incurs the least memory usage