Fast and exact geodesic computation using Edge-based Windows Grouping.

Abstract

Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. As the “Big Data Era” arrives, a fast and accurate solution to the geodesic computation problem on large scale models with constantly increasing resolutions is desired. However, it is still challenging to deal with the speed, memory cost and accuracy of the geodesic computation at the same time. This thesis addresses the aforementioned challenge by proposing the Edge- based Windows Grouping (EWG) technique. With the local geodesic information encoded in a “window”, EWG groups the windows based on the mesh edges and processes them together. Thus, the interrelationships among the grouped windows can be utilized to improve the performance of geodesic computation on triangle meshes. Based on EWG, a novel exact geodesic algorithm is proposed in this thesis, which is fast, accurate and memory-efficient. This algorithm computes the geodesic distances at mesh vertices by propagating the geodesic information from the source over the entire mesh. Its high performance comes from its low computational redundancy and management overhead, which are both introduced by EWG. First, the redundant windows on an edge can be removed by comparing its distance with those of the other windows on the same edge. Second, the windows grouped on an edge usually have similar geodesic distances and can be propagated in batches efficiently. To the best of my knowledge, the proposed exact geodesic algorithm is the fastest and most memory-efficient one among all existing methods. In addition, the proposed exact geodesic algorithm is revised and employed to construct the geodesic-metric-based Voronoi diagram on triangle meshes. In this application, the geodesic computation is the bottleneck in both the time and memory costs. The proposed method achieves low memory cost from the key observation that the Voronoi diagram boundaries usually only cross a minority of the meshes’ triangles and most of the windows stored on edges are redundant. As a result, the proposed method resolves the memory bottleneck of the Voronoi diagram construction without sacrificing its speed.