Practical Reduction of Edge Flip Sequences in Two-Dimensional Triangulations (1310.2586v1)

Abstract: The development of laser scanning techniques has popularized the representation of 3D shapes by triangular meshes with a large number of vertices. Compression techniques dedicated to such meshes have emerged, which exploit the idea that the connectivity of a dense mesh does not deviate much from the connectivity that can be constructed automatically from the vertex positions (while possibly being guided by additional codes). The edge flip is one of the tools that can encode the differences between two meshes, and it is important to control the length of a sequence of flips that transform one triangulation into another. This paper provides a practical solution to this problem. Indeed, the problem of determining a minimal sequence of edge flips between two triangulations is NP-complete for some types of triangulations including manifold triangulations of surfaces, so that it is necessary to develop heuristics. Moreover, it is sometimes difficult to establish a first sequence of flips between two meshes, and we advocate a solution based on the reduction of an existing sequence. The new approach we propose is founded on the assignment of labels to identify the edges, with a property of label transfer during a flip. This gives a meaning to the tracking of an edge in a sequence of flips and offers the exploitation of very simple combinatorial properties. All the operations are performed directly on the sequence of labels denoting the edges to be flipped, almost regardless of the underlying surface, since only local temporary connectivity is involved.