Construction of Injective Mappings Of Meshes.
Yaron Lipman · arXiv (Cornell University) · 2013
This paper introduces three sets of sufficient conditions, for generating injective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it consistently preserves or inverts the orientations of all elements. However, these conditions are insufficient to guarantee injectivity. In this paper we provide additional simple conditions that, together with the above mentioned necessary conditions guarantee injectivity of the simplicial map. The first set of conditions generalizes classical global inversion theorems to the mesh (piecewise-linear) case. That is, proves that in case the boundary simplicial map is bijective and the necessary condition holds the map is a bijection. The second set of conditions is concerned with mapping of a mesh to a polytope and replaces the (often hard) requirement of a bijective boundary map with a collection of linear constraints that guarantees that the resulting map is injective over the interior of the mesh. These linear conditions provide a practical tool for optimizing an injective map of the mesh while allowing the boundary map to adjust freely. Allowing more freedom in the boundary conditions is useful for two reasons: a) it circumvents the hard task of providing a bijective boundary map, and b) it allows optimizing the boundary map together with the simplicial map to achieve lower energy levels. The third set of conditions adds to the second set the requirement that the boundary maps are orientation preserving as-well. This set of conditions guarantees that the map is injective on the boundary of the mesh as-well as its interior. Several experiments using the sufficient conditions are shown for mapping triangular meshes injectively. A secondary goal of this paper is to advocate and develop the tool of degree in the context of geometry processing and modeling of meshes.