Layout Embedding via Combinatorial Optimization

Abstract

Figure 1: We embed a given layout connectivity (left, visualized as a coarse mesh for illustration) into a target surface with prescribed landmark positions (center left) by successively embedding edges as shortest paths in an optimized order. Previous methods choose a greedy sequence of locally optimal decisions, which can lead to severe topological artifacts (center right) due to accidental blocking of subsequently inserted paths. Our branch-and-bound method globally optimizes over all possible edge insertion sequences to find an embedding of shortest total path length, which yields the expected homotopy class (right).

Surface Maps via Adaptive Triangulations

“…The surfaces are clustered into 10 groups of similar developmental stage and the tasks are (1) computing an average surface per group and (2) sequentially interpolating between these average surfaces via a common triangulation. The original experiment in [ESD * 22] was performed via [SCBK20] (initialized with [BSK21]), which only allows computing pairwise maps. Within each group, one surface was mapped to all others to allow averaging, and sequential maps were computed between consecutive pairs of average surfaces.…”

“…Note that this procedure for initial landmark satisfaction is a simple heuristic to a difficult problem that involves choosing a map homotopy class (between the genus-0 input surfaces punctured at the landmark positions, cf. [BSK21]). While it is not in general guaranteed that our penalty term satisfies all landmark constraints, it succeeded within a few seconds in all examples shown here.…”

“…Choosing a map's homotopy class (cf. [FM11]), i.e., how it twists around surface handles, tunnels, and fixed correspondences is an ongoing research effort [KS04, SAPH04, LGQ08, BSK21, BSCK21]. In this paper, we separate our problem setting from these research questions by focusing on the topologically simpler case of mapping between genus‐0 surfaces via the sphere.…”

Hex-Mesh Generation and Processing: a Survey

“…Current methods such as ] are limited to simply discarding a feature line if a nonintersecting chain of edges can be computed, and insert it otherwise, regardless of its geometric deviation from the target curve. The use of more sophisticated heuristics such as [Born et al 2021] may significantly increase the number of features successfully transferred and also help reduce the geometric distortion due to a wiser choice of the feature edges. Alternative methods tailored to operate on the connectivity of hexahedral meshes may also be developed, and coupled with (as local as possible) hex-mesh refinement techniques to resolve intricate configurations.…”

Isogeometric analysis of diffusion problems on random surfaces

“…In practice, the surface representation ( 10) is typically not directly available for more complex geometries. However, the partitioning of surface meshes into conforming quadrilateral patches is well solved, see for example [6,10,33,34,39] and the references therein. The procedure suggested there is based on determining a suitable set of nodes called singularities and connecting these by a set of arcs called separatrices, such that the resulting partition consists of multiple valid conforming quadrilateral patches.…”