Figure 1: Overview our scheme for tetrahedral meshes (illustrated in 2D). (a) We interpret the Morse complex of a simplicial mesh in terms of the primal mesh Σ (solid lines) and its dual Σ d (dashed lines). (b) Encoding the Discrete Morse gradient field entirely with the tetrahedra enables the use of compact topological data structures for morphological extraction. We associate the descending Morse complexes with the cells of Σ (c-d), the ascending Morse complexes with the cells of Σ d (e-f) and the Morse-Smale complex with the dually subdivided tetrahedral mesh Σ S (g), whose hexahedral cells are defined by a tetrahedron and one of its vertices. All relations are encoded strictly in terms of the vertices and tetrahedra of Σ. AbstractWe consider the problem of computing discrete Morse and Morse-Smale complexes on an unstructured tetrahedral mesh discretizing the domain of a 3D scalar field. We use a duality argument to define the cells of the descending Morse complex in terms of the supplied (primal) tetrahedral mesh and those of the ascending complex in terms of its dual mesh. The Morse-Smale complex is then described combinatorially as collections of cells from the intersection of the primal and dual meshes. We introduce a simple compact encoding for discrete vector fields attached to the mesh tetrahedra that is suitable for combination with any topological data structure encoding just the vertices and tetrahedra of the mesh. We demonstrate the effectiveness and scalability of our approach over large unstructured tetrahedral meshes by developing algorithms for computing the discrete gradient field and for extracting the cells of the Morse and Morse-Smale complexes. We compare implementations of our approach on an adjacency-based topological data structure and on the PR-star octree, a compact spatio-topological data structure.

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We propose the PR-star octree as a combined spatial data structure for performing efficient topological queries on tetrahedral meshes. The PR-star octree augments the Point Region octree (PR Octree) with a list of tetrahedra incident to its indexed vertices, i.e. those in the star of its vertices. Thus, each leaf node encodes the minimal amount of information necessary to locally reconstruct the topological connectivity of its indexed elements. This provides the flexibility to efficiently construct the optimal data structure to solve the task at hand using a fraction of the memory required for a corresponding data structure on the global tetrahedral mesh. Due to the spatial locality of successive queries in typical GIS applications, the construction costs of these runtime data structures are amortized over multiple accesses while processing each node. We demonstrate the advantages of the PR-star octree representation in several typical GIS applications, including detection of the domain boundaries, computation of local curvature estimates and mesh simplification.

We address the problem of performing spatial queries on\ud tetrahedral meshes. These latter arise in several application\ud domains including 3D GIS, scientific visualization, finite el-\ud ement analysis. We have defined and implemented a family\ud of spatial indexes, that we call tetrahedral trees. Tetrahedral\ud trees subdivide a cubic domain containing the mesh in an\ud octree or 3D kd-tree fashion, with three different subdivision\ud criteria. Here, we present and compare such indexes, their\ud memory usage, and spatial queries on them

Simplicial complexes are widely used to discretize shapes. In low dimensions, a 3D shape is represented by discretizing its boundary surface, encoded as a triangle mesh, or by discretizing the enclosed volume, encoded as a tetrahedral mesh. High‐dimensional simplicial complexes have recently found their application in topological data analysis. Topological data analysis aims at studying a point cloud P, possibly embedded in a high‐dimensional metric space, by investigating the topological characteristics of the simplicial complexes built on P. Analysing such complexes is not feasible due to their size and dimensions. To this aim, the idea of simplifying a complex while preserving its topological features has been proposed in the literature. Here, we consider the problem of efficiently simplifying simplicial complexes in arbitrary dimensions. We provide a new definition for the edge contraction operator, based on a top‐based data structure, with the objective of preserving structural aspects of a simplicial shape (i.e., its homology), and a new algorithm for verifying the link condition on a top‐based representation. We implement the simplification algorithm obtained by coupling the new edge contraction and the link condition on a specific top‐based data structure, that we use to demonstrate the scalability of our approach.

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