136 research outputs found
Computing a discrete Morse gradient from a watershed decomposition
We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological characterization of the function and of the shape on which it is defined through a Morse decomposition. Unfortunately, discrete Morse theory and related algorithms assume that the input scalar function has no flat areas, whereas such areas are common in real data and are easily handled by watershed algorithms. We propose here a new approach for building a discrete Morse gradient on a triangulated 3D shape endowed by a scalar function starting from the decomposition of the shape induced by the watershed transform. This allows for treating flat areas without adding noise to the data. Experimental results show that our approach has significant advantages over existing ones, which eliminate noise through perturbation: it is faster and always precise in extracting the correct number of critical elements
Topological Decompositions for 3D Non-manifold Simplicial Shapes
Modeling and understanding complex non-manifold shapes is a key issue in several applications including form-feature identification in CAD/CAE, and shape recognition for Web searching. Geometric shapes are commonly discretized as simplicial 2- or 3-complexes embedded in the 3D Euclidean space. The topological structure of a non-manifold simplicial shape can be analyzed through its decomposition into a collection of components with simpler topology. The granularity of the decomposition depends on the combinatorial complexity of the components. In this paper, we present topological tools for structural analysis of three-dimensional non-manifold shapes. This analysis is based on a topological decomposition at two different levels. We discuss the topological properties of the components at each level, and we present algorithms for computing such decompositions. We investigate the relations among the components, and propose a graph-based representation for such relations
A dimension-independent simplicial data structure for non-manifold shapes
We consider the problem of representing and manipulating non-manifold
multi-dimensional shapes, discretized as -dimensional simplicial
Euclidean complexes, for modeling finite element meshes derived from CAD
models. We propose a dimension-independent data structure for simplicial
complexes, that we call the {\em Incidence Simplicial (IS)} data
structure. The IS data structure is scalable to manifold complexes, and
supports efficient traversal and update algorithms for performing
topological modifications, such as hole removal or dimension reduction. It
has the same expressive power and performances as the incidence graph,
commonly used for dimension-independent representation of simplicial and
cell complexes, but it is much more compact. We present efficient
algorithms for traversing, generating and updating a simplicial complex
described as an IS data structure. We compare the IS data structure with
dimension-independent and dimension-specific representations for
simplicial complexes. Finally, we briefly discuss two applications that
the IS data structure supports, namely decomposition of non-manifold
objects for effective geometric reasoning, and multi-resolution modeling
of non-manifold multi-dimensional shapes
Computing discrete Morse complexes from simplicial complexes
We consider the problem of efficiently computing a discrete Morse complex on
simplicial complexes of arbitrary dimension and very large size. Based on a
common graph-based formalism, we analyze existing data structures for
simplicial complexes, and we define an efficient encoding for the discrete
Morse gradient on the most compact of such representations. We theoretically
compare methods based on reductions and coreductions for computing a discrete
Morse gradient, proving that the combination of reductions and coreductions
produces new mutually equivalent approaches. We design and implement a new
algorithm for computing a discrete Morse complex on simplicial complexes. We
show that our approach scales very well with the size and the dimension of the
simplicial complex also through comparisons with the only existing
public-domain algorithm for discrete Morse complex computation. We discuss
applications to the computation of multi-parameter persistent homology and of
extrema graphs for visualization of time-varying 3D scalar fields
The Stellar decomposition: A compact representation for simplicial complexes and beyond
We introduce the Stellar decomposition, a model for efficient topological data structures over a broad range of simplicial and cell complexes. A Stellar decomposition of a complex is a collection of regions indexing the complex’s vertices and cells such that each region has sufficient information to locally reconstruct the star of its vertices, i.e., the cells incident in the region’s vertices. Stellar decompositions are general in that they can compactly represent and efficiently traverse arbitrary complexes with a manifold or non-manifold domain. They are scalable to complexes in high dimension and of large size, and they enable users to easily construct tailored application-dependent data structures using a fraction of the memory required by a corresponding global topological data structure on the complex.
As a concrete realization of this model for spatially embedded complexes, we introduce the Stellar tree, which combines a nested spatial tree with a simple tuning parameter to control the number of vertices in a region. Stellar trees exploit the complex’s spatial locality by reordering vertex and cell indices according to the spatial decomposition and by compressing sequential ranges of indices. Stellar trees are competitive with state-of-the-art topological data structures for manifold simplicial complexes and offer significant improvements for cell complexes and non-manifold simplicial complexes. We conclude with a high-level description of several mesh processing and analysis applications that utilize Stellar trees to process large datasets
IA*: An adjacency-based representation for non-manifold simplicial shapes in arbitrary dimensions
Propriétés topologiques pour la modélisation géométrique de domaines d'études comportant des singularités non-variétés
L’étude de comportement mécanique de
structures et/ou d’écoulements s’appuie fréquemment sur des modèles géométriques perçus
comme des assemblages de volumes, surfaces, lignes, connectés entre eux et comportant
des singularités non-variétés. Une classification d’objets comportant des singularités
non-variétés et des propriétés topologiques globales sont présentées pour accroître
l’efficacité des modeleurs et la génération des contraintes de maillages
TopoCluster: A Localized Data Structure for Topology-based Visualization
Unstructured data are collections of points with irregular topology, often represented through simplicial meshes, such as triangle and tetrahedral meshes. Whenever possible such representations are avoided in visualization since they are computationally demanding if compared with regular grids. In this work, we aim at simplifying the encoding and processing of simplicial meshes. The paper proposes TopoCluster, a new localized data structure for tetrahedral meshes. TopoCluster provides efficient computation of the connectivity of the mesh elements with a low memory footprint. The key idea of TopoCluster is to subdivide the simplicial mesh into clusters. Then, the connectivity information is computed locally for each cluster and discarded when it is no longer needed. We define two instances of TopoCluster. The first instance prioritizes time efficiency and provides only a modest savings in memory, while the second instance drastically reduces memory consumption up to an order of magnitude with respect to comparable data structures. Thanks to the simple interface provided by TopoCluster, we have been able to integrate both data structures into the existing Topological Toolkit (TTK) framework. As a result, users can run any plugin of TTK using TopoCluster without changing a single line of code
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