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Register a new userDimensional operators for mathematical morphology on simplicial complexes
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Added by
Laurent Najman
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Fabio Dias
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In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at url{https://code.google.com/p/math-morpho-simplicial-complexes.}
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We present the `Basic S* algorithm for computing shortest path through a metric simplicial complex. In particular, given a metric graph, $G$, which is constructed as a discrete representation of an underlying configuration space (a larger continuous space/manifold typically of dimension greater than one), we consider the Rips complex, $mathcal{R}(G)$, associated with it. Such a complex, and hence shortest paths in it, represent the underlying metric space more closely than what the graph does. While discrete graph representations of continuous spaces is convenient for motion planning in configuration spaces of robotic systems, the metric induced in them by the ambient configuration space is significantly different from the metric of the configuration space itself. We remedy this problem using the simplicial complex representation. Our algorithm requires only an abstract graph, $G=(V,E)$, and a cost/length function, $d:Erightarrow mathbb{R}_+$, as inputs, and no global information such as an embedding or a global coordinate chart is required. The complexity of the Basic S* algorithm is comparable to that of Dijkstras search, but, as the results presented in this paper demonstrate, the shortest paths obtained using the proposed algorithm represent/approximate the geodesic paths in the original metric space significantly more closely.
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There have been several recent articles studying homology of various types of random simplicial complexes. Several theorems have concerned thresholds for vanishing of homology, and in some cases expectations of the Betti numbers. However little seems known so far about limiting distributions of random Betti numbers. In this article we establish Poisson and normal approximation theorems for Betti numbers of different kinds of random simplicial complex: ErdH{o}s-Renyi random clique complexes, random Vietoris-Rips complexes, and random v{C}ech complexes. These results may be of practical interest in topological data analysis.
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