58 research outputs found
Computing Multidimensional Persistence
The theory of multidimensional persistence captures the topology of a
multifiltration -- a multiparameter family of increasing spaces.
Multifiltrations arise naturally in the topological analysis of scientific
data. In this paper, we give a polynomial time algorithm for computing
multidimensional persistence. We recast this computation as a problem within
computational algebraic geometry and utilize algorithms from this area to solve
it. While the resulting problem is Expspace-complete and the standard
algorithms take doubly-exponential time, we exploit the structure inherent
withing multifiltrations to yield practical algorithms. We implement all
algorithms in the paper and provide statistical experiments to demonstrate
their feasibility.Comment: This paper has been withdrawn by the authors. Journal of
Computational Geometry, 1(1) 2010, pages 72-100.
http://jocg.org/index.php/jocg/article/view/1
The conformal alpha shape filtration
Conformal alpha shapes are a new filtration of the Delaunay triangulation of a finite set of points in â„ťd. In contrast to (ordinary) alpha shapes the new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. The local scale parameter conforms to the local geometry and is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-uniformly sampled surfaces, which is not possible with alpha shape
The conformal alpha shape filtration
Conformal alpha shapes are a new filtration of the Delaunay triangulation of a finite set of points in &Rdbl;d. In contrast to (ordinary) alpha shapes the new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. The local scale parameter conforms to the local geometry and is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-uniformly sampled surfaces, which is not possible with alpha shapes
Conformal alpha shapes
We define a new filtration of the Delaunay triangulation of a finite set of points in &Rdbl;d, similar to the alpha shape filtration. The new filtration is parameterized by a local scale parameter instead of the global scale parameter in alpha shapes. Since our approach shares many properties with the alpha shape filtration and the local scale parameter conforms to the local geometry we call it conformal alpha shape filtration. The local scale parameter is motivated from applications and previous algorithms in surface reconstruction. We show how conformal alpha shapes can be used for surface reconstruction of non-unifomly sampled surf aces, which is not possible with alpha shapes. © The Eurographics Association 2005
Topological Data Analysis
Scientific data is often in the form of a finite set of noisy points, sampled from an unknown space, and embedded in a high-dimensional space. Topological data analysis focuses on recovering the topology of the sampled space. In this chapter, we look at methods for constructing combinatorial representations of point sets, as well as theories and algorithms for effective computation of robust topological invariants. Throughout, we maintain a computational view by applying our techniques to a dataset representing the conformation space of a small molecule
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