Medial Axis Approximation and Unstable Flow Complex

Giesen, Joachim; Ramos, Edgar A.; Sadri, Bardia

doi:10.1142/s0218195908002751

Cited by 6 publications

(15 citation statements)

References 12 publications

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“…Yet, such cascades may prevent the construction from being practical if speed is mandatory. Let us instantiate this conclusion for reconstruction [4] and medial axis approximation [19,20]. For the former, manifolds sought are close from the sample points, and their construction is expected to be affordable.…”

Section: Methodsmentioning

confidence: 99%

“…The construction of stable manifolds of index two saddles in 3D has been presented in [8], with a generalization to any dimension in [4]. The construction of unstable manifolds has been presented in [19,20]. In geometric modeling terms, the flow complex (i.e.…”

Section: Morse Theory and The Flow Complexmentioning

confidence: 99%

“…selected stable and unstable manifold) has interesting approximation properties. So far, stable manifolds have been used for surface reconstruction [4], while unstable manifolds have been involved with the approximation of the medial axis [19] and the identification of special regions (cylinders and flat regions) in [20]. Aside these works which are concerned with manifolds, an example reconstruction of a self-intersecting surface in terms of stable manifolds is presented on Fig.…”

Section: Morse Theory and The Flow Complexmentioning

confidence: 99%

“…The stable manifold of an index one saddle is the Gabriel edge. For the unstable manifold, the construction is presented in [19]. The key primitive consists of computing the image of a (portion of) Voronoi edge.…”

Section: Building the Flow Complexmentioning

confidence: 99%

“…4 for an illustration. The construction of unstable manifold reduces to the computation of the orbit(s) of the Voronoi vertex(ices) of the tetrahedron(a) incident on the facet defining the critical point [19]. The tricky operation is again the algorithm of section 4, used forward this time.…”

Section: Building the Flow Complexmentioning

confidence: 99%

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Robust construction of the three-dimensional flow complex

Cazals

Parameswaran

Pion

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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The Delaunay triangulation and its dual the Voronoi diagram are ubiquitous geometric complexes. From a topological standpoint, the connection has recently been made between these cell complexes and the Morse theory of distance functions. In particular, in the generic setting, algorithms have been proposed to compute the flow complex -the stable and unstable manifolds associated to the critical points of the distance function to a point set. As algorithms ignoring degenerate cases and numerical issues are bound to fail on general inputs, this paper develops the first complete and robust algorithm to compute the flow complex.First, we present complete algorithms for the flow operator, unraveling a delicate interplay between the degenerate cases of Delaunay and those which are flow specific. Second, we sketch how the flow operator unifies the construction of stable and unstable manifolds. Third, we discuss numerical issues related to predicates on cascaded constructions. Finally, we report experimental results with CGAL's filtered kernel, showing that the construction of the flow complex incurs a small overhead w.r.t. the Delaunay triangulation when moderate cascading occurs. These observations provide important insights on the relevance of the flow complex for (surface) reconstruction and medial axis approximation, and should foster flow complex based algorithms.In a broader perspective and to the best of our knowledge, this paper is the first one reporting on the effective implementation of a geometric algorithm featuring cascading.

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Section: Methodsmentioning

confidence: 99%

Section: Morse Theory and The Flow Complexmentioning

confidence: 99%

Section: Morse Theory and The Flow Complexmentioning

confidence: 99%

Section: Building the Flow Complexmentioning

confidence: 99%

Section: Building the Flow Complexmentioning

confidence: 99%

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Robust construction of the three-dimensional flow complex

Cazals

Parameswaran

Pion

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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Approximating the Pathway Axis and the Persistence Diagrams for a Collection of Balls in 3-Space

Yaffe

Halperin

2010

Discrete Comput Geom

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Given a collection B of balls in a three-dimensional space, we wish to explore the cavities, voids, and tunnels in the complement space of B. We introduce the pathway axis of B as a useful subset of the medial axis of the complement of B and prove that it satisfies several desirable geometric properties. We present an algorithm that constructs the pathway graph of B, a piecewise-linear approximation of the pathway axis. At the heart of our approach is an approximation scheme that constructs a collection K of same-size balls that approximate B so that the Hausdorff distance between B and K is bounded by a prescribed parameter. We prove a bound on the ratio between the number of balls in K and the number of balls in B. We employ this bound and the approximation scheme to show how to approximate the persistence diagrams for B, which can be used to extract major topological features such as the large voids and tunnels in the complement of B. We show that our approach is superior in terms of complexity to the standard point-sample approaches for the two problems that we address in this paper: approximating the pathway axis of B and approximating the persistence diagrams for B. In a companion paper we introduce MolAxis, a tool for the identification of channels in macromolecules that demonstrates how the pathway graph and the persistence diagrams are used to identify plausible pathways in the complement of molecules.

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