Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
On the definition and the construction of pockets in macromolecules
Discrete Applied Mathematics - Special volume on computational molecular biology DAM-CMB series volume 2
Interval arithmetic yields efficient dynamic filters for computational geometry
Discrete Applied Mathematics - Special issue 14th European workshop on computational geometry CG'98 Selected papers
The flow complex: a data structure for geometric modeling
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Molecular shape analysis based upon the morse-smale complex and the connolly function
Proceedings of the nineteenth annual symposium on Computational geometry
Alpha-shapes and flow shapes are homotopy equivalent
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Any open bounded subset of Rn has the same homotopy type than its medial axis
SM '03 Proceedings of the eighth ACM symposium on Solid modeling and applications
Weighted alpha shapes
A geometric convection approach of 3-D reconstruction
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Identifying flat and tubular regions of a shape by unstable manifolds
Proceedings of the 2006 ACM symposium on Solid and physical modeling
A generic lazy evaluation scheme for exact geometric computations
Science of Computer Programming
Reconstructing 3D compact sets
Computational Geometry: Theory and Applications
A parallel algorithm for computing the flow complex
Proceedings of the twenty-ninth 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.