Three-dimensional alpha shapes
ACM Transactions on Graphics (TOG)
Smooth surface reconstruction via natural neighbour interpolation of distance functions
Proceedings of the sixteenth annual symposium on Computational geometry
Computing and comprehending topology: persistence and hierarchical morse complexes
Computing and comprehending topology: persistence and hierarchical morse complexes
Stability and Finiteness Properties of Medial Axis and Skeleton
Journal of Dynamical and Control Systems
Provable surface reconstruction from noisy samples
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability and homotopy of a subset of the medial axis
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Topology guaranteeing manifold reconstruction using distance function to noisy data
Proceedings of the twenty-second annual symposium on Computational geometry
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
Provably good sampling and meshing of Lipschitz surfaces
Proceedings of the twenty-second annual symposium on Computational geometry
Shape smoothing using double offsets
Proceedings of the 2007 ACM symposium on Solid and physical modeling
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Reconstruction using witness complexes
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Approximating the pathway axis and the persistence diagram of a collection of balls in 3-space
Proceedings of the twenty-fourth annual symposium on Computational geometry
Normal cone approximation and offset shape isotopy
Computational Geometry: Theory and Applications
Ascending and descending regions of a discrete Morse function
Computational Geometry: Theory and Applications
Contour Reconstruction for Multiple 2D Regions Based on Adaptive Boundary Samples
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Shape reconstruction from unorganized set of points
ICIAR'10 Proceedings of the 7th international conference on Image Analysis and Recognition - Volume Part I
Pattern Recognition Letters
Geometric interoperability via queries
Computer-Aided Design
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In this work, one proves that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it. For this, one introduces the weak feature size (wfs) that generalizes the notion of local feature size. Our results apply to open sets with positive wfs, which include many sets whose boundaries are not smooth and even nowhere smooth. This class includes also the piecewise analytic open sets which cover many cases encountered in practical applications. The proofs are based on the study of distance functions to closed sets and their critical points. As an application, one gives an algorithmic way, thanks to persistent homology techniques, to compute the homology groups of open sets from noisy samples of points on their boundary.