Three-dimensional alpha shapes
VVS '92 Proceedings of the 1992 workshop on Volume visualization
The union of balls and its dual shape
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
The crust and the &Bgr;-Skeleton: combinatorial curve reconstruction
Graphical Models and Image Processing
r-regular shape reconstruction from unorganized points
Computational Geometry: Theory and Applications - special issue on applied computational geometry
Complexity of the delaunay triangulation of points on surfaces the smooth case
Proceedings of the nineteenth annual symposium on Computational geometry
A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces
Discrete & Computational Geometry
Graphical Models
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics)
Complexity of Delaunay triangulation for points on lower-dimensional polyhedra
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension
Proceedings of the twenty-fifth annual symposium on Computational geometry
Technical Section: Fast construction of the Vietoris-Rips complex
Computers and Graphics
The tidy set: a minimal simplicial set for computing homology of clique complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Manifold reconstruction using tangential Delaunay complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Efficient data structure for representing and simplifying simplicial complexes in high dimensions
Proceedings of the twenty-seventh annual symposium on Computational geometry
Smooth surface reconstruction via natural neighbour interpolation of distance functions
Computational Geometry: Theory and Applications
Efficient data structure for representing and simplifying simplicial complexes in high dimensions
Proceedings of the twenty-seventh annual symposium on Computational geometry
Preserving geometric properties in reconstructing regions from internal and nearby points
Computational Geometry: Theory and Applications
Multinerves and helly numbers of acyclic families
Proceedings of the twenty-eighth annual symposium on Computational geometry
Zigzag zoology: rips zigzags for homology inference
Proceedings of the twenty-ninth annual symposium on Computational geometry
Graph induced complex on point data
Proceedings of the twenty-ninth annual symposium on Computational geometry
| Hi-index | 0.00 |
We associate with each compact set X of Rn two real-valued functions cX and hX defined on R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on cP(t) entails that the Rips complex of P at scale r collapses to the Cech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Cech complex of P or with a Rips complex of P. Regarding the reconstruction with Cech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide topologically correct reconstruction of shapes. This may be of some computational interest in high dimensions.