Discrete & Computational Geometry
Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Graphical Models
Tightening non-simple paths and cycles on surfaces
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Topology guaranteeing manifold reconstruction using distance function to noisy data
Proceedings of the twenty-second annual symposium on Computational geometry
Stability of Persistence Diagrams
Discrete & Computational Geometry
Manifold reconstruction in arbitrary dimensions using witness complexes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Geodesic Delaunay triangulation and witness complex in the plane
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Testing contractibility in planar rips complexes
Proceedings of the twenty-fourth annual symposium on Computational geometry
Towards a theoretical foundation for Laplacian-based manifold methods
Journal of Computer and System Sciences
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Approximating loops in a shortest homology basis from point data
Proceedings of the twenty-sixth annual symposium on Computational geometry
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on Computational geometry
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A cut locus of a point p in a compact Riemannian manifold M is defined as the set of points where minimizing geodesics issued from p stop being minimizing. It is known that a cut locus contains most of the topological information of M. Our goal is to utilize this property of cut loci to decipher the topology of M from a point sample. Recently it has been shown that Rips complexes can be built from a point sample P of M systematically to compute the Betti numbers, the rank of the homology groups of M. Rips complexes can be computed easily and therefore are favored over others such as restricted Delaunay, alpha, Cech, and witness complex. However, the sizes of the Rips complexes tend to be large. Since the dimension of a cut locus is lower than that of the manifold M, a subsample of P approximating the cut locus is usually much smaller in size and hence admits a relatively smaller Rips complex. In this paper we explore the above approach for point data sampled from surfaces embedded in any high dimensional Euclidean space. We present an algorithm that computes a subsample P' of a sample P of a 2-manifold where P' approximates a cut locus. Empirical results show that the first Betti number of M can be computed from the Rips complexes built on these subsamples. The sizes of these Rips complexes are much smaller than the one built on the original sample of M.