Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Graphical Models
Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics)
Stability of Persistence Diagrams
Discrete & Computational Geometry
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
On the Nonlinear Statistics of Range Image Patches
SIAM Journal on Imaging Sciences
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on Computational geometry
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Scalar Field Analysis over Point Cloud Data
Discrete & Computational Geometry
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
The simplex tree: an efficient data structure for general simplicial complexes
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the ietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is huge--particularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one-dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.