Recovering structure from r-sampled objects
SGP '09 Proceedings of the Symposium on Geometry Processing
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
SMI 2011: Full Paper: Localized Cocone surface reconstruction
Computers and Graphics
Reconstructing 3D compact sets
Computational Geometry: Theory and Applications
Zigzag zoology: rips zigzags for homology inference
Proceedings of the twenty-ninth annual symposium on Computational geometry
SMI 2013: Minimizing edge length to connect sparsely sampled unstructured point sets
Computers and Graphics
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We present a novel reconstruction algorithm that, given an input point set sampled from an object S, builds a one-parameter family of complexes that approximate S at different scales. At a high level, our method is very similar in spirit to Chew’s surface meshing algorithm, with one notable difference though: the restricted Delaunay triangulation is replaced by the witness complex, which makes our algorithm applicable in any metric space. To prove its correctness on curves and surfaces, we highlight the relationship between the witness complex and the restricted Delaunay triangulation in 2d and in 3d. Specifically, we prove that both complexes are equal in 2d and closely related in 3d, under some mild sampling assumptions.