Guaranteed-quality mesh generation for curved surfaces
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Topological persistence and simplification
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Provably good moving least squares
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Manifold reconstruction from point samples
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Curve reconstruction from noisy samples
Computational Geometry: Theory and Applications - Special issue on the 19th annual symposium on computational geometry - SoCG 2003
Graphical Models
Provably good sampling and meshing of surfaces
Graphical Models - Solid modeling theory and applications
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
Provable surface reconstruction from noisy samples
Computational Geometry: Theory and Applications
Surface reconstruction from noisy point clouds
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Natural neighbor coordinates of points on a surface
Computational Geometry: Theory and Applications
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Weak witnesses for Delaunay triangulations of submanifolds
Proceedings of the 2007 ACM symposium on Solid and physical modeling
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
Towards persistence-based reconstruction in euclidean spaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Analysis of scalar fields over point cloud data
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
From Segmented Images to Good Quality Meshes Using Delaunay Refinement
Emerging Trends in Visual Computing
Persistent cohomology and circular coordinates
Proceedings of the twenty-fifth annual symposium on Computational geometry
A growing self-organizing network for reconstructing curves and surfaces
IJCNN'09 Proceedings of the 2009 international joint conference on Neural Networks
Manifold homotopy via the flow complex
SGP '09 Proceedings of the Symposium on Geometry Processing
Geodesic delaunay triangulations in bounded planar domains
ACM Transactions on Algorithms (TALG)
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
Technical Section: Efficient construction of the Čech complex
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: 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.