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
A simple algorithm for homeomorphic surface reconstruction
Proceedings of the sixteenth annual symposium on Computational geometry
Introduction to Algorithms
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Persistence-sensitive simplification functions on 2-manifolds
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
Stability of Persistence Diagrams
Discrete & Computational Geometry
Shape smoothing using double offsets
Proceedings of the 2007 ACM symposium on Solid and physical modeling
Stability and Computation of Topological Invariants of Solids in ${\Bbb R}^n$
Discrete & Computational Geometry
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
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Cut locus and topology from surface point data
Proceedings of the twenty-fifth annual symposium on Computational geometry
Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes
Discrete & Computational Geometry - 23rd Annual Symposium on Computational Geometry
Homological illusions of persistence and stability
Homological illusions of persistence and stability
Provable surface reconstruction from noisy samples
Computational Geometry: Theory and Applications
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
Smooth surface reconstruction via natural neighbour interpolation of distance functions
Computational Geometry: Theory and Applications
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
Reconstructing shapes with guarantees by unions of convex sets
Proceedings of the twenty-sixth annual symposium on Computational geometry
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Sampling conditions for recovering the homology of a set using topological persistence are much weaker than sampling conditions required by any known polynomial time algorithm for producing a topologically correct reconstruction. Under the former sampling conditions which we call weak sampling conditions, we give an algorithm that outputs a topologically correct reconstruction. Unfortunately, even though the algorithm terminates, its time complexity is unbounded. Motivated by the question of knowing if a polynomial time algorithm for reconstruction exists under the weak sampling conditions, we identify at the heart of our algorithm a test which requires answering the following question: given two 2-dimensional simplicial complexes L ⊂ K, does there exist a simplicial complex containing L and contained in K which realizes the persistent homology of L into K? We call this problem the homological simplification of the pair (K, L) and prove that this problem is NP-complete, using a reduction from 3SAT.