Stability of Persistence Diagrams
Discrete & Computational Geometry
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
Extending Persistence Using Poincaré and Lefschetz Duality
Foundations of Computational Mathematics
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Homological illusions of persistence and stability
Homological illusions of persistence and stability
Quantifying Transversality by Measuring the Robustness of Intersections
Foundations of Computational Mathematics
Optimal Topological Simplification of Discrete Functions on Surfaces
Discrete & Computational Geometry
Alexander duality for functions: the persistent behavior of land and water and shore
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
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
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We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology of some complex H*(X) with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.