Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Persistent homology for kernels, images, and cokernels
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Zigzag persistent homology and real-valued functions
Proceedings of the twenty-fifth annual symposium on Computational geometry
Computing Elevation Maxima by Searching the Gauss Sphere
SEA '09 Proceedings of the 8th International Symposium on Experimental Algorithms
Measuring and computing natural generators for homology groups
Computational Geometry: Theory and Applications
A randomized O(m log m) time algorithm for computing Reeb graphs of arbitrary simplicial complexes
Proceedings of the twenty-sixth annual symposium on Computational geometry
Topological inference via meshing
Proceedings of the twenty-sixth annual symposium on Computational geometry
International Journal of Computer Vision
Coreduction homology algorithm for inclusions and persistent homology
Computers & Mathematics with Applications
Persistent homology under non-uniform error
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Computing elevation maxima by searching the gauss sphere
Journal of Experimental Algorithmics (JEA)
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
Reeb graphs: approximation and persistence
Proceedings of the twenty-seventh annual symposium on 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
Persistent homology and partial similarity of shapes
Pattern Recognition Letters
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Homological reconstruction and simplification in R3
Proceedings of the twenty-ninth annual symposium on Computational geometry
Comparing shapes through multi-scale approximations of the matching distance
Computer Vision and Image Understanding
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Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds.