On the complexity of computing the homology type of a triangulation
SFCS '91 Proceedings of the 32nd annual symposium on Foundations of computer science
Computing Betti numbers via combinatorial Laplacians
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Computing and comprehending topology: persistence and hierarchical morse complexes
Computing and comprehending topology: persistence and hierarchical morse complexes
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Inequalities for the curvature of curves and surfaces
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Vines and vineyards by updating persistence in linear time
Proceedings of the twenty-second annual symposium on Computational geometry
Coordinate-free Coverage in Sensor Networks with Controlled Boundaries via Homology
International Journal of Robotics Research
On the Local Behavior of Spaces of Natural Images
International Journal of Computer Vision
Using persistent homology to recover spatial information from encounter traces
Proceedings of the 9th ACM international symposium on Mobile ad hoc networking and computing
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Multi-scale Representation and Persistency for Shape Description
MDA '08 Proceedings of the 3rd international conference on Advances in Mass Data Analysis of Images and Signals in Medicine, Biotechnology, Chemistry and Food Industry
Towards a theoretical foundation for Laplacian-based manifold methods
Journal of Computer and System Sciences
Extending the notion of AT-model for integer homology computation
GbRPR'07 Proceedings of the 6th IAPR-TC-15 international conference on Graph-based representations in pattern recognition
Metric structures on datasets: stability and classification of algorithms
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part II
Towards a theoretical foundation for laplacian-based manifold methods
COLT'05 Proceedings of the 18th annual conference on Learning Theory
Topological estimation using witness complexes
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
A barcode shape descriptor for curve point cloud data
SPBG'04 Proceedings of the First Eurographics conference on Point-Based Graphics
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We study the homology of a filtered d-dimensional simplicial complex K as a single algebraic entity and establish a correspondence that provides a simple description over fields. Our analysis enables us to derive a natural algorithm for computing persistent homology over an arbitrary field in any dimension. Our study also implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.