Computing Multidimensional Persistence
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Detecting critical regions in multidimensional data sets
Computers & Mathematics with Applications
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
A global method for reducing multidimensional size graphs
GbRPR'11 Proceedings of the 8th international conference on Graph-based representations in pattern recognition
Persistent betti numbers for a noise tolerant shape-based approach to image retrieval
CAIP'11 Proceedings of the 14th international conference on Computer analysis of images and patterns - Volume Part I
Persistence modules, shape description, and completeness
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Stable Comparison of Multidimensional Persistent Homology Groups with Torsion
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
Pattern Recognition Letters
The persistence space in multidimensional persistent homology
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
A study of monodromy in the computation of multidimensional persistence
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
Classification of hepatic lesions using the matching metric
Computer Vision and Image Understanding
PHOG: photometric and geometric functions for textured shape retrieval
SGP '13 Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing
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Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.