Stability of Persistence Diagrams
Discrete & Computational Geometry
The theory of multidimensional persistence
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Describing shapes by geometrical-topological properties of real functions
ACM Computing Surveys (CSUR)
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Proximity of persistence modules and their diagrams
Proceedings of the twenty-fifth annual symposium on Computational geometry
Homological illusions of persistence and stability
Homological illusions of persistence and stability
Computing Multidimensional Persistence
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Lipschitz Functions Have L p -Stable Persistence
Foundations of Computational Mathematics
A new algorithm for computing the 2-dimensional matching distance between size functions
Pattern Recognition Letters
Suspension models for testing shape similarity methods
Computer Vision and Image Understanding
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The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare the rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.