Gromov‐Hausdorff Stable Signatures for Shapes using Persistence

@article{Chazal2009GromovHausdorffSS,
  title={Gromov‐Hausdorff Stable Signatures for Shapes using Persistence},
  author={Fr{\'e}d{\'e}ric Chazal and David Cohen-Steiner and Leonidas J. Guibas and Facundo M{\'e}moli and Steve Oudot},
  journal={Computer Graphics Forum},
  year={2009},
  volume={28}
}
We introduce a family of signatures for finite metric spaces, possibly endowed with real valued functions, based on the persistence diagrams of suitable filtrations built on top of these spaces. We prove the stability of our signatures under Gromov‐Hausdorff perturbations of the spaces. We also extend these results to metric spaces equipped with measures. Our signatures are well‐suited for the study of unstructured point cloud data, which we illustrate through an application in shape… 
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References

SHOWING 1-10 OF 54 REFERENCES
Stability of persistence diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Proximity of persistence modules and their diagrams
TLDR
This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.
On the use of Gromov-Hausdorff Distances for Shape Comparison
TLDR
These reformulations render these distances more amenable to practical computations without sacrificing theoretical underpinnings, and establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms.
Persistence barcodes for shapes
TLDR
This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.
Topological Persistence and Simplification
TLDR
Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
A Gromov-Hausdorff Framework with Diffusion Geometry for Topologically-Robust Non-rigid Shape Matching
TLDR
This paper explores the applicability of diffusion distances within the Gromov-Hausdorff framework and finds that in addition to the relatively low complexity involved in the computation of the diffusion distances between surface points, its recognition and matching performances favorably compare to the classical geodesic distances in the presence of topological changes between the non-rigid shapes.
Metric Structures for Riemannian and Non-Riemannian Spaces
Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-
Gromov-Hausdorff distances in Euclidean spaces
  • F. Mémoli
  • Mathematics
    2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops
  • 2008
TLDR
A connection is uncovered that links the problem of computing GH and EH and the family of Euclidean Distance Matrix completion problems and the pair are comparable in a precise sense that is not the linear behaviour one would expect.
Persistent Homology — a Survey
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical
Stability in multidimensional Size Theory
This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in
...
1
2
3
4
5
...