Barcodes: The persistent topology of data

Ghrist, Robert

doi:10.1090/S0273-0979-07-01191-3

Author: Robert Ghrist
Journal: Bull. Amer. Math. Soc. 45 (2008), 61-75
MSC (2000): Primary 55N35; Secondary 62H35
DOI: https://doi.org/10.1090/S0273-0979-07-01191-3
Published electronically: October 26, 2007
MathSciNet review: 2358377
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Abstract: This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—persistent homology—and a novel representation of this algebraic characterization—barcodes. We sketch an application of these techniques to the classification of natural images.

BK P. Bubenik and P. Kim, “A statistical approach to persistent homology”, preprint (2006), arXiv:math.AT/0607634.

Erik Carlsson, Gunnar Carlsson, and Vin de Silva, An algebraic topological method for feature identification, Internat. J. Comput. Geom. Appl. 16 (2006), no. 4, 291–314. MR 2250511, DOI https://doi.org/10.1142/S021819590600204X

CIDZ G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, “On the local behavior of spaces of natural images”, Intl. J. Computer Vision, in press.

CIMRS G. Carlsson, T. Ishkhanov, F. Mémoli, D. Ringach, and G. Sapiro, “Topological analysis of the responses of neurons in V1”, in preparation (2007).

CZCG G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas, “Persistence barcodes for shapes”, Intl. J. Shape Modeling, 11 (2005), 149-187.

CL F. Chazal and A. Lieutier, “Weak feature size and persistent homology: computing homology of solids in $\mathbb {R}^n$ from noisy data samples”, in Proc. 21st Sympos. Comput. Geom. (2005).

CEH D. Cohen-Steiner, H. Edelsbrunner, and J. Harer, “Stability of persistence diagrams”, in Proc. 21st Sympos. Comput. Geom. (2005), 263–271.

D V. de Silva, “A weak definition of Delaunay triangulation”, preprint (2003).

DC V. de Silva and G. Carlsson, “Topological estimation using witness complexes”, in SPBG’04 Symposium on Point-Based Graphics (2004), 157-166.

DG:p V. de Silva and R. Ghrist, “Coverage in sensor networks via persistent homology”, Alg. & Geom. Topology, 7 (2007), 339–358.

PLEX V. de Silva and P. Perry, PLEX home page, http://math.stanford.edu/comptop/programs/plex/.

Herbert Edelsbrunner, David Letscher, and Afra Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002), no. 4, 511–533. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). MR 1949898, DOI https://doi.org/10.1007/s00454-002-2885-2

EM H. Edelsbrunner and E.P. Mücke, “Three-dimensional alpha shapes”, ACM Transactions on Graphics, 13:1 (1994), 43-72.

GO L. Guibas and S. Oudot, “Reconstruction using witness complexes”, in Proc. 18th ACM-SIAM Sympos. on Discrete Algorithms (2007).

Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, Computational homology, Applied Mathematical Sciences, vol. 157, Springer-Verlag, New York, 2004. MR 2028588
David Mumford, Pattern theory: the mathematics of perception, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 401–422. MR 1989195

MLP D. Mumford, A. Lee, and K. Pedersen, “The nonlinear statistics of high-contrast patches in natural images”, Intl. J. Computer Vision, 54 (2003), 83–103.

B. W. Silverman, Density estimation for statistics and data analysis, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1986. MR 848134

HS J. van Hateren and A. van der Schaff, “Independent Component Filters of Natural Images Compared with Simple Cells in Primary Visual Cortex”, Proc. R. Soc. London, B 265 (1998), 359–366.

L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), no. 1, 454–472 (German). MR 1512371, DOI https://doi.org/10.1007/BF01447877
Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005), no. 2, 249–274. MR 2121296, DOI https://doi.org/10.1007/s00454-004-1146-y

ZC2 A. Zomorodian and G. Carlsson, “Localized homology”, Proc. Shape Modeling International (2007), 189–198.

ZC3 A. Zomorodian and G. Carlsson, “The theory of multidimensional persistence”, Proc. Symposium on Computational Geometry (2007), 184–193.