One-dimensional reduction of multidimensional persistent homology

Cagliari, Francesca; Di Fabio, Barbara; Ferri, Massimo

doi:10.1090/S0002-9939-10-10312-8

One-dimensional reduction of multidimensional persistent homology
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by Francesca Cagliari, Barbara Di Fabio and Massimo Ferri PDF

Proc. Amer. Math. Soc. 138 (2010), 3003-3017 Request permission

Abstract:

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on the $i$-essentiality of homological critical values conclude the paper.

References

S. Biasotti, A. Cerri, P. Frosini, D. Giorgi, and C. Landi, Multidimensional size functions for shape comparison, J. Math. Imaging Vision 32 (2008), no. 2, 161–179. MR 2434687, DOI 10.1007/s10851-008-0096-z
S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo and M. Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys 40, no. 4 (2008), 12:1–12:87.
Francesca Cagliari, Massimo Ferri, and Paola Pozzi, Size functions from a categorical viewpoint, Acta Appl. Math. 67 (2001), no. 3, 225–235. MR 1861130, DOI 10.1023/A:1011923819754
Gunnar Carlsson and Afra Zomorodian, The theory of multidimensional persistence, Computational geometry (SCG’07), ACM, New York, 2007, pp. 184–193. MR 2469164, DOI 10.1145/1247069.1247105
A. Cerri and P. Frosini, Necessary conditions for discontinuities of multidimensional size functions, tech. rept. 2625, Univ. Bologna, Italy (2009), available at http://amsacta. cib.unibo.it/2625/ .
C. Chen and D. Freedman, Quantifying homology classes, 25th Symp. on Theoretical Aspects of Computer Science, Bordeaux, France (2008), 169–180, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, http://drops.dagstuhl. de/opus/volltexte/2008/1347.
David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. MR 2279866, DOI 10.1007/s00454-006-1276-5
M. d’Amico, $\Delta ^*$ reduction of size graphs as a new algorithm for computing size functions of shapes, Proc. Internat. Conf. on Computer Vision, Pattern Recognition and Image Processing, Feb. 27–Mar. 3, 2000, Atlantic City 2 (2000), 107–110.
M. d’Amico, P. Frosini and C. Landi, Natural pseudo-distance and optimal matching between reduced size functions, Acta Applicandae Mathematicae 109, no. 2 (2010), 527–554, available at http://www.springerlink.com/content/cj84327h4n280144/fulltext.pdf.
P. Donatini and P. Frosini, Natural pseudodistances between closed curves, Forum Math. 21, no. 6 (2009), 981–999.
Herbert Edelsbrunner and John Harer, Jacobi sets of multiple Morse functions, Foundations of computational mathematics: Minneapolis, 2002, London Math. Soc. Lecture Note Ser., vol. 312, Cambridge Univ. Press, Cambridge, 2004, pp. 37–57. MR 2189626
Herbert Edelsbrunner and John Harer, Persistent homology—a survey, Surveys on discrete and computational geometry, Contemp. Math., vol. 453, Amer. Math. Soc., Providence, RI, 2008, pp. 257–282. MR 2405684, DOI 10.1090/conm/453/08802
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 10.1007/s00454-002-2885-2
Patrizio Frosini, A distance for similarity classes of submanifolds of a Euclidean space, Bull. Austral. Math. Soc. 42 (1990), no. 3, 407–416. MR 1083277, DOI 10.1017/S0004972700028574
P. Frosini, Measuring shapes by size functions, Proc. of SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, 1607, SPIE, Boston, MA (1991), 122–133.
Patrizio Frosini and Claudia Landi, Size functions and formal series, Appl. Algebra Engrg. Comm. Comput. 12 (2001), no. 4, 327–349. MR 1855041, DOI 10.1007/s002000100078
Patrizio Frosini and Michele Mulazzani, Size homotopy groups for computation of natural size distances, Bull. Belg. Math. Soc. Simon Stevin 6 (1999), no. 3, 455–464. MR 1721761
Robert Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 61–75. MR 2358377, DOI 10.1090/S0273-0979-07-01191-3
Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, Computational homology, Applied Mathematical Sciences, vol. 157, Springer-Verlag, New York, 2004. MR 2028588, DOI 10.1007/b97315
Kevin P. Knudson, A refinement of multi-dimensional persistence, Homology Homotopy Appl. 10 (2008), no. 1, 259–281. MR 2399474, DOI 10.4310/HHA.2008.v10.n1.a11
C. Landi and P. Frosini, New pseudodistances for the size function space, Proc. SPIE, Vol. 3168, Vision Geometry. VI, Robert A. Melter, Angela Y. Wu, Longin J. Latecki (eds.), SPIE (1997), 52–60.
C. Uras and A. Verri, Computing size functions from edge maps, Internat. J. Comput. Vision 23, no. 2 (1997), 169–183.
A. Verri and C. Uras, Metric-topological approach to shape representation and recognition, Image Vision Comput. 14 (1996), 189–207.
A. Verri, C. Uras, P. Frosini and M. Ferri, On the use of size functions for shape analysis, Biol. Cybern. 70 (1993), 99–107.