Reducing complexes in multidimensional persistent homology theory

Forman's discrete Morse theory appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for one-dimensional filtrations. This paper is perhaps the first attempt in the direction of extending such algorithms to multidimensional filtrations. An initial framework related to Morse matchings for the multidimensional setting is proposed, and a matching algorithm given by King, Knudson, and Mramor is extended in this direction. The correctness of the algorithm is proved, and its complexity analyzed. The algorithm is used for establishing a reduction of a simplicial complex to a smaller but not necessarily optimal cellular complex. First experiments with filtrations of triangular meshes are presented.

[1]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[2]  M. Ferri,et al.  One-dimensional reduction of multidimensional persistent homology , 2007, math/0702713.

[3]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[4]  M. Ferri,et al.  Betti numbers in multidimensional persistent homology are stable functions , 2013 .

[5]  Afra Zomorodian,et al.  The Theory of Multidimensional Persistence , 2007, SCG '07.

[6]  坂上 貴之,et al.  書評「T. Kaczynski, K. Mischaikow, and M. Mrozek:Computational Homology (Applied Mathematical Sciences 157, Springer-Verlag, 2004 年, 480 ページ)」 , 2005 .

[7]  貴之 坂上.  T. Kaczynski, K. Mischaikow, M. Mrozek, Computational Homology, Springer Verlag, NY, 2004 , 2005 .

[8]  M. Mrozek,et al.  Homology Computation by Reduction of Chain Complexes , 1998 .

[9]  R. Ho.  Algebraic Topology , 2022 .

[10]  Afra Zomorodian,et al.  Computing Multidimensional Persistence , 2009, J. Comput. Geom..

[11]  Walter G. Kropatsch,et al.  A Global Method for Reducing Multidimensional Size Graphs , 2011, GbRPR.

[12]  R. Forman.  Morse Theory for Cell Complexes , 1998 .

[13]  Henry King,et al.  Generating Discrete Morse Functions from Point Data , 2005, Exp. Math..

[14]  Olaf Delgado-Friedrichs,et al.  Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory , 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  A. B. Kahn,et al.  Topological sorting of large networks , 1962, CACM.

[16]  Peter John Wood,et al.  Ieee Transactions on Pattern Analysis and Machine Intelligence Theory and Algorithms for Constructing Discrete Morse Complexes from Grayscale Digital Images , 2022 .

[17]  Marc E. Pfetsch,et al.  Computing Optimal Morse Matchings , 2006, SIAM J. Discret. Math..

[18]  Konstantin Mischaikow,et al.  Morse Theory for Filtrations and Efficient Computation of Persistent Homology , 2013, Discret. Comput. Geom..

[19]  Patrizio Frosini,et al.  On the use of size functions for shape analysis , 1993, [1993] Proceedings IEEE Workshop on Qualitative Vision.

[20]  坂上 貴之.  書評 Computational Homology , 2005 .

[21]  Pawel Dlotko,et al.  Computing homology and persistent homology using iterated Morse decomposition , 2012, ArXiv.

[22]  Daniela Giorgi,et al.  Multidimensional Size Functions for Shape Comparison , 2008, Journal of Mathematical Imaging and Vision.

[23]  Marc Ethier,et al.  Comparison of persistent homologies for vector functions: From continuous to discrete and back , 2012, Comput. Math. Appl..

[24]  Patrizio Frosini,et al.  Measuring shapes by size functions , 1992, Other Conferences.

[25]  R. Forman.  A USER'S GUIDE TO DISCRETE MORSE THEORY , 2002 .

[26]  P. Dlotko,et al.  SIMPLIFICATION OF COMPLEXES FOR PERSISTENT HOMOLOGY COMPUTATIONS , 2013, 1304.8074.

[27]  Leonidas J. Guibas,et al.  Persistence barcodes for shapes , 2004, SGP '04.

[28]  Marian Mrozek,et al.  Coreduction Homology Algorithm , 2009, Discret. Comput. Geom..

[29]  Reiner Creutzburg,et al.  Virtual tutorials, Wikipedia books, and multimedia-based teaching for blended learning support in a course on algorithms and data structures , 2014, Electronic Imaging.

[30]  Claudia Landi,et al.  Persistent homology and partial similarity of shapes , 2012, Pattern Recognit. Lett..