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Discrete Morse theory

From Wikipedia, the free encyclopedia

Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces,[1] homology computation,[2][3] denoising,[4] mesh compression,[5] and topological data analysis.[6]

Notation regarding CW complexes

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Let be a CW complex and denote by its set of cells. Define the incidence function in the following way: given two cells and in , let be the degree of the attaching map from the boundary of to . The boundary operator is the endomorphism of the free abelian group generated by defined by

It is a defining property of boundary operators that . In more axiomatic definitions[7] one can find the requirement that

which is a consequence of the above definition of the boundary operator and the requirement that .

Discrete Morse functions

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A real-valued function is a discrete Morse function if it satisfies the following two properties:

  1. For any cell , the number of cells in the boundary of which satisfy is at most one.
  2. For any cell , the number of cells containing in their boundary which satisfy is at most one.

It can be shown[8] that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell , provided that is a regular CW complex. In this case, each cell can be paired with at most one exceptional cell : either a boundary cell with larger value, or a co-boundary cell with smaller value. The cells which have no pairs, i.e., whose function values are strictly higher than their boundary cells and strictly lower than their co-boundary cells are called critical cells. Thus, a discrete Morse function partitions the CW complex into three distinct cell collections: , where:

  1. denotes the critical cells which are unpaired,
  2. denotes cells which are paired with boundary cells, and
  3. denotes cells which are paired with co-boundary cells.

By construction, there is a bijection of sets between -dimensional cells in and the -dimensional cells in , which can be denoted by for each natural number . It is an additional technical requirement that for each , the degree of the attaching map from the boundary of to its paired cell is a unit in the underlying ring of . For instance, over the integers , the only allowed values are . This technical requirement is guaranteed, for instance, when one assumes that is a regular CW complex over .

The fundamental result of discrete Morse theory establishes that the CW complex is isomorphic on the level of homology to a new complex consisting of only the critical cells. The paired cells in and describe gradient paths between adjacent critical cells which can be used to obtain the boundary operator on . Some details of this construction are provided in the next section.

The Morse complex

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A gradient path is a sequence of paired cells

satisfying and . The index of this gradient path is defined to be the integer

The division here makes sense because the incidence between paired cells must be . Note that by construction, the values of the discrete Morse function must decrease across . The path is said to connect two critical cells if . This relationship may be expressed as . The multiplicity of this connection is defined to be the integer . Finally, the Morse boundary operator on the critical cells is defined by

where the sum is taken over all gradient path connections from to .

Basic results

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Many of the familiar results from continuous Morse theory apply in the discrete setting.

The Morse inequalities

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Let be a Morse complex associated to the CW complex . The number of -cells in is called the -th Morse number. Let denote the -th Betti number of . Then, for any , the following inequalities[9] hold

, and

Moreover, the Euler characteristic of satisfies

Discrete Morse homology and homotopy type

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Let be a regular CW complex with boundary operator and a discrete Morse function . Let be the associated Morse complex with Morse boundary operator . Then, there is an isomorphism[10] of homology groups

and similarly for the homotopy groups.

Applications

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Discrete Morse theory finds its application in molecular shape analysis,[11] skeletonization of digital images/volumes,[12] graph reconstruction from noisy data,[13] denoising noisy point clouds[14] and analysing lithic tools in archaeology.[15]

See also

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References

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  1. ^ Mori, Francesca; Salvetti, Mario (2011), "(Discrete) Morse theory for Configuration spaces" (PDF), Mathematical Research Letters, 18 (1): 39–57, doi:10.4310/MRL.2011.v18.n1.a4, MR 2770581
  2. ^ Perseus: the Persistent Homology software.
  3. ^ Mischaikow, Konstantin; Nanda, Vidit (2013). "Morse Theory for Filtrations and Efficient computation of Persistent Homology". Discrete & Computational Geometry. 50 (2): 330–353. doi:10.1007/s00454-013-9529-6.
  4. ^ Bauer, Ulrich; Lange, Carsten; Wardetzky, Max (2012). "Optimal Topological Simplification of Discrete Functions on Surfaces". Discrete & Computational Geometry. 47 (2): 347–377. arXiv:1001.1269. doi:10.1007/s00454-011-9350-z.
  5. ^ Lewiner, T.; Lopes, H.; Tavares, G. (2004). "Applications of Forman's discrete Morse theory to topology visualization and mesh compression" (PDF). IEEE Transactions on Visualization and Computer Graphics. 10 (5): 499–508. doi:10.1109/TVCG.2004.18. PMID 15794132. S2CID 2185198. Archived from the original (PDF) on 2012-04-26.
  6. ^ "the Topology ToolKit". GitHub.io.
  7. ^ Mischaikow, Konstantin; Nanda, Vidit (2013). "Morse Theory for Filtrations and Efficient computation of Persistent Homology". Discrete & Computational Geometry. 50 (2): 330–353. doi:10.1007/s00454-013-9529-6.
  8. ^ Forman 1998, Lemma 2.5
  9. ^ Forman 1998, Corollaries 3.5 and 3.6
  10. ^ Forman 1998, Theorem 7.3
  11. ^ Cazals, F.; Chazal, F.; Lewiner, T. (2003). "Molecular shape analysis based upon the morse-smale complex and the connolly function". Proceedings of the nineteenth annual symposium on Computational geometry. ACM Press. pp. 351–360. doi:10.1145/777792.777845. ISBN 978-1-58113-663-0. S2CID 1570976.
  12. ^ Delgado-Friedrichs, Olaf; Robins, Vanessa; Sheppard, Adrian (March 2015). "Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory". IEEE Transactions on Pattern Analysis and Machine Intelligence. 37 (3): 654–666. doi:10.1109/TPAMI.2014.2346172. hdl:1885/12873. ISSN 1939-3539. PMID 26353267. S2CID 7406197.
  13. ^ Dey, Tamal K.; Wang, Jiayuan; Wang, Yusu (2018). Speckmann, Bettina; Tóth, Csaba D. (eds.). Graph Reconstruction by Discrete Morse Theory. 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs). Vol. 99. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. pp. 31:1–31:15. doi:10.4230/LIPIcs.SoCG.2018.31. ISBN 978-3-95977-066-8. S2CID 3994099.
  14. ^ Mukherjee, Soham (2021-09-01). "Denoising with discrete Morse theory". The Visual Computer. 37 (9): 2883–94. doi:10.1007/s00371-021-02255-7. S2CID 237426675.
  15. ^ Bullenkamp, Jan Philipp; Linsel, Florian; Mara, Hubert (2022), "Lithic Feature Identification in 3D based on Discrete Morse Theory", Proceedings of Eurographics Workshop on Graphics and Cultural Heritage (GCH), Delft, Netherlands: Eurographics Association, pp. 55–58, doi:10.2312/VAST/VAST10/131-138, ISBN 9783038681786, ISSN 2312-6124, S2CID 17294591, retrieved 2022-10-05