Discrete Morse theory

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Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman and Kenneth Brown. [1] The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, [2] homology computation, [3] [4] denoising, [5] mesh compression, [6] and topological data analysis. [7]

Contents

Notation regarding CW complexes

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 [8] 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

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 [9] 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

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

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

The Morse inequalities

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 [10] hold

, and

Moreover, the Euler characteristic of satisfies

Discrete Morse homology and homotopy type

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 [11] of homology groups

and similarly for the homotopy groups.

Applications

Discrete Morse theory finds its application in molecular shape analysis, [12] skeletonization of digital images/volumes, [13] graph reconstruction from noisy data, [14] denoising noisy point clouds [15] and analysing lithic tools in archaeology. [16]

See also

References

  1. Brown, Kenneth S. (1992). "The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem". Algorithms and Classification in Combinatorial Group Theory. Mathematical Sciences Research Institute Publications. Vol. 23. New York, NY: Springer New York. pp. 137–163. doi:10.1007/978-1-4613-9730-4_6. ISBN   978-1-4613-9732-8.
  2. 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
  3. Perseus: the Persistent Homology software.
  4. 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 .
  5. 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 .
  6. 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. Bibcode:2004ITVCG..10..499L. doi:10.1109/TVCG.2004.18. PMID   15794132. S2CID   2185198. Archived from the original (PDF) on 2012-04-26.
  7. "the Topology ToolKit". GitHub.io.
  8. 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 .
  9. Forman 1998 , Lemma 2.5
  10. Forman 1998 , Corollaries 3.5 and 3.6
  11. Forman 1998 , Theorem 7.3
  12. 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.
  13. 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. Bibcode:2015ITPAM..37..654D. doi:10.1109/TPAMI.2014.2346172. hdl: 1885/12873 . ISSN   1939-3539. PMID   26353267. S2CID   7406197.
  14. 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.
  15. 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.
  16. 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