Abstract
In this paper, we study a class of discrete Morse functions, coming from Discrete Morse Theory, that are equivalent to a class of simplicial stacks, coming from Mathematical Morphology. We show that, as in Discrete Morse Theory, we can see the gradient vector field of a simplicial stack (seen as a discrete Morse function) as the only relevant information we should consider. Last, but not the least, we also show that the Minimum Spanning Forest of the dual graph of a simplicial stack is induced by the gradient vector field of the initial function. This result allows computing a watershed-cut from a gradient vector field.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The superscripts correspond to the dimensions of the faces.
References
Boutry, N., Géraud, T., Najman, L.: An equivalence relation between morphological dynamics and persistent homology in 1D. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds.) ISMM 2019. LNCS, vol. 11564, pp. 57–68. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-20867-7_5
Boutry, N., Géraud, T., Najman, L.: An equivalence relation between morphological dynamics and persistent homology in n-D. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds.) DGMM 2021. LNCS, vol. 12708, pp. 525–537. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-76657-3_38
Boutry, N., Najman, L., Géraud, T.: Some equivalence relation between persistent homology and morphological dynamics. J. Math. Imaging Vis. (2022). https://doi.org/10.1007/s10851-022-01104-z, https://hal.archives-ouvertes.fr/hal-03676854
Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. 23rd printing (1999)
Couprie, M., Bertrand, G.: New characterizations of simple points in 2D, 3D, and 4D discrete spaces. IEEE Trans. Pattern Anal. Mach. Intell. 31(4), 637–648 (2008)
Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Collapses and watersheds in pseudomanifolds. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 397–410. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10210-3_31
Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Collapses and watersheds in pseudomanifolds of arbitrary dimension. J. Math. Imaging Vis. 50(3), 261–285 (2014). https://doi.org/10.1007/s10851-014-0498-z
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Trans. Pattern Anal. Mach. Intell. 31(8), 1362–1374 (2009)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest path forests, and topological watersheds. IEEE Trans. Pattern Anal. Mach. Intell. 32(5), 925–939 (2010)
De Floriani, L., Fugacci, U., Iuricich, F., Magillo, P.: Morse complexes for shape segmentation and homological analysis: discrete models and algorithms. Comput. Graph. Forum 34(2), 761–785 (2015)
Edelsbrunner, H., Harer, J.: Persistent homology - a survey. Contemp. Math. 453, 257–282 (2008)
Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Morse-Smale complexes for piecewise linear 3-manifolds. In: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pp. 361–370 (2003)
Forman, R.: A Discrete Morse Theory for cell complexes. In: Yau, S.T. (ed.) Geometry. Topology for Raoul Bott. International Press, Somerville MA (1995)
Grimaud, M.: New measure of contrast: the dynamics. In: Image Algebra and Morphological Image Processing III, vol. 1769, pp. 292–306. International Society for Optics and Photonics (1992)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
Munch, E.: A user’s guide to topological data analysis. J. Learn. Anal. 4(2), 47–61 (2017)
Najman, L., Talbot, H.: Mathematical Morphology: From Theory to Applications. Wiley, Hoboken (2013)
Robins, V., Wood, P.J., Sheppard, A.P.: Theory and algorithms for constructing discrete Morse complexes from grayscale digital images. IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1646–1658 (2011)
Scoville, N.A.: Discrete Morse Theory, vol. 90. American Mathematical Soc. (2019)
Tierny, J.: Introduction to Topological Data Analysis. Technical report, Sorbonne University, LIP6, APR team, France (2017). https://hal.archives-ouvertes.fr/cel-01581941
Whitehead, J.H.C.: Simplicial spaces, nuclei and m-groups. Proc. Lond. Math. Soc. 2(1), 243–327 (1939)
Acknowledgements
The authors would like to thank both Julien Tierny and Thierry Géraud, for many insightful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Boutry, N., Bertrand, G., Najman, L. (2022). Gradient Vector Fields of Discrete Morse Functions and Watershed-Cuts. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-19897-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-19896-0
Online ISBN: 978-3-031-19897-7
eBook Packages: Computer ScienceComputer Science (R0)