Abstract
In this paper we provide a new characterization of cell decomposition (called slope complex) of a given 2-dimensional continuous surface. Each patch (cell) in the decomposition must satisfy that there exists a monotonic path for any two points in the cell. We prove that any triangulation of such surface is a slope complex and explain how to obtain new slope complexes with a smaller number of slope regions decomposing the surface. We give the minimal number of slope regions by counting certain bounding edges of a triangulation of the surface obtained from its critical points.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
For “gray value” z, \(g^{-1}(z) = \{p \in \mathfrak {R}^2 \,|\, g(p) = z \}\) is the level set of gray value z.
References
Cerman, M., Gonzalez-Diaz, R., Kropatsch, W.: LBP and irregular graph pyramids. In: Azzopardi, G., Petkov, N. (eds.) CAIP 2015. LNCS, vol. 9257, pp. 687–699. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-23117-4_59
Cerman, M., Janusch, I., Gonzalez-Diaz, R., Kropatsch, W.G.: Topology-based image segmentation using LBP pyramids. Mach. Vis. Appl. 27(8), 1161–1174 (2016)
Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse - Smale complexes for piecewise linear 2-manifolds. Discrete Comput. Geom. 30(1), 87–107 (2003)
Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Morse-Smale complexes for piecewise linear 3-manifolds. Symp. Comput. Geom. 2003, 361–370 (2009)
Edelsbrunner, H., Harer, J.: The persistent Morse complex segmentation of a 3-manifold. In: Magnenat-Thalmann, N. (ed.) 3DPH 2009. LNCS, vol. 5903, pp. 36–50. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10470-1_4
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Latecki, L.J., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Comput. Vis. Image Underst. 61, 70–83 (1995)
Kropatsch, W.G., Casablanca, R.M., Batavia, D., Gonzalez-Diaz, R.: On the space between critical points. Submitted to 21st International Conference on Discrete Geometry for Computer Imagery (2019)
Peltier, S., Ion, A., Haxhimusa, Y., Kropatsch, W.G., Damiand, G.: Computing homology group generators of images using irregular graph pyramids. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 283–294. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72903-7_26
Acknowledgments
This research has been partially supported by MINECO, FEDER/UE under grant MTM2015-67072-P. We thank the anonymous reviewers for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kropatsch, W.G., Casablanca, R.M., Batavia, D., Gonzalez-Diaz, R. (2019). Computing and Reducing Slope Complexes. In: Marfil, R., Calderón, M., Díaz del Río, F., Real, P., Bandera, A. (eds) Computational Topology in Image Context. CTIC 2019. Lecture Notes in Computer Science(), vol 11382. Springer, Cham. https://doi.org/10.1007/978-3-030-10828-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-10828-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10827-4
Online ISBN: 978-3-030-10828-1
eBook Packages: Computer ScienceComputer Science (R0)