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.
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For “gray value” z, \(g^{-1}(z) = \{p \in \mathfrak {R}^2 \,|\, g(p) = z \}\) is the level set of gray value z.
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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.
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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
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DOI: https://doi.org/10.1007/978-3-030-10828-1_2
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