Topology-Preserving Thinning in 2-D Pseudomanifolds
Abstract
Preserving topological properties of objects during thinning procedures is an important issue in the field of image analysis. In the case of 2-D digital images (i.e. images defined on ℤ2) such procedures are usually based on the notion of simple point. By opposition to the case of spaces of higher dimensions (i.e. ℤ n , n ≥ 3), it was proved in the 80’s that the exclusive use of simple points in ℤ2 was indeed sufficient to develop thinning procedures providing an output that is minimal with respect to the topological characteristics of the object. Based on the recently introduced notion of minimal simple set (generalising the notion of simple point), we establish new properties related to topology-preserving thinning in 2-D spaces which extend, in particular, this classical result to more general spaces (the 2-D pseudomanifolds) and objects (the 2-D cubical complexes).
Keywords
Topology preservation simple points simple sets cubical complexes collapse confluence pseudomanifoldsReferences
- 1.Bertrand, G.: On topological watersheds. Journal of Mathematical Imaging and Vision 22(2–3), 217–230 (2005)MathSciNetCrossRefGoogle Scholar
- 2.Bertrand, G.: On critical kernels. Comptes Rendus de l’Académie des Sciences, Série Mathématiques 1(345), 363–367 (2007)Google Scholar
- 3.Bing, R.H.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. Lectures on Modern Mathematics II, pp. 93–128 (1964)Google Scholar
- 4.Davies, E.R., Plummer, A.P.N.: Thinning algorithms: A critique and a new methodology. Pattern Recognition 14(1–6), 53–63 (1981)MathSciNetCrossRefGoogle Scholar
- 5.Giblin, P.: Graphs, surfaces and homology. Chapman and Hall, Boca Raton (1981)zbMATHGoogle Scholar
- 6.Yung Kong, T.: On topology preservation in 2-D and 3-D thinning. International Journal of Pattern Recognition and Artificial Intelligence 9(5), 813–844 (1995)CrossRefGoogle Scholar
- 7.Yung Kong, T.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 3–18. Springer, Heidelberg (1997)CrossRefGoogle Scholar
- 8.Kong, T.Y., Rosenfeld, A.: Digital topology: Introduction and survey. Computer Vision, Graphics, and Image Processing 48(3), 357–393 (1989)CrossRefGoogle Scholar
- 9.Kovalesky, V.A.: Finite topology as applied to image analysis. Computer Vision, Graphics, and Image Processing 46(2), 141–161 (1989)CrossRefGoogle Scholar
- 10.Maunder, C.R.F.: Algebraic topology. Dover, New York (1996)zbMATHGoogle Scholar
- 11.Passat, N., Couprie, M., Bertrand, G.: Minimal simple pairs in the 3-D cubic grid. Journal of Mathematical Imaging and Vision 32(3), 239–249 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Passat, N., Couprie, M., Mazo, L., Bertrand, G.: Topological properties of thinning in 2-D pseudomanifolds. Technical Report IGM2008-5, Université de Marne-la-Vallée (2008)Google Scholar
- 13.Passat, N., Mazo, L.: An introduction to simple sets. Technical Report LSIIT-2008-1, Université de Strasbourg (2008)Google Scholar
- 14.Ronse, C.: A topological characterization of thinning. Theoretical Computer Science 43(1), 31–41 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Rosenfeld, A.: Connectivity in digital pictures. Journal of the Association for Computer Machinery 17(1), 146–160 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Zeeman, E.C.: Seminar on Combinatorial Topology. In: IHES (1963)Google Scholar