# Topology on Digital Label Images

- 352 Downloads
- 3 Citations

## Abstract

In digital imaging, after several decades devoted to the study of topological properties of binary images, there is an increasing need of new methods enabling to take into (topological) consideration *n*-ary images (also called label images). Indeed, while binary images enable to handle one object of interest, label images authorise to simultaneously deal with a plurality of objects, which is a frequent requirement in several application fields. In this context, one of the main purposes is to propose topology-preserving transformation procedures for such label images, thus extending the ones (e.g., growing, reduction, skeletonisation) existing for binary images. In this article, we propose, for a wide range of digital images, a new approach that permits to locally modify a label image, while preserving not only the topology of each label set, but also the topology of any arrangement of the labels understood as the topology of any union of label sets. This approach enables in particular to unify and extend some previous attempts devoted to the same purpose.

## Keywords

Digital imaging Topology Label images Homotopy Simple points## References

- 1.Buneman, O.P.: A grammar for the topological analysis of plane figures. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 5, pp. 383–393 (1969) Google Scholar
- 2.Kong, T.Y.: A digital fundamental group. Comput. Graph.
**13**(2), 159–166 (1989) CrossRefGoogle Scholar - 3.González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math.
**147**, 245–263 (2005). Advances in Discrete Geometry and Topology MathSciNetMATHCrossRefGoogle Scholar - 4.Hilditch, C.J.: Linear skeletons from square cupboards. In: Meltzer, B., Michie, D. (eds.) Machine Intelligence, vol. 4, pp. 403–420 (1969) Google Scholar
- 5.Bertrand, G.: On P-simple points. C. R. Acad. Sci., Sér. Math.
**1**(321), 1077–1084 (1995) MathSciNetGoogle Scholar - 6.Ronse, C.: A topological characterization of thinning. Theor. Comput. Sci.
**43**(0), 31–41 (1986) MathSciNetMATHCrossRefGoogle Scholar - 7.Passat, N., Mazo, L.: An introduction to simple sets. Pattern Recognit. Lett.
**30**(15), 1366–1377 (2009) CrossRefGoogle Scholar - 8.Mangin, J.-F., Frouin, V., Bloch, I., Régis, J., López-Krahe, J.: From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations. J. Math. Imaging Vis.
**5**(4), 297–318 (1995) CrossRefGoogle Scholar - 9.Miri, S., Passat, N., Armspach, J.-P.: Topology-preserving discrete deformable model: application to multi-segmentation of brain MRI. In: International Conference on Image and Signal Processing—ICISP 2008. Lecture Notes in Computer Science, vol. 5099, pp. 67–75. Springer, Berlin (2008) Google Scholar
- 10.Poupon, F., Mangin, J.-F., Hasboun, D., Poupon, C., Magnin, I., Frouin, V.: Multi-object deformable templates dedicated to the segmentation of brain deep structures. In: MICCAI’98: Proceedings of the First International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 1134–1143. Springer, Berlin (1998) Google Scholar
- 11.Bazin, P.-L., Pham, D.: Topology-preserving tissue classification of magnetic resonance brain images. IEEE Trans. Med. Imaging
**26**(4), 487–496 (2007) CrossRefGoogle Scholar - 12.Liu, J., Huang, S., Nowinski, W.: Registration of brain atlas to MR images using topology preserving front propagation. J. Signal Process. Syst.
**55**(1), 209–216 (2009) CrossRefGoogle Scholar - 13.Rosenfeld, A.: Connectivity in digital pictures. J. Assoc. Comput. Mach.
**17**(1), 146–160 (1970) MathSciNetMATHCrossRefGoogle Scholar - 14.Duda, O., Hart, P.E., Munson, J.H.: Graphical data processing research study and experimental investigation. Tech. Rep. AD650926. Stanford Research Institute (1967) Google Scholar
- 15.Damiand, G., Dupas, A., Lachaud, J.-O.: Fully deformable 3D digital partition model with topological control. Pattern Recognit. Lett.
**32**, 1374–1383 (2011) CrossRefGoogle Scholar - 16.Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recognit. Lett.
**15**, 1003–1011 (1994) CrossRefGoogle Scholar - 17.Latecki, L.J.: Multicolor well-composed pictures. Pattern Recognit. Lett.
**16**(4), 425–431 (1995) CrossRefGoogle Scholar - 18.Siqueira, S., Latecki, L., Tustison, N., Gallier, J., Gee, J.: Topological repairing of 3D digital images. J. Math. Imaging Vis.
**30**(3), 249–274 (2008) MathSciNetCrossRefGoogle Scholar - 19.Cointepas, Y., Bloch, I., Garnero, L.: A cellular model for multi-objects multi-dimensional homotopic deformations. Pattern Recognit.
**34**, 1785–1798 (2001) MATHCrossRefGoogle Scholar - 20.Bazin, P.-L., Ellingsen, L., Pham, D.: Digital homeomorphisms in deformable registration. In: Karssemeijer, N., Lelieveldt, B. (eds.) IPMI. Lecture Notes in Computer Science, vol. 4584, pp. 211–222. Springer, Berlin (2007) Google Scholar
- 21.Whitehead, J.H.C.: Simplicial Spaces, Nuclei and
*m*-Groups. Proc. Lond. Math. Soc.**s2-45**, 243–327 (1939) CrossRefGoogle Scholar - 22.Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: a unified topological framework. Journal of Mathematical Imaging and Vision. doi: 10.1007/s10851-011-0308-9
- 23.Bertrand, G., Couprie, M.: Two-dimensional thinning algorithms based on critical kernels. J. Math. Imaging Vis.
**31**(1), 35–56 (2008) MathSciNetCrossRefGoogle Scholar - 24.Mazo, L., Passat, N., Couprie, M., Ronse, C.: Paths, homotopy and reduction in digital images. Acta Appl. Math.
**113**(2), 167–193 (2011) MathSciNetMATHCrossRefGoogle Scholar - 25.Bing, R.: Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. Lectures on Modern Mathematics
**II**, 93–128 (1964) MathSciNetGoogle Scholar - 26.Zeeman, E.: On the dunce hat. Topology
**2**, 341–358 (1964) MathSciNetCrossRefGoogle Scholar - 27.Whitehead, J.H.C.: Combinatorial homotopy. I. Bull. Am. Math. Soc.
**55**, 213–245 (1949) MathSciNetMATHCrossRefGoogle Scholar - 28.Fourey, S., Malgouyres, R.: A concise characterization of 3D simple points. Discrete Appl. Math.
**125**(1), 59–80 (2003) MathSciNetMATHCrossRefGoogle Scholar - 29.Maunder, C.R.F.: Algebraic Topology. Dover, New York (1996) Google Scholar
- 30.Munkres, J.: Elements of Algebraic Topology. Westview Press, Boulder (1996) Google Scholar
- 31.May, A.: A Concise Course in Algebraic Topology. University Chicago Press, Chicago (1999) MATHGoogle Scholar
- 32.Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
- 33.Giblin, P.: Graphs, Surfaces and Homology. Cambridge University Press, Cambridge (2010) MATHCrossRefGoogle Scholar
- 34.Alexandroff, P.: Diskrete Räume, Rec. Math. [Mat. Sbornik] N.S. 501–519 (1937) Google Scholar
- 35.Birkhoff, G.: Rings of sets. Duke Math. J.
**3**(3), 443–454 (1937) MathSciNetCrossRefGoogle Scholar - 36.McCord, M.: Singular homology groups and homotopy groups of finite topological spaces. Duke Math. J.
**33**(3), 465–474 (1966) MathSciNetMATHCrossRefGoogle Scholar - 37.Bertrand, G.: New notions for discrete topology. In: DCGI’99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, pp. 218–228 (1999) CrossRefGoogle Scholar
- 38.Stong, R.E.: Finite topological spaces. Trans. Am. Math. Soc.
**123**(25), 325–340 (1966) MathSciNetMATHCrossRefGoogle Scholar - 39.May, J.P.: Finite topological spaces (lecture notes). url: www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf (2008)
- 40.Barmak, J.A., Minian, E.G.: Simple homotopy types and finite spaces. Adv. Math.
**218**, 87–104 (2008) MathSciNetMATHCrossRefGoogle Scholar - 41.Barmak, J.A., Minian, E.G.: One-point reductions of finite spaces,
*h*-regular*CW*-complexes and collapsibility, Algebraic & Geometric. Topology**8**(3), 1763–1780 (2008) MathSciNetMATHGoogle Scholar - 42.Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process.
**48**, 357–393 (1989) CrossRefGoogle Scholar - 43.Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process.
**46**(2), 141–161 (1989) CrossRefGoogle Scholar - 44.Kovalevsky, V.: Axiomatic digital topology. J. Math. Imaging Vis.
**26**(1), 41–58 (2006) MathSciNetCrossRefGoogle Scholar - 45.Kovalesky, V.: Geometry of Locally Finite Spaces. Publishing House Dr. Baerbel Kovalevski (2008) Google Scholar
- 46.Herman, G.T.: Geometry of Digital Spaces. Birkhäuser, Basel (1998) MATHGoogle Scholar
- 47.Kronheimer, E.: The topology of digital images. Topol. Appl.
**46**, 279–303 (1992) MathSciNetMATHCrossRefGoogle Scholar - 48.Ronse, C., Agnus, V.: Morphology on label images: flat-type operators and connections. J. Math. Imaging Vis.
**22**(2), 283–307 (2005) MathSciNetCrossRefGoogle Scholar - 49.Ronse, C., Agnus, V.: Geodesy on label images, and applications to video sequence processing. J. Vis. Commun. Image Represent.
**19**, 392–408 (2008) CrossRefGoogle Scholar - 50.Ayala, R., Domínguez, E., Francés, A., Quintero, A.: Digital lighting functions. In: Procs. Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1347, pp. 139–150. Springer, Berlin (1997) Google Scholar
- 51.Ronse, C.: An isomorphism for digital images, Journal of Combinatorial Theory, Series A 39(2) Google Scholar
- 52.Spanier, E.H.: Algebraic Topology. Springer, Berlin (1994) Google Scholar
- 53.Kong, T.Y.: Topology-preserving deletion of 1’s from 2-, 3- and 4-dimensional binary images. In: DGCI’97: Proceedings of the 7th International Workshop on Discrete Geometry for Computer Imagery, pp. 3–18. Springer, Berlin (1997) Google Scholar
- 54.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 (2009) CrossRefGoogle Scholar - 55.Blyth, T.: Lattices and Ordered Algebraic Structures. Springer, London (2005) MATHGoogle Scholar
- 56.Grätzer, G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (2003) MATHGoogle Scholar