Journal of Mathematical Imaging and Vision

, Volume 49, Issue 2, pp 418–433 | Cite as

Topology-Preserving Conditions for 2D Digital Images Under Rigid Transformations

  • Phuc Ngo
  • Yukiko Kenmochi
  • Nicolas Passat
  • Hugues Talbot


In the continuous domain \(\mathbb{R}^{n}\), rigid transformations are topology-preserving operations. Due to digitization, this is not the case when considering digital images, i.e., images defined on \(\mathbb{Z}^{n}\). In this article, we begin to investigate this problem by studying conditions for digital images to preserve their topological properties under all rigid transformations on \(\mathbb{Z}^{2}\). Based on (i) the recently introduced notion of DRT graph, and (ii) the notion of simple point, we propose an algorithm for evaluating digital images topological invariance.


Rigid transformation 2D digital image Discrete topology Simple point DRT graph 



The research leading to these results has received partial funding from the French Agence Nationale de la Recherche (Grant Agreement ANR-10-BLAN-0205 03).


  1. 1.
    Zitová, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(11), 977–1000 (2003) CrossRefGoogle Scholar
  2. 2.
    Yilmaz, A., Javed, O., Shah, M.: Object tracking: a survey. ACM Comput. Surv. 38(4), 1–45 (2006) CrossRefGoogle Scholar
  3. 3.
    Jain, V., Bollmann, B., Richardson, M., Berger, D., Helmstaedter, M., Briggman, K., Denk, W., Bowden, J., Mendenhall, J., Abraham, W., Harris, K., Kasthuri, N., Hayworth, K., Schalek, R., Tapia, J., Lichtman, J., Seung, S.: Boundary learning by optimization with topological constraints. In: CVPR, Proceedings, pp. 2488–2495. IEEE, New York (2010) Google Scholar
  4. 4.
    Faisan, S., Passat, N., Noblet, V., Chabrier, R., Meyer, C.: Topology preserving warping of 3-D binary images according to continuous one-to-one mappings. IEEE Trans. Image Process. 20(8), 2135–2145 (2011) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dawant, B., Hartmann, S., Thirion, J., Maes, F., Vandermeulen, D., Demaerel, P.: Automatic 3-D segmentation of internal structures of the head in MR images using a combination of similarity and free-form deformations: Part I, methodology and validation on normal subjects. IEEE Trans. Med. Imaging 18(10), 902–916 (1999) CrossRefGoogle Scholar
  6. 6.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Sufficient conditions for topological invariance of 2D digital images under rigid transformations. In: DGCI, Proceedings. Lecture Notes in Computer Science, vol. 7749, pp. 155–168. Springer, Berlin (2013) Google Scholar
  7. 7.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Combinatorial structure of rigid transformations in 2D digital images. Comput. Vis. Image Underst. 117(4), 393–408 (2013) CrossRefGoogle Scholar
  8. 8.
    Jacob, M.-A., Andres, E.: On discrete rotations. In: DGCI, Proceedings, pp. 161–174 (1995) Google Scholar
  9. 9.
    Amir, A., Kapah, O., Tsur, D.: Faster two-dimensional pattern matching with rotations. Theor. Comput. Sci. 368(3), 196–204 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Amir, A., Landau, G.M., Vishkin, U.: Efficient pattern matching with scaling. J. Algorithms 13(1), 2–32 (1992) zbMATHCrossRefGoogle Scholar
  11. 11.
    Amir, A., Butman, A., Lewenstein, M., Porat, E.: Real two dimensional scaled matching. Algorithmica 53(3), 314–336 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hundt, C., Liśkiewicz, M., Nevries, R.: A combinatorial geometrical approach to two-dimensional robust pattern matching with scaling and rotation. Theor. Comput. Sci. 410(51), 5317–5333 (2009) zbMATHCrossRefGoogle Scholar
  13. 13.
    Hundt, C., Liśkiewicz, M.: On the complexity of affine image matching. In: STACS, Proceedings. Lecture Notes in Computer Science, vol. 4393, pp. 284–295. Springer, Berlin (2007) Google Scholar
  14. 14.
    Hundt, C.: Affine image matching is uniform TC0-complete. In: CPM, Proceedings. Lecture Notes in Computer Science, vol. 6129, pp. 13–25. Springer, Berlin (2010) Google Scholar
  15. 15.
    Hundt, C., Liśkiewicz, M.: Combinatorial bounds and algorithmic aspects of image matching under projective transformations. In: MFCS, Proceedings. Lecture Notes in Computer Science, vol. 5162, pp. 395–406. Springer, Berlin (2008) Google Scholar
  16. 16.
    Reveillès, J.-P.: Géométrie discrète, calcul en nombres entiers et algorithmique, Thèse d’État. Université Strasbourg 1 (1991) Google Scholar
  17. 17.
    Andres, E.: The quasi-shear rotation. In: DGCI, Proceedings. Lecture Notes in Computer Science, vol. 1176, pp. 307–314. Springer, Berlin (1996) Google Scholar
  18. 18.
    Richman, M.S.: Understanding discrete rotations. In: ICASSP, Proceedings, vol. 3, pp. 2057–2060. IEEE, New York (1997) Google Scholar
  19. 19.
    Nouvel, B.: Rotations discrètes et automates cellulaires. Ph.D. thesis, École Normale Supérieure de Lyon (2006) Google Scholar
  20. 20.
    Nouvel, B., Rémila, E.: Incremental and transitive discrete rotations. In: IWCIA, Proceedings. Lecture Notes in Computer Science, vol. 4040, pp. 199–213. Springer, Berlin (2006) Google Scholar
  21. 21.
    Thibault, Y., Kenmochi, Y., Sugimoto, A.: Computing upper and lower bounds of rotation angles from digital images. Pattern Recognit. 42(8), 1708–1717 (2009) zbMATHCrossRefGoogle Scholar
  22. 22.
    Bertrand, G.: On critical kernels. C. R. Acad. Sci., Sér. 1 Math. 345, 363–367 (2007) zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Rosenfeld, A.: Connectivity in digital pictures. J. ACM 17(1), 146–160 (1970) zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989) CrossRefGoogle Scholar
  25. 25.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Paths, homotopy and reduction in digital images. Acta Appl. Math. 113(2), 167–193 (2011) zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: a unified topological framework. J. Math. Imaging Vis. 44(1), 19–37 (2012) zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Khalimsky, E.: Topological structures in computer science. J. Appl. Math. Simul. 1(1), 25–40 (1987) zbMATHMathSciNetGoogle Scholar
  28. 28.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989) CrossRefGoogle Scholar
  29. 29.
    Bertrand, G., Malandain, G.: A new characterization of three-dimensional simple points. Pattern Recognit. Lett. 15(2), 169–175 (1994) zbMATHCrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Ronse, C.: A topological characterization of thinning. Theor. Comput. Sci. 43(1), 31–41 (2007) MathSciNetGoogle Scholar
  32. 32.
    Bertrand, G.: On P-simple points. C. R. Acad. Sci., Sér. 1 Math. 321, 1077–1084 (1995) zbMATHMathSciNetGoogle Scholar
  33. 33.
    Passat, N., Mazo, L.: An introduction to simple sets. Pattern Recognit. Lett. 30(15), 1366–1377 (2009) CrossRefGoogle Scholar
  34. 34.
    Couprie, M., Bezerra, F.N., Bertrand, G.: Topological operators for grayscale image processing. J. Electron. Imaging 10(4), 1003–1015 (2001) CrossRefGoogle Scholar
  35. 35.
    Latecki, L.J.: Multicolor well-composed pictures. Pattern Recognit. Lett. 16(4), 425–431 (1997) CrossRefGoogle Scholar
  36. 36.
    Damiand, G., Dupas, A., Lachaud, J.-O.: Fully deformable 3D digital partition model with topological control. Pattern Recognit. Lett. 32(9), 1374–1383 (2011) CrossRefGoogle Scholar
  37. 37.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Topology on digital label images. J. Math. Imaging Vis. 44(3), 254–281 (2012) zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Pham, D., Bazin, P.-L., Prince, J.: Digital topology in brain imaging. IEEE Signal Process. Mag. 27(4), 51–59 (2010) CrossRefGoogle Scholar
  39. 39.
    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
  40. 40.
    Han, X., Xu, C., Prince, J.L.: A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Anal. Mach. Intell. 25(6), 755–768 (2003) CrossRefGoogle Scholar
  41. 41.
    Bazin, P.-L., Ellingsen, L.M., Pham, D.L.: Digital homeomorphisms in deformable registration. In: IPMI, Proceedings. Lecture Notes in Computer Science, vol. 4584, pp. 211–222. Springer, Berlin (2007) Google Scholar
  42. 42.
    Ayala, R., Domínguez, E., Francés, A.R., Quintero, A.: Homotopy in digital spaces. Discrete Appl. Math. 125(1), 3–24 (2003) zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Bertrand, G., Couprie, M., Passat, N.: A note on 3-D simple points and simple-equivalence. Inf. Process. Lett. 109(13), 700–704 (2009) zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Nouvel, B., Rémila, E.: Configurations induced by discrete rotations: periodicity and quasi-periodicity properties. Discrete Appl. Math. 147(2–3), 325–343 (2005) zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    Thibault, Y.: Rotations in 2D and 3D discrete spaces. Ph.D. thesis, Université Paris-Est (2010) Google Scholar
  46. 46.
    Latecki, L.J., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Comput. Vis. Image Underst. 61(1), 70–83 (1995) CrossRefGoogle Scholar
  47. 47.
    Mazo, L.: A framework for label images. In: CTIC, Proceedings. Lecture Notes in Computer Science, vol. 7309, pp. 1–10. Springer, Berlin (2012) Google Scholar
  48. 48.
    Berenstein, C., Lavine, D.: On the number of digital straight line segments. IEEE Trans. Pattern Anal. Mach. Intell. 10(6), 880–887 (1988) zbMATHCrossRefGoogle Scholar
  49. 49.
    Nagy, B.: An algorithm to find the number of the digitizations of discs with a fixed radius. Electron. Notes Discrete Math. 20, 607–622 (2005) CrossRefGoogle Scholar
  50. 50.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, San Diego (1983) Google Scholar
  51. 51.
    Heijmans, H.J.A.M.: Discretization of morphological operators. J. Vis. Commun. Image Represent. 3(2), 182–193 (1992) CrossRefGoogle Scholar
  52. 52.
    Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8(2), 131–159 (1998) zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Phuc Ngo
    • 1
  • Yukiko Kenmochi
    • 1
  • Nicolas Passat
    • 2
  • Hugues Talbot
    • 1
  1. 1.LIGM, UPEMLV-ESIEE-CNRSUniversité Paris-EstParisFrance
  2. 2.CReSTICUniversité de Reims Champagne-ArdenneReimsFrance

Personalised recommendations