A Tutorial on Well-Composedness

Article
  • 50 Downloads

Abstract

Due to digitization, usual discrete signals generally present topological paradoxes, such as the connectivity paradoxes of Rosenfeld. To get rid of those paradoxes, and to restore some topological properties to the objects contained in the image, like manifoldness, Latecki proposed a new class of images, called well-composed images, with no topological issues. Furthermore, well-composed images have some other interesting properties: for example, the Euler number is locally computable, boundaries of objects separate background from foreground, the tree of shapes is well defined. Last, but not the least, some recent works in mathematical morphology have shown that very nice practical results can be obtained thanks to well-composed images. Believing in its prime importance in digital topology, we then propose this state of the art of well-composedness, summarizing its different flavors, the different methods existing to produce well-composed signals, and the various topics that are related to well-composedness.

Keywords

Well-composedness Digital topology Mathematical morphology Critical configurations Discrete surfaces Manifolds 

References

  1. 1.
    Al Faqheri, W., Mashohor, S.: A real-time Malaysian automatic license plate recognition (M-ALPR) using hybrid fuzzy. IJCSNS Int. J. Comput. Sci. Netw. Secur. 9(2), 333–340 (2009)Google Scholar
  2. 2.
    Aleksandrov, P.S.: Combinatorial Topology, vol. 1. Courier Corporation, North Chelmsford (1956)Google Scholar
  3. 3.
    Alexander, J.W.: A proof and extension of the Jordan–Brouwer separation theorem. Trans. Am. Math. Soc. 23(4), 333–349 (1922)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Alexandroff, P., Hopf, H.: Topologie I: Erster Band. Grundbegriffe der Mengentheoretischen, Topologie ,Topologie der Komplexe. Topologische Invarianzsätze und Anschliessende Begriffsbildungen. Verschlingungen im n-Dimensionalen Euklidischen Raum Stetige Abbildungen von Polyedern. Springer (2013)Google Scholar
  5. 5.
    Alexandrov, O., Santosa, F.: A topology-preserving level set method for shape optimization. J. Comput. Phys. 204(1), 121–130 (2005)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Alexandrov, P.S.Ã.: Diskrete RÃd’ume. Matematicheskij Sbornik 2(44), 501–519 (1937)Google Scholar
  7. 7.
    Arcelli, C.: Pattern thinning by contour tracing. Comput. Graph. Image Process. 17(2), 130–144 (1981)CrossRefGoogle Scholar
  8. 8.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Springer, Berlin (2009)MATHCrossRefGoogle Scholar
  9. 9.
    Ballester, C., Caselles, V., Monasse, P.: The tree of shapes of an image. ESAIM: Control Optim. Calc. Var. 9, 1–18 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bazin, P.-L., Ellingsen, L.M., Pham, D.L.: Digital homeomorphisms in deformable registration. In: International Conference on Information Processing in Medical Imaging. Springer, pp. 211–222 (2007)Google Scholar
  11. 11.
    Bazin, P.-L., Pham, D.L.: Topology-preserving tissue classification of magnetic resonance brain images. IEEE Trans. Med. Imaging 26(4), 487–496 (2007)CrossRefGoogle Scholar
  12. 12.
    Berg, G.O., Julian, W.H., Mines, R., Richman, F.: The constructive Jordan curve theorem. Rocky Mt. J. Math. 5(2), 225–236 (1975)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Bertrand, G.: Simple points, topological numbers and geodesic neighborhoods in cubic grids. Pattern Recogn. Lett. 15(10), 1003–1011 (1994)CrossRefGoogle Scholar
  14. 14.
    Bertrand, G.: A Boolean characterization of three-dimensional simple points. Pattern Recogn. Lett. 17(2), 115–124 (1996)CrossRefGoogle Scholar
  15. 15.
    Bertrand, G.: New notions for discrete topology. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 1568, pp. 218–228. Springer (1999)Google Scholar
  16. 16.
    Bertrand, G.: On topological watersheds. J. Math. Imaging Vis. 22(2–3), 217–230 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bertrand, G., Everat, J.-C., Couprie, M.: Topological approach to image segmentation. In: SPIE’s 1996 International Symposium on Optical Science, Engineering, and Instrumentation, pp. 65–76. International Society for Optics and Photonics (1996)Google Scholar
  18. 18.
    Bertrand, G., Everat, J.-C., Couprie, M.: Image segmentation through operators based on topology. J. Electron. Imaging 6(4), 395–405 (1997)CrossRefGoogle Scholar
  19. 19.
    Bertrand, G., Malandain, G.: A new characterization of three-dimensional simple points. Pattern Recogn. Lett. 15(2), 169–175 (1994)MATHCrossRefGoogle Scholar
  20. 20.
    Beucher, S., Lantuéjoul, C.: Use of watersheds in contour detection. Real-time Edge and Motion Detection/Estimation, International Workshop on Image Processing (1979)Google Scholar
  21. 21.
    Beucher, S., Meyer, F.: The morphological approach to segmentation: the watershed transformation. Opt. Eng. N. Y. Marcel Dekker Inc. 34, 433–433 (1992)Google Scholar
  22. 22.
    Bieri, H., Nef, W.: Algorithms for the Euler characteristic and related additive functionals of digital objects. Comput. Vis. Graph. Image Process. 28(2), 166–175 (1984)MATHCrossRefGoogle Scholar
  23. 23.
    Bishop, E., Bridges, D.S.: Constructive Analysis, vol. 279. Springer, Berlin (2012)MATHGoogle Scholar
  24. 24.
    Bloch, E.D.: A First Course in Geometric Topology and Differential Geometry. Springer, Berlin (1997)MATHCrossRefGoogle Scholar
  25. 25.
    Bloch, I., Heijmans, H., Ronse, C.: Mathematical morphology. In: Aiello, M., Pratt-Hartmann, I., Van Benthem, J. (eds.) Handbook of Spatial Logics, pp. 857–944. Springer (2007)Google Scholar
  26. 26.
    Boutry, N.: A study of well-composedness in \(n-\text{D}\). PhD thesis, Université Paris-Est, France (2016)Google Scholar
  27. 27.
    Boutry, N., Géraud, T., Najman, L.: On making \(n\)D images well-composed by a self-dual local interpolation. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) Discrete Geometry for Computer Imagery—Proceedings of the 18th International Conference on Discrete Geometry for Computer Imagery (DGCI), volume 8668 of Lecture Notes in Computer Science, pp. 320–331, Siena, Italy, September 2014. Springer (2014)Google Scholar
  28. 28.
    Boutry, N., Géraud, T., Najman, L.: How to make images well-composed in \(n\text{ D }\) without interpolation. In: Proceedings of the 22nd IEEE International Conference on Image Processing (ICIP), pp. 2149–2153, Québec City, Canada (2015)Google Scholar
  29. 29.
    Boutry, N., Géraud, T., Najman, L.: How to make \(n-\text{ D }\) functions digitally well-composed in a self-dual way. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing – Proceedings of the 12th International Symposium on Mathematical Morphology (ISMM), volume 9082 of Lecture Notes in Computer Science, pp. 561–572, Reykjavik, Iceland. Springer (2015)Google Scholar
  30. 30.
    Braquelaire, J.-P., Brun, L.: Image segmentation with topological maps and inter-pixel representation. J. Vis. Commun. Image Represent. 9(1), 62–79 (1998)CrossRefGoogle Scholar
  31. 31.
    Busta, M., Neumann, L., Matas, J.: Fastext: Efficient unconstrained scene text detector. In: IEEE International Conference on Computer Vision, pp. 1206–1214. IEEE (2015)Google Scholar
  32. 32.
    Caissard, T., Coeurjolly, D., Lachaud, J.-O., Roussillon, T.: Heat kernel Laplace-Beltrami operator on digital surfaces. In: International Conference on Discrete Geometry for Computer Imagery, pp. 241–253. Springer (2017)Google Scholar
  33. 33.
    Caselles, V., Monasse, P.: Grain filters. J. Math. Imaging Vis. 17(3), 249–270 (2002)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Caselles, V., Monasse, P.: Geometric description of images as topographic maps, ser. Lecture Notes in Mathematics. Springer, 1984 (2009)Google Scholar
  35. 35.
    Cecil, T.C.: Numerical methods for partial differential equations involving discontinuities. PhD thesis, University of California Los Angeles (2003)Google Scholar
  36. 36.
    Chen, L.: Genus computing for 3D digital objects: Algorithm and implementation. arXiv preprint arXiv:0912.4936 (2009)
  37. 37.
    Chen, L.: Algorithms for computing topological invariants in 2D and 3D digital spaces. arXiv preprint arXiv:1309.4109 (2013)
  38. 38.
    Coates, A., Carpenter, B., Case, C., Satheesh, S., Suresh, B., Wang, T., Wu, D.J., Ng, A.Y.: Text detection and character recognition in scene images with unsupervised feature learning. In: International Conference on Document Analysis and Recognition, pp. 440–445. IEEE (2011)Google Scholar
  39. 39.
    Cousty, J., Bertrand, G.: Uniqueness of the perfect fusion grid on \(\mathbb{Z}^d\). J. Math. Imaging Vis. 34(3), 291–306 (2009)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Cousty, J., Bertrand, G., Couprie, M., Najman, L.: Fusion graphs: merging properties and watersheds. J. Math. Imaging Vis. 30(1), 87–104 (2008)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Cousty, J., Couprie, M., Najman, L., Bertrand, G.: Grayscale watersheds on perfect fusion graphs. In: International Workshop on Combinatorial Image Analysis, pp. 60–73. Springer (2006)Google Scholar
  42. 42.
    Crozet, S., Géraud, T.: A first parallel algorithm to compute the morphological tree of shapes of \(n\text{ D }\) images. In: Proceedings of the 21st IEEE International Conference on Image Processing (ICIP), pp. 2933–2937, Paris, France (2014)Google Scholar
  43. 43.
    Daragon, X.: Surfaces discrètes et frontières d’objets dans les ordres. PhD thesis, Université de Marne-la-Vallée (2005)Google Scholar
  44. 44.
    Daragon, X., Couprie, M., Bertrand, G.: Marching chains algorithm for Alexandroff–Khalimsky spaces. In: International Symposium on Optical Science and Technology, pp. 51–62. International Society for Optics and Photonics (2002)Google Scholar
  45. 45.
    Daragon, X., Couprie, M., Bertrand, G.: Discrete surfaces and frontier orders. J. Math. Imaging Vis. 23(3), 379–399 (2005)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators in nd. preprint, the Caltech Multi-Res Modeling Group (2000)Google Scholar
  47. 47.
    Deseilligny, M.P., Stamon, G., Suen, C.Y.: Veinerization: a new shape description for flexible skeletonization. IEEE Trans. Pattern Anal. Mach. Intell. 20(5), 505–521 (1998)CrossRefGoogle Scholar
  48. 48.
    Dey, T.K., Guha, S.: Computing homology groups of simplicial complexes in \(\mathbb{R}^3\). J. ACM 45(2), 266–287 (1998)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Diaz-deLeon, J.L., SossaAzuela, J.H.: On the computation of the Euler number of a binary object. Pattern Recogn. 29(3), 471–476 (1996)CrossRefGoogle Scholar
  50. 50.
    Duda, R.O., Munson, J.H.: Graphical-data-processing research study and experimental investigation. Technical report, DTIC Document (1967)Google Scholar
  51. 51.
    Dyer, C.R.: Computing the Euler number of an image from its quadtree. Comput. Graph. Image Process. 13(3), 270–276 (1980)CrossRefGoogle Scholar
  52. 52.
    Eckhardt, U., Latecki, L.J.: Digital Topology. Institut für Angewandte Mathematik (1994)Google Scholar
  53. 53.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. In: Foundations of Computer Science, 2000. Proceedings. 41st Annual Symposium on, pp. 454–463. IEEE (2000)Google Scholar
  54. 54.
    Epshtein, B., Ofek, E., Wexler, Y.: Detecting text in natural scenes with stroke width transform. In: IEEE Conference on Computer Vision and Pattern Recognition, pp. 2963–2970. IEEE (2010)Google Scholar
  55. 55.
    Evako, A.V., Kopperman, R., Mukhin, Y.V.: Dimensional properties of graphs and digital spaces. J. Math. Imaging Vis. 6(2–3), 109–119 (1996)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Fabrizio, J., Robert-Seidowsky, M., Dubuisson, S., Calarasanu, S., Boissel, R.: Textcatcher: a method to detect curved and challenging text in natural scenes. Int. J. Doc. Anal. Recogn. 19(2), 99–117 (2016)CrossRefGoogle Scholar
  57. 57.
    Faisan, S., Passat, N., Noblet, V., Chabrier, R., Meyer, C.: Topology preserving warping of 3D binary images according to continuous one-to-one mappings. IEEE Trans. Image Process. 20(8), 2135–2145 (2011)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Fedorov, E.S.: Course of Crystallography. Published by RK Rikker, Saint-Petersburg (1901). (in Russian)Google Scholar
  59. 59.
    Fiorio, C.: Approche interpixel en analyse dâǍŹimages, une topologie et des algorithmes de segmentation. PhD thesis, Technische Universität Wien (1995)Google Scholar
  60. 60.
    Fiorio, C.: A topologically consistent representation for image analysis: the frontiers topological graph. In: Discrete Geometry for Computer Imagery, pp. 151–162. Springer (1996)Google Scholar
  61. 61.
    Forman, R.: A discrete Morse theory for cell complexes. In: Geometry, Topology 6 Physics for Raoul Bott. Citeseer (1995)Google Scholar
  62. 62.
    Géraud, T., Carlinet, E., Crozet, S.: Self-duality and discrete topology: links between the morphological tree of shapes and well-composed gray-level images. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 12th International Symposium on Mathematical Morphology (ISMM), volume 9082 of Lecture Notes in Computer Science, pp. 573–584, Reykjavik, Iceland. Springer (2015)Google Scholar
  63. 63.
    Géraud, T., Carlinet, E., Crozet, S., Najman, L.: A quasi-linear algorithm to compute the tree of shapes of \(n-\text{ D }\) images. In: Hendriks, C.L. Luengo, Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 11th International Symposium on Mathematical Morphology (ISMM), volume 7883 of Lecture Notes in Computer Science, pp. 98–110, Uppsala, Sweden. Springer (2013)Google Scholar
  64. 64.
    Géraud, T., Xu, Y., Carlinet, E., Boutry, N.: Introducing the Dahu pseudo-distance. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 13th International Symposium on Mathematical Morphology (ISMM), volume 10225 of Lecture Notes in Computer Science, pp. 55–67, Fontainebleau, France. Springer (2017)Google Scholar
  65. 65.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Cohomology ring of 3D cubical complexes. In: IWCIA Special Track on Applications, pp. 139–150 (2009)Google Scholar
  66. 66.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Cubical cohomology ring of 3D photographs. Int. J. Imaging Syst. Technol. 21(1), 76–85 (2011)CrossRefGoogle Scholar
  67. 67.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Well-composed cell complexes. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 6607, pp. 153–162. Springer (2011)Google Scholar
  68. 68.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: 3D well-composed polyhedral complexes. Discrete Appl. Math. 183, 59–77 (2015)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    González-Díaz, R., Jiménez, M.J., Medrano, B.: Encoding specific 3D polyhedral complexes using 3D binary images. In: Discrete Geometry for Computer Imagery, pp. 268–281. Springer (2016)Google Scholar
  70. 70.
    González-Díaz, R., Lamar, J., Umble, R.: Cup products on polyhedral approximations of 3D digital images. In: International Workshop on Combinatorial Image Analysis, pp. 107–119. Springer (2011)Google Scholar
  71. 71.
    González-Díaz, R., Real, P.: Towards digital cohomology. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 2886, pp. 92–101. Springer (2003)Google Scholar
  72. 72.
    González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2), 245–263 (2005)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Grady, L.J., Polimeni, J.R.: Discrete Calculus: Applied Analysis on Graphs for Computational Science. Springer, Berlin (2010)MATHCrossRefGoogle Scholar
  74. 74.
    Gray, S.B.: Local properties of binary images in two dimensions. IEEE Trans. Comput. 100(5), 551–561 (1971)MATHCrossRefGoogle Scholar
  75. 75.
    Greenberg, M.J.: Lectures on Algebraic Topology, vol. 9. W A Benjamin, New York (1967)MATHGoogle Scholar
  76. 76.
    Gross, A., Latecki, L.J.: Digitizations preserving topological and differential geometric properties. Comput. Vis. Image Underst. 62(3), 370–381 (1995)CrossRefGoogle Scholar
  77. 77.
    Han, X., Xu, C., Prince, J.L.: A topology preserving deformable model using level sets. In: IEEE Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 765–770. IEEE (2001)Google Scholar
  78. 78.
    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
  79. 79.
    Heijmans, H.: Theoretical aspects of gray-level morphology. IEEE Trans. Pattern Anal. Mach. Intell. 13(6), 568–582 (1991)CrossRefGoogle Scholar
  80. 80.
    Heijmans, H.: Morphological Image Operators. Advances in Electronics and Electron Physics, vol. 25, pp. 198–202. Academic Press (1994)Google Scholar
  81. 81.
    Huỳnh, L.D., Xu, Y., Géraud, T.: Morphology-based hierarchical representation with application to text segmentation in natural images. In: Proceedings of the 23rd International Conference on Pattern Recognition (ICPR), pp. 4029–4034, Cancun, Mexico. IEEE Computer Society (2016)Google Scholar
  82. 82.
    Janos, L., Rosenfeld, A.: Digital connectedness: an algebraic approach. Pattern Recogn. Lett. 1(3), 135–139 (1983)MATHCrossRefGoogle Scholar
  83. 83.
    Jordan, C.: Cours d’Analyse de l’Ecole Polytechnique, vol. 3. Gauthier-Villars, Paris (1887)MATHGoogle Scholar
  84. 84.
    Ju, T., Losasso, F., Schaefer, S., Warren, J.: Dual contouring of Hermite data. ACM Trans. Graph. 21(3), 339–346 (2002)CrossRefGoogle Scholar
  85. 85.
    Kelley, J.L.: General topology. The university series in higher mathematics (1955)Google Scholar
  86. 86.
    Khalimsky, E.: Applications of connected ordered topological spaces in topology. Conference of Math, Department of Povolosia (1970)Google Scholar
  87. 87.
    Khalimsky, E.: Ordered topological spaces. Naukova Dumka Press, Kiev (1977)MATHGoogle Scholar
  88. 88.
    Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36(1), 1–17 (1990)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Kirby, R.M., Pascucci, V., Silva, C.T., Peters, T.J., Tierny, J., Scheidegger, C., Nonato, L.G., Etiene, T.: Topology verification for isosurface extraction. IEEE Trans. Vis. Comput. Graph. 18(6), 952–965 (2012)CrossRefGoogle Scholar
  90. 90.
    Kong, Y.T., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989)CrossRefGoogle Scholar
  91. 91.
    Kong, Y.T., Rosenfeld, A.: If we use 4-or 8-connectedness for both the objects and the background, the Euler characteristics is not locally computable. Pattern Recogn. Lett. 11(4), 231–232 (1990)MATHCrossRefGoogle Scholar
  92. 92.
    Kopperman, R.: The Khalimsky line as a foundation for digital topology. In: O, YL., Toet, A., Foster, D., Heijmans, H.J.A.M., Meer, P. (eds.) Shape in Picture. Computer and Systems Sciences, vol. 126, pp. 3–20. Springer (1994)Google Scholar
  93. 93.
    Kopperman, R., Meyer, P.R., Wilson, R.G.: A Jordan surface theorem for three-dimensional digital spaces. Discrete Comput. Geom. 6(2), 155–161 (1991)MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Köthe, U.: Generische Programmierung für die Bildverarbeitung. BoD–Books on Demand (2000)Google Scholar
  95. 95.
    Kovalevsky, V.: Axiomatic digital topology. J. Math. Imaging Vis. 26(1), 41–58 (2006)MathSciNetCrossRefGoogle Scholar
  96. 96.
    Kovalevsky, V.A.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46(2), 141–161 (1989)CrossRefGoogle Scholar
  97. 97.
    Krahnstoever, N., Lorenz, C.: Computing curvature-adaptive surface triangulations of three-dimensional image data. Vis. Comp. 20(1), 17–36 (2004)CrossRefGoogle Scholar
  98. 98.
    Lachaud, J.-O., Montanvert, A.: Continuous analogs of digital boundaries: a topological approach to iso-surfaces. Graph. Models Image Process. 62(3), 129–164 (2000)CrossRefGoogle Scholar
  99. 99.
    Lachaud, J.-O., Thibert, B.: Properties of gauss digitized shapes and digital surface integration. J. Math. Imaging Vis. 54(2), 162–180 (2016)MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Latecki, L.J.: 3D well-composed pictures. Graph. Models Image Process. 59(3), 164–172 (1997)CrossRefGoogle Scholar
  101. 101.
    Latecki, L.J.: Discrete Representation of Spatial Objects in Computer Vision, vol. 11. Springer, Berlin (1998)MATHGoogle Scholar
  102. 102.
    Latecki, L.J.: Well-composed sets. Adv. Electron. Electron Phys. 112, 95–163 (2000)Google Scholar
  103. 103.
    Latecki, L.J., Conrad, C., Gross, A.: Preserving topology by a digitization process. J. Math. Imaging Vis. 8(2), 131–159 (1998)MathSciNetMATHCrossRefGoogle Scholar
  104. 104.
    Latecki, L.J., Eckhardt, U., Rosenfeld, A.: Well-composed sets. Comput. Vis. Image Underst. 61(1), 70–83 (1995)CrossRefGoogle Scholar
  105. 105.
    Le Guyader, C., Vese, L.A.: Self-repelling snakes for topology-preserving segmentation models. IEEE Trans. Image Process. 17(5), 767–779 (2008)MathSciNetCrossRefGoogle Scholar
  106. 106.
    Lee, C.-N., Poston, T., Rosenfeld, A.: Winding and Euler numbers for 2D and 3D digital images. Graph. Models Image Process. 53(6), 522–537 (1991)MATHCrossRefGoogle Scholar
  107. 107.
    Lee, C.-N., Rosenfeld, A.: Computing the Euler number of a 3D image. Center for Automation Research Technical Report CAR-TR-205 (1986)Google Scholar
  108. 108.
    Lee, J.-J., Lee, P.-H., Lee, S.-W., Yuille, A.L., Koch, C.: Adaboost for text detection in natural scene. In: ICDAR, pp. 429–434 (2011)Google Scholar
  109. 109.
    Lévy, B.: Laplace–Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: IEEE International Conference on Shape Modeling and Applications, 2006. SMI 2006, pp. 13–13. IEEE (2006)Google Scholar
  110. 110.
    Lima, E.L.: The Jordan–Brouwer separation theorem for smooth hypersurfaces. Am. Math. Mon. 95(1), 39–42 (1988)MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Liu, J., Huang, S., Nowinski, W.L.: Registration of brain atlas to MR images using topology preserving front propagation. J. Signal Process. Syst. 55(1–3), 209–216 (2009)CrossRefGoogle Scholar
  112. 112.
    Lorensen, W.E., Cline, H.E.: Marching cubes: a high resolution 3D surface construction algorithm. In: Special Interest Group on Computer GRAPHics and Interactive Techniques, 21(3), pp. 163–169. ACM (1987)Google Scholar
  113. 113.
    Lunscher, W.H.H.J., Beddoes, M.P.: Fast binary-image boundary extraction. Comput. Vis. Graph. Image Process. 38((3), 229–257 (1987)CrossRefGoogle Scholar
  114. 114.
    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
  115. 115.
    Marchadier, J., Arquès, D., Michelin, S.: Thinning grayscale well-composed images. Pattern Recogn. Lett. 25(5), 581–590 (2004)MATHCrossRefGoogle Scholar
  116. 116.
    Mazo, L.: Déformations homotopiques dans les images digitales n-aires. PhD thesis, Université de Strasbourg (2011)Google Scholar
  117. 117.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Digital imaging: a unified topological framework. J. Math. Imaging Vis. 44(1), 19–37 (2012)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    Mazo, L., Passat, N., Couprie, M., Ronse, C.: Topology on digital label images. J. Math. Imaging Vis. 44(3), 254–281 (2012)MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    Meinhardt-Llopis, E.: Morphological and statistical techniques for the analysis of 3D images. PhD thesis, Universitat Pompeu Fabra, Spain (2011)Google Scholar
  120. 120.
    Meyer, F.: Skeletons and perceptual graphs. Signal Process. 16(4), 335–363 (1989)MathSciNetCrossRefGoogle Scholar
  121. 121.
    Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics, vol. 3, pp. 35–57. Springer (2003)Google Scholar
  122. 122.
    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, pp. 67–75. Springer (2008)Google Scholar
  123. 123.
    Monasse, P., Guichard, F.: Fast computation of a contrast-invariant image representation. IEEE Trans. Image Process. 9(5), 860–872 (2000)CrossRefGoogle Scholar
  124. 124.
    Mylopoulos, J.P., Pavlidis, T.: On the topological properties of quantized spaces, I. The notion of dimension. J. ACM 18(2), 239–246 (1971)MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Mylopoulos, J.P., Pavlidis, T.: On the topological properties of quantized spaces, II. Connectivity and order of connectivity. J. ACM 18(2), 247–254 (1971)MathSciNetMATHCrossRefGoogle Scholar
  126. 126.
    Najman, L., Couprie, M.: Watershed algorithms and contrast preservation. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 2886, pp. 62–71. Springer (2003)Google Scholar
  127. 127.
    Najman, L., Géraud, T.: Discrete set-valued continuity and interpolation. In: Hendriks, C.L. Luengo, Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 11th International Symposium on Mathematical Morphology (ISMM), volume 7883 of Lecture Notes in Computer Science, pp. 37–48, Heidelberg. Springer (2013)Google Scholar
  128. 128.
    Najman, L., Romon, P. (eds.) Discrete curvature: theory and applications, volume 3 of Actes des rencontres du CIRM, France. CEDRAM (2014)Google Scholar
  129. 129.
    Najman, L., Romon, P. (eds.) Modern Approaches to Discrete Curvature, volume 2184 of Lecture Notes in Mathematics. Springer International Publishing (2017)Google Scholar
  130. 130.
    Najman, L., Schmitt, M.: Watershed of a continuous function. Signal Process. 38(1), 99–112 (1994)CrossRefGoogle Scholar
  131. 131.
    Najman, L., Talbot, H.: Mathematical Morphology. Wiley, New York (2013)MATHCrossRefGoogle Scholar
  132. 132.
    Nakahara, M.: Geometry, Topology and Physics. CRC Press, Boca Raton (2003)MATHGoogle Scholar
  133. 133.
    Newman, M.H.A.: Elements of the Topology of Plane Sets of Points. Cambridge University Press, Cambridge (1939)MATHGoogle Scholar
  134. 134.
    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)MATHCrossRefGoogle Scholar
  135. 135.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Sufficient conditions for topological invariance of 2D images under rigid transformations. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 7749, pp. 155–168. Springer (2013)Google Scholar
  136. 136.
    Ngo, P., Kenmochi, Y., Passat, N., Talbot, H.: Topology-preserving conditions for 2d digital images under rigid transformations. J. Math. Imaging Vis. 49(2), 418–433 (2014)MathSciNetMATHCrossRefGoogle Scholar
  137. 137.
    Ngo, P., Passat, N., Kenmochi, Y., Talbot, H.: Well-composed images and rigid transformations. In: IEEE International Conference on Image Processing, pp. 3035–3039 (2013)Google Scholar
  138. 138.
    Ngo, P., Passat, N., Kenmochi, Y., Talbot, H.: Topology-preserving rigid transformation of 2D digital images. IEEE Trans. Image Process. 23(2), 885–897 (2014)MathSciNetMATHCrossRefGoogle Scholar
  139. 139.
    Papert, S., Minsky, M.: Perceptrons: An Introduction to Computational Geometry. The MIT Press, Cambridge (1969)MATHGoogle Scholar
  140. 140.
    Pavlidis, T.: Algorithms for Graphics and Image Processing. Springer, Berlin (2012)MATHGoogle Scholar
  141. 141.
    Pham, D.L., Bazin, P.-L., Prince, J.L.: Digital topology in brain imaging. Signal Process. Mag. 27(4), 51–59 (2010)CrossRefGoogle Scholar
  142. 142.
    Poupon, F., Mangin, J., Hasboun, D., Poupon, C., Magnin, I., Frouin, V.: Multi-object deformable templates dedicated to the segmentation of brain deep structures. Medical Image Computing and Computer-Assisted Intervention, MICCAI ’98, pp. 1134–1143 (1998)Google Scholar
  143. 143.
    Roerdink, J.B.T.M., Meijster, A.: The watershed transform: definitions, algorithms and parallelization strategies. Fundam. Inf. 41(1–2), 187–228 (2000)MathSciNetMATHGoogle Scholar
  144. 144.
    Ronse, C.: Flat morphological operators on arbitrary power lattices. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. Lecture Notes in Computer Science, vol. 2616, pp. 1–21. Springer (2003)Google Scholar
  145. 145.
    Rosenfeld, A.: Connectivity in digital pictures. J. ACM 17(1), 146–160 (1970)MathSciNetMATHCrossRefGoogle Scholar
  146. 146.
    Rosenfeld, A.: Arcs and curves in digital pictures. J. ACM 20(1), 81–87 (1973)MathSciNetMATHCrossRefGoogle Scholar
  147. 147.
    Rosenfeld, A.: Adjacency in digital pictures. Inf. Control 26(1), 24–33 (1974)MathSciNetMATHCrossRefGoogle Scholar
  148. 148.
    Rosenfeld, A.: Digital topology. Am. Math. Mon. 86(8), 621–630 (1979)MathSciNetMATHCrossRefGoogle Scholar
  149. 149.
    Rosenfeld, A.: Fuzzy digital topology. Inf. Control 40(1), 76–87 (1979)MathSciNetMATHCrossRefGoogle Scholar
  150. 150.
    Rosenfeld, A.: Picture Languages-Formal Model of Picture Recognition. Academic Press, New York (1979)MATHGoogle Scholar
  151. 151.
    Rosenfeld, A.: On connectivity properties of grayscale pictures. Pattern Recogn. 16(1), 47–50 (1983)CrossRefGoogle Scholar
  152. 152.
    Rosenfeld, A., Kong, T.Y., Nakamura, A.: Topology-preserving deformations of two-valued digital pictures. Graph. Models Image Process. 60(1), 24–34 (1998)CrossRefGoogle Scholar
  153. 153.
    Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. J. ACM 13(4), 471–494 (1966)MATHCrossRefGoogle Scholar
  154. 154.
    Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Proceedings of the fifth Eurographics symposium on Geometry processing, pp. 225–233. Eurographics Association (2007)Google Scholar
  155. 155.
    Saha, P.K., Strand, R., Borgefors, G.: Digital topology and geometry in medical imaging: a survey. IEEE Trans. Med. Imaging 34(9), 1940–1964 (2015)CrossRefGoogle Scholar
  156. 156.
    Salembier, P., Serra, J.: Flat zones filtering, connected operators, and filters by reconstruction. IEEE Trans. Image Process. 4(8), 1153–1160 (1995)CrossRefGoogle Scholar
  157. 157.
    Ségonne, F.: Active contours under topology control—genus preserving level sets. Int. J. Comput. Vis. 79(2), 107–117 (2008)CrossRefGoogle Scholar
  158. 158.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, Cambridge (1982)MATHGoogle Scholar
  159. 159.
    Serra, J., Kiran, B.R.: Digitization of partitions and tessellations. In: Normand, N., Guédon, J., Autrusseau, F. (eds.) Discrete Geometry for Computer Imagery Lecture Notes in Computer Science, vol. 9647, pp. 323–334. Springer (2016)Google Scholar
  160. 160.
    Shekhar, R., Fayyad, E., Yagel, R., Cornhill, J. F.: Octree-based decimation of marching cubes surfaces. In: Visualization’96, pp. 335–342. IEEE (1996)Google Scholar
  161. 161.
    Shi, Y., Lai, R., Gill, R., Pelletier, D., Mohr, D., Sicotte, N., Toga, A.W.: Conformal metric optimization on surface (CMOS) for deformation and mapping in Laplace-Beltrami embedding space. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 327–334. Springer (2011)Google Scholar
  162. 162.
    Siqueira, M., Latecki, L.J., Gallier, J.: Making 3D binary digital images well-composed. In: Electronic Imaging 2005, pp. 150–163. International Society for Optics and Photonics (2005)Google Scholar
  163. 163.
    Siqueira, M., Latecki, L.J., Tustison, N., Gallier, J., Gee, J.: Topological repairing of 3D digital images. J. Math. Imaging Vis. 30(3), 249–274 (2008)MathSciNetCrossRefGoogle Scholar
  164. 164.
    Snidaro, L., Foresti, G.L.: Real-time thresholding with Euler numbers. Pattern Recogn. Lett. 24(9), 1533–1544 (2003)MATHCrossRefGoogle Scholar
  165. 165.
    Soille, P.: Morphological Image Analysis: Principles and Applications. Springer, Berlin (2013)MATHGoogle Scholar
  166. 166.
    Soille, P., Pesaresi, M.: Advances in mathematical morphology applied to geoscience and remote sensing. IEEE Trans. Geosci. Remote Sens. 40(9), 2042–2055 (2002)CrossRefGoogle Scholar
  167. 167.
    Sossa-Azuela, J.H., Santiago-Montero, R., Pérez-Cisneros, M., Rubio-Espino, E.: Computing the Euler number of a binary image based on a vertex codification. J. Appl. Res. Technol. 11(3), 360–370 (2013)CrossRefGoogle Scholar
  168. 168.
    Stelldinger, P.: Image Digitization and Its Influence on Shape Properties in Finite Dimensions, vol. 312. IOS Press, Amsterdam (2008)MATHGoogle Scholar
  169. 169.
    Stelldinger, P., Köthe, U.: Towards a general sampling theory for shape preservation. Image Vis. Comput. 23(2), 237–248 (2005)MATHCrossRefGoogle Scholar
  170. 170.
    Stelldinger, P., Latecki, L.J.: 3D object digitization: majority interpolation and marching cubes. In: IEEE International Conference on Pattern Recognition, vol. 2, pp. 1173–1176. IEEE (2006)Google Scholar
  171. 171.
    Stelldinger, P., Latecki, L.J., Siqueira, M.: Topological equivalence between a 3D object and the reconstruction of its digital image. IEEE Trans. Pattern Anal. Mach. Intell. 29(1), 126–140 (2007)CrossRefGoogle Scholar
  172. 172.
    Stelldinger, P., Strand, R.: Topology preserving digitization with FCC and BCC grids. In: International Workshop on Combinatorial Image Analysis, pp. 226–240. Springer (2006)Google Scholar
  173. 173.
    Stout, L.N.: Two discrete forms of the Jordan curve theorem. Am. Math. Mon. 95(4), 332–336 (1988)MathSciNetMATHCrossRefGoogle Scholar
  174. 174.
    Sundaramoorthi, G., Yezzi, A.: Global regularizing flows with topology preservation for active contours and polygons. IEEE Trans. Image Process. 16(3), 803–812 (2007)MathSciNetCrossRefGoogle Scholar
  175. 175.
    Tajine, M., Ronse, C.: Topological properties of Hausdorff discretizations. In: Goutsias, J., Vincent, L., Bloomberg, D.S. (eds.) Mathematical Morphology and Its Applications to Image and Signal Processing. Computational Imaging and Vision, vol. 18, pp. 41–50. Springer (2002)Google Scholar
  176. 176.
    Toriwaki, J., Yoshida, H.: Fundamentals of Three-Dimensional Digital Image Processing. Springer, Berlin (2009)MATHCrossRefGoogle Scholar
  177. 177.
    Tu, W.-C., He, S., Yang, Q., Chien, S.-Y.: Real-time salient object detection with a minimum spanning tree. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2334–2342 (2016)Google Scholar
  178. 178.
    Tustison, N.J., Avants, B.B., Siqueira, M., Gee, J.C.: Topological well-composedness and glamorous glue: a digital gluing algorithm for topologically constrained front propagation. IEEE Trans. Image Process. 20(6), 1756–1761 (2011)MathSciNetCrossRefGoogle Scholar
  179. 179.
    Tverberg, H.: A proof of the Jordan curve theorem. Bull. Lond. Math. Soc. 12(1), 34–38 (1980)MathSciNetMATHCrossRefGoogle Scholar
  180. 180.
    Van Vliet, L.J., Young, I.T., Beckers, G.L.: An edge detection model based on non-linear Laplace filtering. In: International Workshop on Pattern Recognition and Artificial Intelligence, towards an Integration (1988)Google Scholar
  181. 181.
    Voss, K.: Images, objects, and surfaces in \(\mathbb{Z}^n\). Int. J. Pattern Recognit. Artif. Intell. 5(05), 797–808 (1991)CrossRefGoogle Scholar
  182. 182.
    Wang, T., Wu, D.J., Coates, A., Ng, A.Y.: End-to-end text recognition with convolutional neural networks. In: International Conference on Pattern Recognition, pp. 3304–3308. IEEE (2012)Google Scholar
  183. 183.
    Wang, Y., Bhattacharya, P.: Digital connectivity and extended well-composed sets for gray images. Comput. Vis. Image Underst. 68(3), 330–345 (1997)CrossRefGoogle Scholar
  184. 184.
    Wilson, P.R.: Euler formulas and geometric modeling. IEEE Comput. Graph. Appl. 8(5), 24–36 (1985)CrossRefGoogle Scholar
  185. 185.
    Xu, Y., Carlinet, E., Géraud, T., Najman, L.: Efficient computation of attributes and saliency maps on tree-based image representations. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 12th International Symposium on Mathematical Morphology (ISMM), volume 9082 of Lecture Notes in Computer Science, pp. 693–704, Reykjavik, Iceland. Springer (2015)Google Scholar
  186. 186.
    Yongchao, X., Carlinet, E., Géraud, T., Najman, L.: Hierarchical segmentation using tree-based shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 39(3), 457–469 (2017)CrossRefGoogle Scholar
  187. 187.
    Xu, Y., Géraud, T., Najman, L.: Context-based energy estimator: application to object segmentation on the tree of shapes. In: Proceedings of the 19th IEEE International Conference on Image Processing (ICIP), pp. 1577–1580, Orlando, Florida, USA (2012)Google Scholar
  188. 188.
    Xu, Y., Géraud, T., Najman, L.: Morphological filtering in shape spaces: applications using tree-based image representations. In: Proceedings of the 21st International Conference on Pattern Recognition (ICPR), pp. 485–488, Tsukuba Science City, Japan. IEEE Computer Society (2012)Google Scholar
  189. 189.
    Xu, Y., Géraud, T., Najman, L.: Two applications of shape-based morphology: blood vessels segmentation and a generalization of constrained connectivity. In: Hendriks, C.L. Luengo., Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing—Proceedings of the 11th International Symposium on Mathematical Morphology (ISMM), volume 7883 of Lecture Notes in Computer Science, pp. 390–401, Heidelberg. Springer (2013)Google Scholar
  190. 190.
    Yongchao, X., Géraud, T., Najman, L.: Connected filtering on tree-based shape-spaces. IEEE Trans. Pattern Anal. Mach. Intell. 38(6), 1126–1140 (2016)CrossRefGoogle Scholar
  191. 191.
    Yongchao, X., Monasse, P., Géraud, T., Najman, L.: Tree-based morse regions: a topological approach to local feature detection. IEEE Trans. Image Process. 23(12), 5612–5625 (2014)MathSciNetCrossRefGoogle Scholar
  192. 192.
    Yang, H.S., Sengupta, S.: Intelligent shape recognition for complex industrial tasks. IEEE Control Syst. Mag. 8(3), 23–30 (1988)CrossRefGoogle Scholar
  193. 193.
    Ye, Q., Doermann, D.: Text detection and recognition in imagery: A survey. IEEE Trans. Pattern Anal. Mach. Intell. 37(7), 1480–1500 (2015)CrossRefGoogle Scholar
  194. 194.
    Yokoi, S., Toriwaki, J.-I., Fukumura, T.: An analysis of topological properties of digitized binary pictures using local features. Comput. Graph. Image Process. 4(1), 63–73 (1975)MathSciNetCrossRefGoogle Scholar
  195. 195.
    Zhang, J., Sclaroff, S., Lin, Z., Shen, X., Price, B., Mech, R.: Minimum barrier salient object detection at 80 fps. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 1404–1412 (2015)Google Scholar
  196. 196.
    Zhu, Y., Yao, C., Bai, X.: Scene text detection and recognition: recent advances and future trends. Front. Comput. Sci. 10(1), 19–36 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.EPITA Research and Development Laboratory (LRDE)Le Kremlin-BicêtreFrance
  2. 2.Université Paris-Est, LIGM, Équipe A3SI, ESIEEMarne-la-ValléeFrance

Personalised recommendations