Advertisement

Digital Analytical Geometry: How Do I Define a Digital Analytical Object?

  • Eric Andres
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9448)

Abstract

This paper is meant as a short survey on analytically defined digital geometric objects. We will start by giving some elements on digitizations and their relations to continuous geometry. We will then explain how, from simple assumptions about properties a digital object should have, one can build mathematically sound digital objects. We will end with open problems and challenges for the future.

Keywords

Digital analytical geometry Digital objects 

References

  1. 1.
    Andres, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. GMIP 59(5), 302–309 (1997)Google Scholar
  2. 2.
    Andres, E., Jacob, M.A.: The discrete analytical hyperspheres. IEEE Trans. Vis. Comp. Graphics 3(1), 75–86 (1997)CrossRefGoogle Scholar
  3. 3.
    Andres, E., Nehlig, P., Françon, J.: Supercover of straight lines, planes and triangles. In: Ahronovitz, E., Fiorio, C. (eds.) DGCI 1997. LNCS, vol. 1347. Springer, Heidelberg (1997) CrossRefGoogle Scholar
  4. 4.
    Andres, E., Nehlig, P., Francon, J.: Tunnel-free supercover 3D polygons and polyhedra. In: Eurographics 1997. Computer Graphics Forum, vol. 16, pp. C3–C13 (1997)Google Scholar
  5. 5.
    Andrès, É.: Defining discrete objects for polygonalization: the standard model. In: Braquelaire, A., Lachaud, J.-O., Vialard, A. (eds.) DGCI 2002. LNCS, vol. 2301, pp. 313–325. Springer, Heidelberg (2002) CrossRefGoogle Scholar
  6. 6.
    Andres, E.: Discrete linear objects in dimension n: the standard model. Graph. Models 65(1–3), 92–111 (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    Andres, E.: The supercover of an m-flat is a discrete analytical object. Theor. Comput. Sci. 406(1–2), 8–14 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Andres, E., Largeteau-Skapin, G., Rodríguez, M.: Generalized perpendicular bisector and exhaustive discrete circle recognition. Graph. Models 73(6), 354–364 (2011)CrossRefGoogle Scholar
  9. 9.
    Andres, E., Roussillon, T.: Analytical description of digital circles. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 235–246. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  10. 10.
    Berthé, V., Jamet, D., Jolivet, T., Provençal, X.: Critical connectedness of thin arithmetical discrete planes. In: Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B. (eds.) DGCI 2013. LNCS, vol. 7749, pp. 107–118. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Berthé, V., Labbé, S.: An arithmetic and combinatorial approach to three-dimensional discrete lines. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 47–58. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  12. 12.
    Bresenham, J.: Algorithm for computer control of a digital plotter. IBM Syst. J. 4(1), 25–30 (1965)CrossRefGoogle Scholar
  13. 13.
    Bresenham, J.: A linear algorithm for incremental digital display of circular arcs. Commun. ACM 20(2), 100–106 (1977)zbMATHCrossRefGoogle Scholar
  14. 14.
    Brimkov, V.E., Andres, E., Barneva, R.P.: Object discretizations in higher dimensions. Pattern Recogn. Lett. 23(6), 623–636 (2002)zbMATHCrossRefGoogle Scholar
  15. 15.
    Brimkov, V.E., Barneva, R.P.: Graceful planes and thin tunnel-free meshes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 53–64. Springer, Heidelberg (1999) CrossRefGoogle Scholar
  16. 16.
    Brimkov, V.E., Barneva, R.P.: Graceful planes and lines. Theor. Comput. Sci. 283(1), 151–170 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Brimkov, V.E., Barneva, R.P.: Connectivity of discrete planes. Theor. Comput. Sci. 319(1–3), 203–227 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Brimkov, V.E., Coeurjolly, D., Klette, R.: Digital planarity - a review. Discrete Appl. Math. 155(4), 468–495 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Brons, R.: Linguistic methods for the description of a straight line on a grid. CGIP 3(1), 48–62 (1974)MathSciNetGoogle Scholar
  20. 20.
    Chassery, J.M., Montanvert, A.: Géométrie discrète en imagerie. Ed. Hermès, Paris (1987)Google Scholar
  21. 21.
    Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Insight in discrete geometry and computational content of a discrete model of the continuum. Pattern Recogn. 42(10), 2220–2228 (2009)zbMATHCrossRefGoogle Scholar
  22. 22.
    Jordan, C.: Remarques sur les intégrales définies. Journal de Mathématiques, 4ème série, T.8, pp. 69–99 (1892)Google Scholar
  23. 23.
    Coeurjolly, D., Blot, V., Jacob-Da Col, M.-A.: Quasi-Affine transformation in 3-D: theory and algorithms. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 68–81. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  24. 24.
    Cohen-Or, D., Kaufman, A.E.: Fundamentals of surface voxelization. CVGIP 57(6), 453–461 (1995)Google Scholar
  25. 25.
    Coven, E.M., Hedlund, G.: Sequences with minimal block growth. Math. Syst. Theory 7(2), 138–153 (1973)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Dachille, F., Kaufman, A.E.: Incremental triangle voxelization. In: Proceeding Graphics Interface, pp. 205–212. Canadian Human-Computer Communications Society, Montréal (2000)Google Scholar
  27. 27.
    Debled-Renesson, I., Reveillès, J.P.: A new approach to digital planes. In: SPIE Vision Geometry III, vol. 2356, Boston (1994)Google Scholar
  28. 28.
    Debled-Rennesson, I.: Etude et reconnaissance des droites et plans discrets, PhD Thesis. Ph.D. thesis, Université Louis Pasteur, Strasbourg, France (1995)Google Scholar
  29. 29.
    Debled-Rennesson, I., Remy, J., Rouyer-Degli, J.: Segmentation of discrete curves into fuzzy segments. Elect. Notes Discrete Math. 12, 372–383 (2003)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Feschet, F., Reveillès, J.-P.: A generic approach for n-dimensional digital lines. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 29–40. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  31. 31.
    Figueiredo, O., Reveillès, J.: A contribution to 3D digital lines. In: 5th DGCI, pp. 187–198, Clermont-Ferrand (1995)Google Scholar
  32. 32.
    Fiorio, C., Jamet, D., Toutant, J.L.: Discrete circles: an arithmetical approach with non-constant thickness. In: Proceeding SPIE Vision Geometry XIV, vol. 6066, pp. 1–12 (2006)Google Scholar
  33. 33.
    Francon, J.: Arithmetic planes and combinatorial manifolds. In: 5th DGCI, pp. 209–217, Clermont-Ferrand (1995)Google Scholar
  34. 34.
    Francon, J.: Discrete combinatorial surfaces. CVGIP 57(1), 20–26 (1995)Google Scholar
  35. 35.
    Francon, J.: Sur la topologie d’un plan arithmétique. Theor. Comput. Sci. 156(1&2), 159–176 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Gérard, Y., Provot, L., Feschet, F.: Introduction to digital level layers. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 83–94. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  37. 37.
    Heijmans, H.J.A.M.: Morphological image operators. Academy Press, Boston (1994)zbMATHGoogle Scholar
  38. 38.
    Herman, G.T.: Discrete multidimensional jordan surfaces. CVGIP 54(6), 507–515 (1992)MathSciNetGoogle Scholar
  39. 39.
    Jamet, D., Toutant, J.: Minimal arithmetic thickness connecting discrete planes. Discrete Appl. Math. 157(3), 500–509 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Kaufman, A.E.: Efficient algorithms for 3D scan-conversion of parametric curves, surfaces, and volumes. In: Proceeding 14th SIGGRAPH, pp. 171–179 (1987)Google Scholar
  41. 41.
    Kaufman, A.E.: Efficient algorithms for scan-converting 3D polygons. Comput. Graph. 12(2), 213–219 (1988)CrossRefGoogle Scholar
  42. 42.
    Kim, C.E.: Three-dimensional digital line segments. IEEE Trans. PAMI 5(2), 231–234 (1983)zbMATHCrossRefGoogle Scholar
  43. 43.
    Klette, R., Rosenfeld, A.: Digital straightness - a review. Discrete Appl. Math. 139(1–3), 197–230 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. CVGIP 48(3), 357–393 (1989)Google Scholar
  45. 45.
    Kovalesky, V.: Finite topology and image analysis. Adv. Electron. Electron Phys. 84, 197–259 (1992)CrossRefGoogle Scholar
  46. 46.
    Lincke, C., Wüthrich, C.A.: Surface digitizations by dilations which are tunnel-free. Discrete Appl. Math. 125(1), 81–91 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    McIlroy, M.D.: Best approximate circles on integer grids. ACM Trans. Graph. 2(4), 237–263 (1983)zbMATHCrossRefGoogle Scholar
  48. 48.
    McIlroy, M.D.: Getting raster ellipses right. ACM Trans. Graph. 11(3), 259–275 (1992)zbMATHCrossRefGoogle Scholar
  49. 49.
    Montanari, U.: On limit properties in digitization schemes. J. ACM 17(2), 348–360 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Mora, F., Ruillet, G., Andres, E., Vauzelle, R.: Pedagogic discrete visualization of electromagnetic waves. In: Eurographics 2003, Interactive Demos and Posters, pp. 123–126 (2003)Google Scholar
  51. 51.
    Morgenthaler, D.G., Rosenfeld, A.: Surfaces in three-dimensional digital images. Inf. Control 51(3), 227–247 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Reveillès, J.P.: Calcul en Nombres Entiers et Algorithmique. Ph.D. thesis, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  53. 53.
    Reveillès, J., Richard, D.: Back and forth between continuous and discrete for the working computer scientist. Ann. Math. Artif. Intell. 16, 89–152 (1996)zbMATHCrossRefGoogle Scholar
  54. 54.
    Ronse, C., Tajine, M.: Hausdorff discretization for cellular distances and its relation to cover and supercover discretizations. J. Vis. Commun. Image Represent. 12(2), 169–200 (2001)CrossRefGoogle Scholar
  55. 55.
    Rosenfeld, A.: Digital topology. Amer. Math. Monthly 86, 621–630 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Rosenfeld, A., Kong, T.Y., Wu, A.Y.: Digital surfaces. GMIP 53(4), 305–312 (1991)zbMATHGoogle Scholar
  57. 57.
    Sankar, P.: Grid intersect quantization schemes for solid object digitization. Comput. Graphics Image Process. 8(1), 25–42 (1978)CrossRefGoogle Scholar
  58. 58.
    Sekiya, F., Sugimoto, A.: On connectivity of discretized 2D explicit curve. In: Mathematical Progress in Expressive Image Synthesis, Symposium MEIS 2014, pp. 16–25, Japan (2014)Google Scholar
  59. 59.
    Stelldinger, P., Terzic, K.: Digitization of non-regular shapes in arbitrary dimensions. Image Vision Comput. 26(10), 1338–1346 (2008)CrossRefGoogle Scholar
  60. 60.
    Tajine, M., Ronse, C.: Topological properties of hausdorff discretization, and comparison to other discretization schemes. Theor. Comput. Sci. 283(1), 243–268 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    Taubin, G.: Rasterizing algebraic curves and surfaces. IEEE Comput. Graphics 14(2), 14–22 (1994)CrossRefGoogle Scholar
  62. 62.
    Toutant, J.-L., Andres, E., Largeteau-Skapin, G., Zrour, R.: Implicit digital surfaces in arbitrary dimensions. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) DGCI 2014. LNCS, vol. 8668, pp. 332–343. Springer, Heidelberg (2014) Google Scholar
  63. 63.
    Toutant, J., Andres, E., Roussillon, T.: Digital circles, spheres and hyperspheres: from morphological models to analytical characterizations and topological properties. Discrete Appl. Math. 161(16–17), 2662–2677 (2013)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Laboratoire XLIM, SIC, UMR CNRS 7252Université de PoitiersFuturoscope ChasseneuilFrance

Personalised recommendations