Digital Geometry — The Birth of a New Discipline

  • Reinhard Klette
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 628)


Basic concepts of digital geometry are introduced, with emphasis on digitized Euclidean geometry of curves and surfaces. Topics covered include connectedness and distance transforms, integer metrics, digitization models, multigrid convergence, digital straight line segments and planar patches, and approximation of curves and surfaces.


Grid Point Cell Complex Medial Axis Euclidean Plane Chain Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. S. Aleksandrov. Combinatorial Topology, volume 2. Graylock Press, Rochester, 1957.zbMATHGoogle Scholar
  2. [2]
    P. Alexandroff and H. Hopf. Topologie—Erster Band. Die Grundlehren der mathemat. Wiss. in Einzeldarstellungen, Band XLV. Verlag von Julius Springer, Berlin, 1935.Google Scholar
  3. [3]
    T. A. Anderson and C. E. Kim. Representation of digital line segments and their preimages. Computer Vision, Graphics, Image Processing, 30:279–288, 1985.zbMATHCrossRefGoogle Scholar
  4. [4]
    E. Andres. Cercles discrèts et rotations discrètes. Thèse de Doctorat, Centre de Recherche en Informatique, Université Louis Pasteur, Strasbourg, 1992.Google Scholar
  5. [5]
    E. Andres. Discrete circles, rings and spheres. Computers & Graphics, 18:695–706, 1994.CrossRefGoogle Scholar
  6. [6]
    E. Andres, R. Acharya, and C. Sibata. Discrete analytical hyper-planes. Graphical Models Image Processing, 59:302–309, 1997.CrossRefGoogle Scholar
  7. [7]
    C. Arcelli and A. Massarotti. Regular arcs in digital contours. Computer Graphics Image Processing, 4:339–360, 1975.CrossRefGoogle Scholar
  8. [8]
    E. Artzy, G. Frieder, and G. T. Herman. The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. Computer Vision, Graphics, Image Processing, 15:1–24, 1981.CrossRefGoogle Scholar
  9. [9]
    P. Bhattacharya and A. Rosenfeld, a-convexity. Pattern Recognition Letters, 21:955–957, 2000.CrossRefGoogle Scholar
  10. [10]
    H. Blum. A transformation for extracting new descriptors of shape. In W. Wathen-Dunn, editor, Models for the Perception of Speech and Visual Form, pages 362–380. MIT Press, Cambridge, 1964.Google Scholar
  11. [11]
    L. M. Blumenthal. Theory and Applications of Distance Geometry. Clarendon Press, Oxford, 1953.zbMATHGoogle Scholar
  12. [12]
    J. Böhm and E. Hertel. Jenaer Beiträge zur diskreten Geometrie. In Wiss. Zeitschrift der Friedrich-Schiller-Universität Jena, Heft 1, pages 10–21. 1987.Google Scholar
  13. [13]
    G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, Image Processing, 34:344–371, 1986.CrossRefGoogle Scholar
  14. [14]
    G. Borgefors, I. Nyström, and G. Sanniti di Baja, editors. Discrete Geometry for Computer Imagery (9th Intl. Conf., Uppsala), LNCS 1953. Springer, Berlin, 2000.zbMATHGoogle Scholar
  15. [15]
    J. Bresenham. An incremental algorithm for digital plotting. In ACM Natl Conf., 1963.Google Scholar
  16. [16]
    J. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems J., 4:25–30, 1965.CrossRefGoogle Scholar
  17. [17]
    R. Brons. Linguistic methods for description of a straight line on a grid. Computer Graphics Image Processing, 2:48–62, 1974.MathSciNetCrossRefGoogle Scholar
  18. [18]
    A. M. Bruckstein. Self-similarity properties of digitized straight lines. Contemporary Math., 119:1–20, 1991.MathSciNetCrossRefGoogle Scholar
  19. [19]
    J. M. Chassery. Connectivity and consecutivity in digital pictures. Computer Graphics Image Processing, 9:294–300, 1979.CrossRefGoogle Scholar
  20. [20]
    J. M. Chassery. Discrete convexity: definition, parametrization and compatibility with continuous convexity. Computer Vision, Graphics, Image Processing, 21:326–344, 1983.zbMATHCrossRefGoogle Scholar
  21. [21]
    J.-M. Chassery and A. Montanvert. Géométrie discrète en imagerie. Ed. Hermès, Paris, 1991.Google Scholar
  22. [22]
    M. Chleík and F. Sloboda. Approximation of surfaces by minimal surfaces with obstacles. Technical report, Institute of Control Theory and Robotics, Slovak Academy of Sciences, Bratislava, 2000.Google Scholar
  23. [23]
    I. Debled-Rennesson and J.-P. Reveillés. A linear algorithm for segmentation of digital curves. Intl. J. Pattern Recognition Artificial Intelligence, 9:635–662, 1995.CrossRefGoogle Scholar
  24. [24]
    G. P. Dinneen. Programming pattern recognition. In Proc. Western Joint Computer Conf., pages 94–100, 1955.Google Scholar
  25. [25]
    L. Dorst and A. W. M. Smeulders. Discrete representations of straight lines. IEEE Trans. PAMI, 6:450–462, 1984.zbMATHCrossRefGoogle Scholar
  26. [26]
    L. Dorst and A. W. M. Smeulders. Decomposition of discrete curves into piecewise segments in linear time. Contemporary Math., 119:169–195, 1991.MathSciNetCrossRefGoogle Scholar
  27. [27]
    U. Eckhardt and G. Maderlechner. Invariant thinning. Intl. J. Pattern Recognition Artificial Intelligence, 7:1115–1144, 1993.CrossRefGoogle Scholar
  28. [28]
    O. D. Faugeras, M. Hebert, P. Mussi, and J. D. Boissonnat. Polyhedral approximation of 3-D objects without holes. Computer Vision, Graphics, Image Processing, 25:169–183, 1984.CrossRefGoogle Scholar
  29. [29]
    L. Fejes Tóth. Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin, 1953.zbMATHGoogle Scholar
  30. [30]
    J. Françon. Discrete combinatorial surfaces. Graphical Models Image Processing, 57:20–26, 1995.CrossRefGoogle Scholar
  31. [31]
    J. Françon and L. Papier. Polyhedrization of the boundary of a voxel object. In G. Bertrand, M. Couprie, and L. Perroton, editors, Proc. DGCI, LNCS 1568, pages 425–434. Springer, Berlin, 1999.Google Scholar
  32. [32]
    J. Françon, J.-M. Schramm, and M. Tajine. Recognizing arithmetic straight lines and planes. In Proc. DGCI, LNCS 1176, pages 141–150. Springer, Berlin, 1996.Google Scholar
  33. [33]
    H. Freeman. Techniques for the digital computer analysis of chain-encoded arbitrary plane curves. In Proc. Natl. Elect. Conf., volume 17, pages 421–432, 1961.Google Scholar
  34. [34]
    H. Freeman. A review of relevant problems in the processing of line-drawing data. In A. Grasselli, editor, Automatic Interpretation and Classification of Images, pages 155–174. Academic Press, New York, 1969.Google Scholar
  35. [35]
    H. Freeman. Boundary encoding and processing. In B. S. Lipkin and A. Rosenfeld, editors, Picture Processing and Psychopictorics, pages 241–263. Academic Press, New York, 1970.Google Scholar
  36. [36]
    A. Grzegorczyk. An Outline of Mathematical Logic. D. Reidel, Dordrecht, 1974.zbMATHCrossRefGoogle Scholar
  37. [37]
    H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperime-trie. Springer, Berlin, 1957.CrossRefGoogle Scholar
  38. [38]
    G. Herman. Geometry of Digital Spaces. Birkhäuser, Boston, 1998.zbMATHGoogle Scholar
  39. [39]
    C. J. Hilditch. Linear skeletons from square cupboards. In B. Meitzer and D. Michie, editors, Machine Intelligence 4, pages 403–420. Edinburgh University Press, 1969.Google Scholar
  40. [40]
    L. Hodes. Discrete approximation of continuous convex blobs. SIAMJ. Appl. Math., 19:477–485, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [41]
    A. Hübler. Diskrete Geometrie für die digitale Bildverarbeitung. Habil.-Schrift, Friedrich-Schiller-Universität Jena, 1989.Google Scholar
  42. [42]
    A. Hübler. Motions in the discrete plane. In A. Hübler, W. Nagel, B. D. Ripley, and G. Werner, editors, Proc. Geobild, pages 29–36. Akademie-verlag, Berlin, 1989.Google Scholar
  43. [43]
    A. Hübler, R. Klette, and K. Voss. Determination of the convex hull of a finite set of planar points within linear time. EIK, 17:121–140, 1981.zbMATHGoogle Scholar
  44. [44]
    S. H. Y. Hung. On the straightness of digital arcs. IEEE Trans. PAMI, 7:1264–1269, 1985.CrossRefGoogle Scholar
  45. [45]
    D. P. Huttenlocher and W. J. Rucklidge. A multi-resolution technique for comparing images using the Hausdorff distance. In Proc. IEEE Computer Vision and Pattern Recognition, pages 705–706, 1993.CrossRefGoogle Scholar
  46. [46]
    M. N. Huxley. Exponential sums and lattice points. Proc. London Math. Soc., 60:471–502, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    A. Imiya and U. Eckhardt. The Euler characteristics of discrete objects and discrete quasi-objects. Computer Vision Image Understanding, 75:307–318, 1999.CrossRefGoogle Scholar
  48. [48]
    M. Y. Jaisimha, R. M. Haralick, and D. Dori. Quantitative performance evaluation of thinning algorithms in the presence of noise. In C. Arcelli, L. P. Cordelia, and G. Sanniti di Baja, editors, Aspects of Visual Form Processing, pages 261–286. World Scientific, Singapore, 1994.Google Scholar
  49. [49]
    E. G. Johnston and A. Rosenfeld. Geometrical operations on digital pictures. In B. S. Lipkin and A. Rosenfeld, editors, Picture Processing and Psychopictorics, pages 217–240. Academic Press, New York, 1970.Google Scholar
  50. [50]
    A. Jonas and N. Kiryati. Digital representation schemes for 3-D curves. Pattern Recognition, 30:1803–1816, 1997.CrossRefGoogle Scholar
  51. [51]
    A. Jonas and N. Kiryati. Length estimation in 3-D using cube quantization. J. Mathematical Imaging and Vision, 8:215–238, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    C. Jordan. Remarques sur les intégrales définies. Journal de Mathématiques (4e série), 8:69–99, 1892.Google Scholar
  53. [53]
    Y. Kenmochi and R. Klette. Surface area estimation for digitized regular solids. In Proc. Vision Geometry IX, SPIE 4117, pages 100–111, 2000.Google Scholar
  54. [54]
    E. Keppel. Approximation of complex surfaces by triangulation of contour lines. IBM J. Res. Develop., 19:2–11, 1975.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [55]
    N. Keskes and O. Faugeras. Surface simple dans Z3. In Proc. 3ème Congrès Reconaissance des Formes et d’Intelligence Artificielle, pages 718–729, 1981.Google Scholar
  56. [56]
    E. Khalimsky. Motion, deformation, and homotopy in finite spaces. In Proc. IEEE Intl. Conf. on Systems, Man and Cybernetics, pages 227–234, 1987.Google Scholar
  57. [57]
    S. Khuller, A. Rosenfeld, and A. Wu. Centers of sets of pixels. Discrete Applied Mathematics, 103:297–306, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  58. [58]
    C. E. Kim and A. Rosenfeld. On the convexity of digital regions. In Proc. 5th Intl. Conf. on Pattern Recognition, pages 1010–1015, 1980.Google Scholar
  59. [59]
    C. E. Kim and A. Rosenfeld. Convex digital solids. IEEE Trans. PAMI, 4:612–618, 1982.zbMATHCrossRefGoogle Scholar
  60. [60]
    C. E. Kim and A. Rosenfeld. Digital straight lines and convexity of digital regions. IEEE Trans. PAMI, 4:149–153, 1982.zbMATHCrossRefGoogle Scholar
  61. [61]
    R. A. Kirsch, L. Cahn, L. C. Ray, and G. H. Urban. Experiments in processing pictorial information with a digital computer. In Proc. Eastern Joint Computer Conf., pages 221–229, 1957.Google Scholar
  62. [62]
    R. Klette. The m-dimensional grid point space. Computer Vision, Graphics, Image Processing, 30:1–12, 1985.zbMATHCrossRefGoogle Scholar
  63. [63]
    R. Klette. Cell complexes through time. In Proc. Vision Geometry IX, SPIE 4117, pages 134–145, 2000.Google Scholar
  64. [64]
    R. Klette and E. V. Krishnamurthy. Algorithms for testing convexity of digital polygons. Computer Graphics Image Processing, 16:177–184, 1981.CrossRefGoogle Scholar
  65. [65]
    R. Klette and B. Yip. The length of digital curves. Machine Graph-ics & Vision, 9:673–703, 2000. Extended version of: R. Klette, V. V. Kovalevsky, and B. Yip. Length estimation of digital curves. In Proc. Vision Geometry VIII, SPIE 3811, pages 117–129, 1999.Google Scholar
  66. [66]
    R. Klette and J. Zunic. Interactions between number theory and image analysis. In Proc. Vision Geometry IX, SPIE 4117, pages 210–221, 2000.Google Scholar
  67. [67]
    R. Klette and J. Zunic. Multigrid convergence of calculated features in image analysis. J. Mathematical Imaging Vision, 13:173–191, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    V. Kovalevsky and S. Fuchs. Theoretical and experimental analysis of the accuracy of perimeter estimates. In W. Förstner and S. Ruwiedel, editors, Robust Computer Vision, pages 218–242. Wichmann, Karlsruhe, 1992.Google Scholar
  69. [69]
    V. A. Kovalevsky. Discrete topology and contour definition. Pattern Recognition Letters, 2:281–288, 1984.CrossRefGoogle Scholar
  70. [70]
    V. A. Kovalevsky. Finite topology as applied to image analysis. Computer Vision, Graphics, Image Processing, 46:141–161, 1989.CrossRefGoogle Scholar
  71. [71]
    V. A. Kovalevsky. New definition and fast recognition of digital straight segments and arcs. In Proc. IEEE Intl. Conf. on Pattern Recognition, pages 31–34, 1990.CrossRefGoogle Scholar
  72. [72]
    Z. Kulpa. Area and perimeter measurements of blobs in discrete binary pictures. Computer Graphics Image Processing, 6:434–454, 1977.MathSciNetCrossRefGoogle Scholar
  73. [73]
    E. Landau. Ausgewählte Abhandlungen zur Gitterpunktlehre. Deutscher Verlag der Wissenschaften, Berlin, 1962.zbMATHGoogle Scholar
  74. [74]
    L. J. Latecki. Discrete Representation of Spatial Objects in Computer Vision. Kluwer, Dordrecht, 1998.Google Scholar
  75. [75]
    L. J. Latecki and A. Rosenfeld. Supportedness and tameness: Differentialless geometry of plane curves. Pattern Recognition, 31:607–622, 1998.CrossRefGoogle Scholar
  76. [76]
    J. B. Listing. Der Census räumlicher Complexe oder Verallgemeinerungen des Euler’schen Satzes von den Polyëdern. Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 10:97–182, 1861 and 1862.Google Scholar
  77. [77]
    J. Loeb. Communication theory of transmission of simple drawings. In W. Jackson, editor, Communication Theory, pages 317–327. Butterworths, London, 1953.Google Scholar
  78. [78]
    H. Minkowski. Geometrie der Zahlen. Teubner, Leipzig, 1910.zbMATHGoogle Scholar
  79. [79]
    M. Minsky and S. Papert. Perceptrons—An Introduction to Computational Geometry. MIT Press, Cambridge, 1969.zbMATHGoogle Scholar
  80. [80]
    G. U. Montanari. A note on minimal length polygonal approximation to a digitized contour. Comm. ACM, 13:41–47, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  81. [81]
    D. G. Morgenthaler and A. Rosenfeld. Surfaces in three-dimensional images. Information and Control, 51:227–247, 1981.MathSciNetzbMATHCrossRefGoogle Scholar
  82. [82]
    T. Pavlidis. Structural Pattern Recognition. Springer, New York, 1977.zbMATHGoogle Scholar
  83. [83]
    J. L. Pfaltz and A. Rosenfeld. Computer representation of planar regions by their skeletons. Comm. ACM, 10:119–122, 125, 1967.Google Scholar
  84. [84]
    O. Philbrick. A study of shape recognition using the medial axis transform. Air Force Cambridge Research Laboratories, Boston, 1966.Google Scholar
  85. [85]
    G. Pick. Geometrisches zur Zahlentheorie. Zeitschrift des Vereins ‘Lotos’, 1899.Google Scholar
  86. [86]
    H. Poincaré. Analysis situs. J. Ecole Polytech. (2), 1:1–121, 1895. Also: Œuvres, vol. 6, pages 193–288, Gauthier-Villars, 1953.Google Scholar
  87. [87]
    K. Reidemeister. Topologie der Polyeder und kombinatorische Topologie der Komplexe. Geest & Portig, Leipzig, 1953.zbMATHGoogle Scholar
  88. [88]
    J.-P. Reveillès. Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, 1991.zbMATHGoogle Scholar
  89. [89]
    W. Rinow. Lehrbuch der Topologie. Deutscher Verlag der Wissenschaften, Berlin, 1975.zbMATHGoogle Scholar
  90. [90]
    C. Ronse. A simple proof of Rosenfeld’s characterization of digital straight line segments. Pattern Recognition Letters, 3:323–326, 1985.zbMATHCrossRefGoogle Scholar
  91. [91]
    C. Ronse. A strong chord property for 4-connected convex digital sets. Computer Vision, Graphics, Image Processing, 35:259–269, 1986.zbMATHCrossRefGoogle Scholar
  92. [92]
    C. Ronse. A topological characterization of thinning. Theoretical Computer Science, 43:31–41, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  93. [93]
    C. Ronse. Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Applied Mathematics, 21:67–79, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  94. [94]
    C. Ronse. A bibliography on digital and computational convexity (1961–1988). IEEE Trans. PAMI, 11:181–190, 1989.zbMATHCrossRefGoogle Scholar
  95. [95]
    C. Ronse and M. Tajine. Discretization in Hausdorff space. J. Mathematical Imaging Vision, 12:219–242, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  96. [96]
    A. Rosenfeld. Connectivity in digital pictures. J. ACM, 17:146–160, 1970.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [97]
    A. Rosenfeld. Arcs and curves in digital pictures. J. ACM, 20:81–87, 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  98. [98]
    A. Rosenfeld. Digital straight line segments. IEEE Trans. Computers, 23:1264–1269, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  99. [99]
    A. Rosenfeld. A note on perimeter and diameter in digital pictures. Information and Control, 24:384–388, 1974.MathSciNetzbMATHCrossRefGoogle Scholar
  100. [100]
    A. Rosenfeld. Geodesies in digital pictures. Information and Control, 36:74–84, 1978.MathSciNetzbMATHCrossRefGoogle Scholar
  101. [101]
    A. Rosenfeld. Picture Languages (chapter: Digital geometry). Academic Press, New York, 1979.zbMATHGoogle Scholar
  102. [102]
    A. Rosenfeld. Digital geometry—Introduction and bibliography. In R. Klette, A. Rosenfeld, and F. Sloboda, editors, Advances in Digital and Computational Geometry, pages 1–85. Springer, Singapore, 1998.Google Scholar
  103. [103]
    A. Rosenfeld and C. E. Kim. How a digital computer can tell whether a line is straight. American Mathematical Monthly, 89:230–235, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  104. [104]
    A. Rosenfeld and R. Klette. Degree of adjacency or surrounded-ness. Pattern Recognition, 18:169–177, 1985.MathSciNetzbMATHCrossRefGoogle Scholar
  105. [105]
    A. Rosenfeld and R. A. Melter. Digital geometry. Technical Report CAR-TR-323, Center for Automation Research, University of Maryland, College Park, 1987.Google Scholar
  106. [106]
    A. Rosenfeld and J. L. Pfaltz. Sequential operations in digital picture processing. J. ACM, 13:471–494, 1966.zbMATHCrossRefGoogle Scholar
  107. [107]
    A. Rosenfeld and J. L. Pfaltz. Distance functions on digital pictures. Pattern Recognition, 1:33–61, 1968.MathSciNetCrossRefGoogle Scholar
  108. [108]
    J. Rothstein and C. Weiman. Parallel and sequential specification of a context sensitive language for straight line grids. Computer Graphics Image Processing, 5:106–124, 1976.CrossRefGoogle Scholar
  109. [109]
    T. Saito and J. Toriwaki. New algorithms for n-dimensional Euclidean distance transformation. Pattern Recognition, 27:1551–1565, 1994.CrossRefGoogle Scholar
  110. [110]
    T. Saito and J. Toriwaki. A sequential thinning algorithm for three dimensional digital pictures using the Euclidean distance transformation. In Proc. 9th Scandinavian Conf. on Image Analysis, pages 507–516, 1995.Google Scholar
  111. [111]
    P. V. Sankar. Grid intersect quantization schemes for solid object digitization. Computer Graphics Image Processing, 8:25–42, 1978.CrossRefGoogle Scholar
  112. [112]
    W. Scherrer. Ein Satz über Gitter und Volumen. Mathematische Annalen, 86:99–107, 1922.MathSciNetzbMATHCrossRefGoogle Scholar
  113. [113]
    J. Serra. Image Analysis and Mathematical Morphology. Academic Press, New York, 1982.zbMATHGoogle Scholar
  114. [114]
    H. Simon, K. Kunze, K. Voss, and W. R. Herrmann. Automatische Bildverarbeitung in Medizin und Biologie. Verlag Theodor Steinkopff, Dresden, 1975.Google Scholar
  115. [115]
    J. Sklansky. Recognition of convex blobs. Pattern Recognition, 2:3–10, 1970.CrossRefGoogle Scholar
  116. [116]
    J. Sklansky. Measuring concavity on a rectangular mosaic. IEEE Trans. Computers, 21:1355–1364, 1972.MathSciNetzbMATHCrossRefGoogle Scholar
  117. [117]
    F. Sloboda and B. Zaťko. On polyhedral form for surface representation. Technical report, Institute of Control Theory and Robotics, Slovak Academy of Sciences, Bratislava, 2000.Google Scholar
  118. [118]
    F. Sloboda, B. Zaťko, and J. Stoer. On approximation of planar one-dimensional continua. In R. Klette, A. Rosenfeld, and F. Sloboda, editors, Advances in Digital and Computational Geometry, pages 113–160. Springer, Singapore, 1998.Google Scholar
  119. [119]
    M. B. Smyth. Region-based discrete geometry. J. Universal Computer Science, 6:447–459, 2000.MathSciNetzbMATHGoogle Scholar
  120. [120]
    E. Steinitz. Beiträge zur Analysis. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 7:29–49, 1908.Google Scholar
  121. [121]
    I. E. Sutherland. Sketchpad: A man-machine graphical communication system. In Proc. SJCC, volume 23, pages 329–346. Spartan Books, 1963.Google Scholar
  122. [122]
    S. Thompson and A. Rosenfeld. Discrete, nonlinear curvature-dependent contour evolution. Pattern Recognition, 31:1949–1959, 1998.CrossRefGoogle Scholar
  123. [123]
    J. Toriwaki, N. Kato, and T. Fukumura. Parallel local operations for a new distance transformation of a line pattern and their applications. IEEE Trans. SMC, 9:628–643, 1979.MathSciNetzbMATHGoogle Scholar
  124. [124]
    J. Toriwaki, S. Yokoi, T. Yonekura, and T. Fukumura. Topological properties and topology-preserving transformation of a three-dimensional binary picture. In Proc. 6th ICPR, pages 414–419, 1982.Google Scholar
  125. [125]
    A. W. Tucker. An abstract approach to manifolds. Annals of Math., 34:191–243, 1933.CrossRefGoogle Scholar
  126. [126]
    S. H. Unger. A computer oriented toward spatial problems. Proc. IRE, 46:1744–1750, 1958.CrossRefGoogle Scholar
  127. [127]
    P. Veelaert. On the flatness of digital hyperplanes. J. Mathematical Imaging Vision, 3:205–221, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  128. [128]
    P. Veelaert. Digital planarity of rectangular surface segments, IEEE Trans. PAMI, 16:647–652, 1994.CrossRefGoogle Scholar
  129. [129]
    K. Voss. Digitalisierungseffekte in der automatischen Bildverarbeitung. EIK, 11:469–477, 1975.Google Scholar
  130. [130]
    K. Voss. Coding of digital straight lines by continued fractions. Computers and Artificial Intelligence, 10:75–80, 1991.MathSciNetGoogle Scholar
  131. [131]
    K. Voss. Discrete Images, Objects, and Functions in Z n. Springer, Berlin, 1993.zbMATHCrossRefGoogle Scholar
  132. [132]
    K. Voss and R. Klette. On the maximal number of edges of convex digital polygons included into a square. Computers and Artificial Intelligence, 1:549–558, 1982.Google Scholar
  133. [133]
    A. M. Vossepoel and A. W. M. Smeulders. Vector code probability and metrication error in the representation of straight lines of finite length. Computer Graphics Image Processing, 20:347–364, 1982.CrossRefGoogle Scholar
  134. [134]
    J. H. C. Whitehead. On the readability of homotopy groups. Annals of Math., 50:261–263, 1949. Also: The Mathematical Works of J. H. C. Whitehead, Volume 3, pages 221–223, Pergamon Press, Oxford, 1962.MathSciNetzbMATHCrossRefGoogle Scholar
  135. [135]
    A. Y. Wu and A. Rosenfeld. Geodesic visibility in graphs. Information Sciences, 108:5–12, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  136. [136]
    L. D. Wu. On the chain code of a line. IEEE Trans. PAMI, 4:347–353, 1982.zbMATHCrossRefGoogle Scholar
  137. [137]
    S. Yokoi, J. Toriwaki, and T. Fukumura. An analysis of topological properties of digitized binary pictures using local features. Computer Graphics Image Processing, 4:63–73, 1975.MathSciNetCrossRefGoogle Scholar
  138. [138]
    S. Yokoi, J. Toriwaki, and T. Fukumura. On generalized distance transformation of digitized pictures. IEEE Trans. PAMI, 3:424–443, 1981.zbMATHCrossRefGoogle Scholar
  139. [139]
    D. P. Young, R. G. Melvin, M. B. Bieterman, F. T. Johnson, and S. S. Samant. A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics. J. Computational Physics, 82:1–66, 1991.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Reinhard Klette
    • 1
  1. 1.University of AucklandNew Zealand

Personalised recommendations