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Foundations of Image Understanding pp 33–71Cite as

Digital Geometry — The Birth of a New Discipline

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

Abstract

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.

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References

  1. P. S. Aleksandrov. Combinatorial Topology, volume 2. Graylock Press, Rochester, 1957.

    MATH  Google Scholar 

  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. T. A. Anderson and C. E. Kim. Representation of digital line segments and their preimages. Computer Vision, Graphics, Image Processing, 30:279–288, 1985.

    Article  MATH  Google Scholar 

  4. E. Andres. Cercles discrèts et rotations discrètes. Thèse de Doctorat, Centre de Recherche en Informatique, Université Louis Pasteur, Strasbourg, 1992.

    Google Scholar 

  5. E. Andres. Discrete circles, rings and spheres. Computers & Graphics, 18:695–706, 1994.

    Article  Google Scholar 

  6. E. Andres, R. Acharya, and C. Sibata. Discrete analytical hyper-planes. Graphical Models Image Processing, 59:302–309, 1997.

    Article  Google Scholar 

  7. C. Arcelli and A. Massarotti. Regular arcs in digital contours. Computer Graphics Image Processing, 4:339–360, 1975.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  9. P. Bhattacharya and A. Rosenfeld, a-convexity. Pattern Recognition Letters, 21:955–957, 2000.

    Article  Google Scholar 

  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. L. M. Blumenthal. Theory and Applications of Distance Geometry. Clarendon Press, Oxford, 1953.

    MATH  Google Scholar 

  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. G. Borgefors. Distance transformations in digital images. Computer Vision, Graphics, Image Processing, 34:344–371, 1986.

    Article  Google Scholar 

  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.

    MATH  Google Scholar 

  15. J. Bresenham. An incremental algorithm for digital plotting. In ACM Natl Conf., 1963.

    Google Scholar 

  16. J. Bresenham. Algorithm for computer control of a digital plotter. IBM Systems J., 4:25–30, 1965.

    Article  Google Scholar 

  17. R. Brons. Linguistic methods for description of a straight line on a grid. Computer Graphics Image Processing, 2:48–62, 1974.

    Article  MathSciNet  Google Scholar 

  18. A. M. Bruckstein. Self-similarity properties of digitized straight lines. Contemporary Math., 119:1–20, 1991.

    Article  MathSciNet  Google Scholar 

  19. J. M. Chassery. Connectivity and consecutivity in digital pictures. Computer Graphics Image Processing, 9:294–300, 1979.

    Article  Google Scholar 

  20. J. M. Chassery. Discrete convexity: definition, parametrization and compatibility with continuous convexity. Computer Vision, Graphics, Image Processing, 21:326–344, 1983.

    Article  MATH  Google Scholar 

  21. J.-M. Chassery and A. Montanvert. Géométrie discrète en imagerie. Ed. Hermès, Paris, 1991.

    Google Scholar 

  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. 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.

    Article  Google Scholar 

  24. G. P. Dinneen. Programming pattern recognition. In Proc. Western Joint Computer Conf., pages 94–100, 1955.

    Google Scholar 

  25. L. Dorst and A. W. M. Smeulders. Discrete representations of straight lines. IEEE Trans. PAMI, 6:450–462, 1984.

    Article  MATH  Google Scholar 

  26. L. Dorst and A. W. M. Smeulders. Decomposition of discrete curves into piecewise segments in linear time. Contemporary Math., 119:169–195, 1991.

    Article  MathSciNet  Google Scholar 

  27. U. Eckhardt and G. Maderlechner. Invariant thinning. Intl. J. Pattern Recognition Artificial Intelligence, 7:1115–1144, 1993.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  29. L. Fejes Tóth. Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin, 1953.

    MATH  Google Scholar 

  30. J. Françon. Discrete combinatorial surfaces. Graphical Models Image Processing, 57:20–26, 1995.

    Article  Google Scholar 

  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. 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. 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. 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. 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. A. Grzegorczyk. An Outline of Mathematical Logic. D. Reidel, Dordrecht, 1974.

    Book  MATH  Google Scholar 

  37. H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperime-trie. Springer, Berlin, 1957.

    Book  Google Scholar 

  38. G. Herman. Geometry of Digital Spaces. Birkhäuser, Boston, 1998.

    MATH  Google Scholar 

  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. L. Hodes. Discrete approximation of continuous convex blobs. SIAMJ. Appl. Math., 19:477–485, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Hübler. Diskrete Geometrie für die digitale Bildverarbeitung. Habil.-Schrift, Friedrich-Schiller-Universität Jena, 1989.

    Google Scholar 

  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. 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.

    MATH  Google Scholar 

  44. S. H. Y. Hung. On the straightness of digital arcs. IEEE Trans. PAMI, 7:1264–1269, 1985.

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  46. M. N. Huxley. Exponential sums and lattice points. Proc. London Math. Soc., 60:471–502, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  47. A. Imiya and U. Eckhardt. The Euler characteristics of discrete objects and discrete quasi-objects. Computer Vision Image Understanding, 75:307–318, 1999.

    Article  Google Scholar 

  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. 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. A. Jonas and N. Kiryati. Digital representation schemes for 3-D curves. Pattern Recognition, 30:1803–1816, 1997.

    Article  Google Scholar 

  51. A. Jonas and N. Kiryati. Length estimation in 3-D using cube quantization. J. Mathematical Imaging and Vision, 8:215–238, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  52. C. Jordan. Remarques sur les intégrales définies. Journal de Mathématiques (4e série), 8:69–99, 1892.

    Google Scholar 

  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. E. Keppel. Approximation of complex surfaces by triangulation of contour lines. IBM J. Res. Develop., 19:2–11, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. S. Khuller, A. Rosenfeld, and A. Wu. Centers of sets of pixels. Discrete Applied Mathematics, 103:297–306, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  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. C. E. Kim and A. Rosenfeld. Convex digital solids. IEEE Trans. PAMI, 4:612–618, 1982.

    Article  MATH  Google Scholar 

  60. C. E. Kim and A. Rosenfeld. Digital straight lines and convexity of digital regions. IEEE Trans. PAMI, 4:149–153, 1982.

    Article  MATH  Google Scholar 

  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. R. Klette. The m-dimensional grid point space. Computer Vision, Graphics, Image Processing, 30:1–12, 1985.

    Article  MATH  Google Scholar 

  63. R. Klette. Cell complexes through time. In Proc. Vision Geometry IX, SPIE 4117, pages 134–145, 2000.

    Google Scholar 

  64. R. Klette and E. V. Krishnamurthy. Algorithms for testing convexity of digital polygons. Computer Graphics Image Processing, 16:177–184, 1981.

    Article  Google Scholar 

  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. 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. R. Klette and J. Zunic. Multigrid convergence of calculated features in image analysis. J. Mathematical Imaging Vision, 13:173–191, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  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. V. A. Kovalevsky. Discrete topology and contour definition. Pattern Recognition Letters, 2:281–288, 1984.

    Article  Google Scholar 

  70. V. A. Kovalevsky. Finite topology as applied to image analysis. Computer Vision, Graphics, Image Processing, 46:141–161, 1989.

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  72. Z. Kulpa. Area and perimeter measurements of blobs in discrete binary pictures. Computer Graphics Image Processing, 6:434–454, 1977.

    Article  MathSciNet  Google Scholar 

  73. E. Landau. Ausgewählte Abhandlungen zur Gitterpunktlehre. Deutscher Verlag der Wissenschaften, Berlin, 1962.

    MATH  Google Scholar 

  74. L. J. Latecki. Discrete Representation of Spatial Objects in Computer Vision. Kluwer, Dordrecht, 1998.

    Google Scholar 

  75. L. J. Latecki and A. Rosenfeld. Supportedness and tameness: Differentialless geometry of plane curves. Pattern Recognition, 31:607–622, 1998.

    Article  Google Scholar 

  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. J. Loeb. Communication theory of transmission of simple drawings. In W. Jackson, editor, Communication Theory, pages 317–327. Butterworths, London, 1953.

    Google Scholar 

  78. H. Minkowski. Geometrie der Zahlen. Teubner, Leipzig, 1910.

    MATH  Google Scholar 

  79. M. Minsky and S. Papert. Perceptrons—An Introduction to Computational Geometry. MIT Press, Cambridge, 1969.

    MATH  Google Scholar 

  80. G. U. Montanari. A note on minimal length polygonal approximation to a digitized contour. Comm. ACM, 13:41–47, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  81. D. G. Morgenthaler and A. Rosenfeld. Surfaces in three-dimensional images. Information and Control, 51:227–247, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  82. T. Pavlidis. Structural Pattern Recognition. Springer, New York, 1977.

    MATH  Google Scholar 

  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. O. Philbrick. A study of shape recognition using the medial axis transform. Air Force Cambridge Research Laboratories, Boston, 1966.

    Google Scholar 

  85. G. Pick. Geometrisches zur Zahlentheorie. Zeitschrift des Vereins ‘Lotos’, 1899.

    Google Scholar 

  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. K. Reidemeister. Topologie der Polyeder und kombinatorische Topologie der Komplexe. Geest & Portig, Leipzig, 1953.

    MATH  Google Scholar 

  88. J.-P. Reveillès. Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, 1991.

    MATH  Google Scholar 

  89. W. Rinow. Lehrbuch der Topologie. Deutscher Verlag der Wissenschaften, Berlin, 1975.

    MATH  Google Scholar 

  90. C. Ronse. A simple proof of Rosenfeld’s characterization of digital straight line segments. Pattern Recognition Letters, 3:323–326, 1985.

    Article  MATH  Google Scholar 

  91. C. Ronse. A strong chord property for 4-connected convex digital sets. Computer Vision, Graphics, Image Processing, 35:259–269, 1986.

    Article  MATH  Google Scholar 

  92. C. Ronse. A topological characterization of thinning. Theoretical Computer Science, 43:31–41, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  93. C. Ronse. Minimal test patterns for connectivity preservation in parallel thinning algorithms for binary digital images. Discrete Applied Mathematics, 21:67–79, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  94. C. Ronse. A bibliography on digital and computational convexity (1961–1988). IEEE Trans. PAMI, 11:181–190, 1989.

    Article  MATH  Google Scholar 

  95. C. Ronse and M. Tajine. Discretization in Hausdorff space. J. Mathematical Imaging Vision, 12:219–242, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  96. A. Rosenfeld. Connectivity in digital pictures. J. ACM, 17:146–160, 1970.

    Article  MathSciNet  MATH  Google Scholar 

  97. A. Rosenfeld. Arcs and curves in digital pictures. J. ACM, 20:81–87, 1973.

    Article  MathSciNet  MATH  Google Scholar 

  98. A. Rosenfeld. Digital straight line segments. IEEE Trans. Computers, 23:1264–1269, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  99. A. Rosenfeld. A note on perimeter and diameter in digital pictures. Information and Control, 24:384–388, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  100. A. Rosenfeld. Geodesies in digital pictures. Information and Control, 36:74–84, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  101. A. Rosenfeld. Picture Languages (chapter: Digital geometry). Academic Press, New York, 1979.

    MATH  Google Scholar 

  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. A. Rosenfeld and C. E. Kim. How a digital computer can tell whether a line is straight. American Mathematical Monthly, 89:230–235, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  104. A. Rosenfeld and R. Klette. Degree of adjacency or surrounded-ness. Pattern Recognition, 18:169–177, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  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. A. Rosenfeld and J. L. Pfaltz. Sequential operations in digital picture processing. J. ACM, 13:471–494, 1966.

    Article  MATH  Google Scholar 

  107. A. Rosenfeld and J. L. Pfaltz. Distance functions on digital pictures. Pattern Recognition, 1:33–61, 1968.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  109. T. Saito and J. Toriwaki. New algorithms for n-dimensional Euclidean distance transformation. Pattern Recognition, 27:1551–1565, 1994.

    Article  Google Scholar 

  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. P. V. Sankar. Grid intersect quantization schemes for solid object digitization. Computer Graphics Image Processing, 8:25–42, 1978.

    Article  Google Scholar 

  112. W. Scherrer. Ein Satz über Gitter und Volumen. Mathematische Annalen, 86:99–107, 1922.

    Article  MathSciNet  MATH  Google Scholar 

  113. J. Serra. Image Analysis and Mathematical Morphology. Academic Press, New York, 1982.

    MATH  Google Scholar 

  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. J. Sklansky. Recognition of convex blobs. Pattern Recognition, 2:3–10, 1970.

    Article  Google Scholar 

  116. J. Sklansky. Measuring concavity on a rectangular mosaic. IEEE Trans. Computers, 21:1355–1364, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. M. B. Smyth. Region-based discrete geometry. J. Universal Computer Science, 6:447–459, 2000.

    MathSciNet  MATH  Google Scholar 

  120. E. Steinitz. Beiträge zur Analysis. Sitzungsberichte der Berliner Mathematischen Gesellschaft, 7:29–49, 1908.

    Google Scholar 

  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. S. Thompson and A. Rosenfeld. Discrete, nonlinear curvature-dependent contour evolution. Pattern Recognition, 31:1949–1959, 1998.

    Article  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  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. A. W. Tucker. An abstract approach to manifolds. Annals of Math., 34:191–243, 1933.

    Article  Google Scholar 

  126. S. H. Unger. A computer oriented toward spatial problems. Proc. IRE, 46:1744–1750, 1958.

    Article  Google Scholar 

  127. P. Veelaert. On the flatness of digital hyperplanes. J. Mathematical Imaging Vision, 3:205–221, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  128. P. Veelaert. Digital planarity of rectangular surface segments, IEEE Trans. PAMI, 16:647–652, 1994.

    Article  Google Scholar 

  129. K. Voss. Digitalisierungseffekte in der automatischen Bildverarbeitung. EIK, 11:469–477, 1975.

    Google Scholar 

  130. K. Voss. Coding of digital straight lines by continued fractions. Computers and Artificial Intelligence, 10:75–80, 1991.

    MathSciNet  Google Scholar 

  131. K. Voss. Discrete Images, Objects, and Functions in Z n. Springer, Berlin, 1993.

    Book  MATH  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  135. A. Y. Wu and A. Rosenfeld. Geodesic visibility in graphs. Information Sciences, 108:5–12, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  136. L. D. Wu. On the chain code of a line. IEEE Trans. PAMI, 4:347–353, 1982.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  138. S. Yokoi, J. Toriwaki, and T. Fukumura. On generalized distance transformation of digitized pictures. IEEE Trans. PAMI, 3:424–443, 1981.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

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  1. University of Auckland, New Zealand

    Reinhard Klette

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  1. University of Maryland, USA

    Larry S. Davis

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Klette, R. (2001). Digital Geometry — The Birth of a New Discipline. In: Davis, L.S. (eds) Foundations of Image Understanding. The Springer International Series in Engineering and Computer Science, vol 628. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1529-6_2

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