The Community for Technology Leaders
RSS Icon
Issue No.04 - April (2008 vol.30)
pp: 577-590
In this paper we define and study digital manifolds of arbitrary dimension, and provide (in particular)a general theoretical basis for curve or surface tracing in picture analysis. The studies involve propertiessuch as one-dimensionality of digital curves and (n-1)-dimensionality of digital hypersurfaces thatmakes them discrete analogs of corresponding notions in continuous topology. The presented approachis fully based on the concept of adjacency relation and complements the concept of dimension ascommon in combinatorial topology. This work appears to be the first one on digital manifolds based ona graph-theoretical definition of dimension. In particular, in the n-dimensional digital space, a digitalcurve is a one-dimensional object and a digital hypersurface is an (n-1)-dimensional object, as it isin the case of curves and hypersurfaces in the Euclidean space. Relying on the obtained properties ofdigital hypersurfaces, we propose a uniform approach for studying good pairs defined by separationsand obtain a classification of good pairs in arbitrary dimension. We also discuss possible applicationsof the presented definitions and results.
digital geometry, digital topology, discrete dimension, digital manifold, digital curve, digital hypersurface, good pair
Valentin Brimkov, Reinhard Klette, "Border and SurfaceTracing - Theoretical Foundations", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.30, no. 4, pp. 577-590, April 2008, doi:10.1109/TPAMI.2007.70725
[1] J.C. Alexander and A.I. Thaler, “The Boundary Count of Digital Pictures,” J. ACM, vol. 18, pp. 105-112, 1971.
[2] P. Alexandroff and H. Hopf, Topologie—Erster Band. Julius Springer, 1935.
[3] E. Andres, R. Acharya, and C. Sibata, “Discrete Analytical Hyperplanes,” Graphical Models Image Processing, vol. 59, pp. 302-309, 1997.
[4] G. Bertrand and R. Malgouyres, “Some Topological Properties of Surfaces in ${\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}^{3}$ ,” J. Math. Imaging Vision, vol. 11, pp. 207-221, 1999.
[5] V.E. Brimkov, E. Andres, and R.P. Barneva, “Object Discretizations in Higher Dimensions,” Pattern Recognition Letters, vol. 23, pp. 623-636, 2002.
[6] V.E. Brimkov and R.P. Barneva, “Plane Digitization and Related Combinatorial Problems,” Discrete Applied Math., vol. 147, pp. 169-186, 2005.
[7] V.E. Brimkov, D. Coeurjolly, and R. Klette, “Digital Planarity—A Review,” Discrete Applied Math., vol. 155, pp. 468-495, 2007.
[8] V.E. Brimkov and R. Klette, “Curves, Hypersurfaces, and Good Pairs of Adjacency Relations,” Proc. Int'l Workshop Combinatorial Image Analysis, pp. 270-284, 2004.
[9] L. Chen, Discrete Surfaces and Manifolds: A Theory of Digital-Discrete Geometry and Topology. Scientific and Practical Computing, 2004.
[10] L. Chen, “Gradually Varied Surfaces and Gradually Varied Functions,” Technical Report CITR-TR 156, Univ. of Auckland, May 2005.
[11] L. Chen, D.H. Cooley, and J. Zhang, “The Equivalence between Two Definitions of Digital Surfaces,” Information Sciences, vol. 115, pp. 201-220, 1999.
[12] L. Chen and J. Zhang, “Digital Manifolds: An Intuitive Definition and Some Properties,” Proc. ACM/SIGGRAPH Symp. Solid Modeling Applications, pp. 459-460, 1993.
[13] D. Cohen-Or, A. Kaufman, and T.Y. Kong, “On the Soundness of Surface Voxelizations,” Topological Algorithms for Digital Image Processing, T.Y. Kong and A. Rosenfeld, eds., pp. 181-204, Elsevier, 1996.
[14] M. Couprie and G. Bertrand, “Tessellations by Connection,” Pattern Recognition Letters, vol. 23, pp. 637-647, 2002.
[15] X. Daragon, M. Couprie, and G. Bertrand, “Discrete Surfaces and Frontier Orders,” J. Math. Imaging and Vision, vol. 23, no. 3, pp.379-399, 2005.
[16] E.E. Domínguez and A.R. Francés, “An Axiomatic Approach to Digital Topology,” Lecture Notes in Computer Scince, vol. 2243, pp.3-16, 2001.
[17] R.O. Duda, P.E. Hart, and J.H. Munson, “Graphical-Data-Processing Research Study and Experimental Investigation,” Technical Report ECOM-01901-26, Stanford Research Inst., Mar. 1967.
[18] U. Eckhardt and L. Latecki, “Topologies for the Digital Spaces ${\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}^{2}$ and ${\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}^{3}$ ,” Computer Vision Image Understanding, vol. 90, pp. 295-312, 2003.
[19] J. Françon, “Discrete Combinatorial Surfaces,” Graphical Models Image Processing, vol. 57, pp. 20-26, 1995.
[20] G.T. Herman, “Boundaries in Digital Spaces: Basic Theory,” Topological Algorithms for Digital Image Processing, T.Y. Kong and A. Rosenfeld, eds., pp. 233-261, Elsevier, 1996.
[21] Y. Kenmochi, A. Imiya, and A. Ichikawa, “Discrete Combinatorial Geometry,” Pattern Recognition, vol. 30, pp. 1719-1728, 1997.
[22] C.E. Kim, “Three-Dimensional Digital Line Segments,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 5, pp. 231-234, 1983.
[23] G. Klette, “Branch Voxels and Junctions in 3D Skeletons,” Proc. Int'l Workshop Combinatorial Image Analysis, pp. 34-44, 2006.
[24] G. Klette, “Topologic, Geometric, or Graph-Theoretic Properties of Skeletal Curves,” PhD dissertation, Groningen Univ., 2007.
[25] R. Klette, “Algorithms for Picture Analysis, Lecture 25,” www. MK2004Algorithms.htm, Mar. 2005.
[26] R. Klette and A. Rosenfeld, Digital Geometry—Geometric Methods for Digital Picture Analysis. Morgan Kaufmann, 2004.
[27] R. Klette and H.-J. Sun, “Digital Planar Segment Based Polyhedrization for Surface Area Estimation,” Visual Form, C. Arcelli, L.P.Cordella, and G. Sanniti di Baja, eds., pp. 356-366, Springer, 2001.
[28] T.Y. Kong, “Digital Topology,” Foundations of Image Understanding, L.S. Davis, ed., pp. 33-71, Kluwer Academic, 2001.
[29] T.Y. Kong, “Topological Adjacency Relations on ${\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}^{n}$ ,” Theoretical Computer Science, vol. 283, pp. 3-28, 2002.
[30] T.Y. Kong, A.W. Roscoe, and A. Rosenfeld, “Concepts of Digital Topology,” Topology and Its Applications, vol. 46, pp. 219-262, 1992.
[31] R. Kopperman, P.R. Meyer, and R. Wilson, “A Jordan Surface Theorem for Three-Dimensional Digital Spaces,” Discrete Computational Geometry, vol. 6, pp. 155-161, 1991.
[32] V.A. Kovalevsky, “Finite Topology as Applied to Image Analysis,” Computer Vision, Graphics, and Image Processing, vol. 46, no. 2, pp. 141-161, 1989.
[33] V. Kovalevsky, “Multidimensional Cell Lists for Investigating 3-Manifolds,” Discrete Applied Math., vol. 125, pp. 25-44, 2003.
[34] J.-O. Lachaud and A. Montanvert, “Continuous Analogs of Digital Boundaries: A Topological Approach to Isosurfaces,” Graphical Models, vol. 62, pp. 129-164, 2000.
[35] L.J. Latecki, Discrete Representations of Spatial Objects in Computer Vision. Kluwer Academic Publisher, 1998.
[36] R. Malgouyres, “A Definition of Surfaces of ${\hbox{\rlap{Z}\kern 2.0pt{\hbox{Z}}}}^{3}$ : A New 3D Discrete Jordan Theorem,” Theoretical Computer Science, vol. 186, pp. 1-41, 1997.
[37] K. Menger, Kurventheorie. Teubner, 1932.
[38] D.G. Morgenthaler and A. Rosenfeld, “Surfaces in Three-Dimensional Digital Images,” Information Control, vol. 51, pp. 227-247, 1981.
[39] J.P. Mylopoulos and T. Pavlidis, “On the Topological Properties of Quantized Spaces, I. The Notion of Dimension,” J. ACM, vol. 18, pp. 239-246, 1971.
[40] J. Ohser and K. Schladitz, Image Processing and Analysis. Clarendon Press, 2006.
[41] J.-P. Reveillès, Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique, Thèse d'état, Université Louis Pasteur, 1991.
[42] A. Rosenfeld, “Adjacency in Digital Pictures,” Information and Control, vol. 26, pp. 24-33, 1974.
[43] A. Rosenfeld, “Compact Figures in Digital Pictures,” IEEE Trans. Systems, Man, and Cybernetics, vol. 4, pp. 221-223, 1974.
[44] A. Rosenfeld, T.Y. Kong, and A.Y. Wu, “Digital Surfaces,” Computer Vision, Graphics, and Image Processing: Graphical Models Image Processing, vol. 53, pp. 305-312, 1991.
[45] A. Rosenfeld and J.L. Pfaltz, “Sequential Operations in Digital Picture Processing,” J. ACM, vol. 13, pp. 471-494, 1966.
[46] M. Siguera, L.J. Latecki, and J. Gallier, “Making 3D Binary Digital Images Well-Composed,” Proc. Vision Geometry, pp. 150-163, 2005.
[47] G. Tourlakis, “Homological Methods for the Classification of Discrete Euclidean Structures,” SIAM J. Applied Math., vol. 33, pp.51-54, 1977.
[48] G. Tourlakis and J. Mylopoulos, “Some Results in Computational Topology,” J. ACM, vol. 20, pp. 430-455, 1973.
[49] J.K. Udupa, “Connected, Oriented, Closed Boundaries in Digital Spaces: Theory and Algorithms,” Topological Algorithms for Digital Image Processing, T.Y. Kong and A. Rosenfeld, eds., pp.205-231, Elsevier, 1996.
[50] P. Urysohn, “Über die allgemeinen Cantorischen Kurven,” Proc. Ann. Meeting, Deutsche Mathematiker Vereinigung, 1923.
[51] D.J.A. Welsh, Matroid Theory. Academic Press, 1976.
[52] F. Wyse, “A Special Topology for the Integers,” Am. Math. Monthly, vol. 77, p. 1119, 1970.
17 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool