Preserving Topology by a Digitization Process

The main task of digital image processing is to recognize properties of real objects based on their digital images. These images are obtained by some sampling device, like a CCD camera, and represented as finite sets of points that are assigned some value in a gray-level or color scale. Based on technical properties of sampling devices, these points are usually assumed to form a square grid and are modeled as finite subsets of Z2. Therefore, a fundamental question in digital image processing is which features in the digital image correspond, under certain conditions, to properties of the underlying objects. In practical applications this question is mostly answered by visually judging the obtained digital images. In this paper we present a comprehensive answer to this question with respect to topological properties. In particular, we derive conditions relating properties of real objects to the grid size of the sampling device which guarantee that a real object and its digital image are topologically equivalent. These conditions also imply that two digital images of a given object are topologically equivalent. This means, for example, that shifting or rotating an object or the camera cannot lead to topologically different images, i.e., topological properties of obtained digital images are invariant under shifting and rotation.

[1]  Lawrence N. Stout.  Two discrete forms of the Jordan curve theorem , 1988 .

[2]  A. W. Roscoe,et al.  Continuous analogs of axiomatized digital surfaces , 1984, Comput. Vis. Graph. Image Process..

[3]  Jayaram K. Udupa,et al.  A justification of a fast surface tracking algorithm , 1992, CVGIP Graph. Model. Image Process..

[4]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[5]  Longin Jan Latecki,et al.  Semi-proximity continuous functions in digital images , 1995, Pattern Recognit. Lett..

[6]  Theo Pavlidis,et al.  Algorithms for Graphics and Imag , 1983 .

[7]  Christopher Habel,et al.  Propositional and Depictorial Representations of Spatial Knowledge: The Case of Path-Concepts , 1989, Natural Language and Logic.

[8]  Longin Jan Latecki,et al.  Nearly continuous functions in digital images , 1995, Other Conferences.

[9]  Longin Jan Latecki,et al.  Digital Topology , 1994 .

[10]  M. Docarmo.  Differential geometry of curves and surfaces , 1976 .

[11]  Azriel Rosenfeld,et al.  Surfaces in Three-Dimensional Digital Images , 1981, Inf. Control..

[12]  Azriel Rosenfeld,et al.  Three-Dimensional Digital Topology , 1981, Inf. Control..

[13]  Azriel Rosenfeld,et al.  Digital Picture Processing , 1976 .

[14]  Longin Jan Latecki,et al.  A Realistic Digitization Model of Straight Lines , 1997, Comput. Vis. Image Underst..

[15]  G. Naber.  Topological Methods in Euclidean Spaces , 1980 .

[16]  L Chen,et al.  Topological Structure in the Perception of Apparent Motion , 1985, Perception.

[17]  Manfredo P. do Carmo,et al.  Differential geometry of curves and surfaces , 1976 .

[18]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[19]  T. Pavlidis.  Algorithms for Graphics and Image Processing , 1981, Springer Berlin Heidelberg.

[20]  L Chen,et al.  Topological structure in visual perception. , 1982, Science.

[21]  R. Fenn.  GEOMETRIC TOPOLOGY IN DIMENSIONS 2 AND 3 , 1978 .

[22]  Ari D. Gross.  Analyzing generalized tubes , 1994, Other Conferences.

[23]  A. W. Roscoe,et al.  A theory of binary digital pictures , 1985, Comput. Vis. Graph. Image Process..

[24]  Laurence Boxer,et al.  Digitally continuous functions , 1994, Pattern Recognit. Lett..

[25]  Azriel Rosenfeld,et al.  'Continuous' functions on digital pictures , 1986, Pattern Recognit. Lett..

[26]  Longin Jan Latecki,et al.  Digitizations preserving topological and differential geometric properties , 1995, Other Conferences.

[27]  Azriel Rosenfeld,et al.  Well-Composed Sets , 1995, Comput. Vis. Image Underst..

[28]  P. R. Meyer,et al.  Computer graphics and connected topologies on finite ordered sets , 1990 .

[29]  L. Chen,et al.  Holes and wholes: A reply to Rubin and Kanwisher , 1990, Perception & psychophysics.

[30]  Gabor T. Herman,et al.  Discrete multidimensional Jordan surfaces , 1992, CVGIP Graph. Model. Image Process..

[31]  Marvin Minsky,et al.  An introduction to computational geometry , 1969 .

[32]  Azriel Rosenfeld,et al.  Digital topology: Introduction and survey , 1989, Comput. Vis. Graph. Image Process..