Thinning on cell complexes from polygonal tilings

This paper provides a theoretical foundation of a thinning method due to Kovalevsky for 2D digital binary images modelled by cell complexes or, equivalently, by Alexandroff T"0 topological spaces, whenever these are constructed from polygonal tilings. We analyze the relation between local and global simplicity of cells, and prove their equivalence under certain conditions. For the proof we apply a digital Jordan theorem due to Neumann-Lara/Wilson which is valid in any connected planar locally Hamiltonian graph. Therefore we first prove that the incidence graph of the cell complex constructed from any polygonal tiling has these properties, showing that it is a triangulation of the plane. Moreover, we prove that the parallel performance of Kovalevsky's thinning method preserves topology in the sense that the numbers of connected components, for both the object and of the background, remain the same.

[1]  Vladimir Kovalevsky,et al.  Algorithms and Data Structures for Computer Topology , 2000, Digital and Image Geometry.

[2]  Edward S. Deutsch,et al.  Thinning algorithms on rectangular, hexagonal, and triangular arrays , 1972, Commun. ACM.

[3]  Azriel Rosenfeld,et al.  Strongly normal sets of contractible tiles in N dimensions , 2007, Pattern Recognit..

[4]  Petra Wiederhold,et al.  Dimension for Alexandrov spaces , 1993, Other Conferences.

[5]  Vladimir Kovalevsky,et al.  Axiomatic Digital Topology , 2006, Journal of Mathematical Imaging and Vision.

[6]  Vladimir Kovalevsky,et al.  Algorithms in Digital Geometry Based on Cellular Topology , 2004, IWCIA.

[7]  Ulrich Eckhardt,et al.  Parallel reduction of digital sets , 1988 .

[8]  Willi Rinow,et al.  Lehrbuch der Topologie , 1975 .

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

[10]  Julian Webster.  Cell Complexes and Digital Convexity , 2000, Digital and Image Geometry.

[11]  Vladimir A. Kovalevsky,et al.  Finite topology as applied to image analysis , 1989, Comput. Vis. Graph. Image Process..

[12]  E. H. Kronheimer.  The topology of digital images , 1992 .

[13]  Hans Jarchow,et al.  Topological Vector Spaces , 1981 .

[14]  Victor Neumann-Lara,et al.  Digital Jordan curves — a graph-theoretical approach to a topological theorem , 1992 .

[15]  B. Krauskopf,et al.  Proc of SPIE , 2003 .

[16]  T. Yung Kong,et al.  A topological approach to digital topology , 1991 .

[17]  L. Marton,et al.  Advances in Electronics and Electron Physics , 1958 .

[18]  Azriel Rosenfeld,et al.  A Characterization of Parallel Thinning Algorithms , 1975, Inf. Control..

[19]  Azriel Rosenfeld,et al.  Digital geometry - geometric methods for digital picture analysis , 2004 .

[20]  Peter Braß,et al.  On strongly normal tesselations , 1999, Pattern Recognit. Lett..

[21]  Jim R. Parker,et al.  Algorithms for image processing and computer vision , 1996 .

[22]  Petra Wiederhold,et al.  Thinning on Quadratic, Triangular, and Hexagonal Cell Complexes , 2008, IWCIA.

[23]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[24]  Garrett Birkhoff,et al.  Lattice Theory Revised Edition , 1948 .

[25]  Vladimir Kovalevsky.  Geometry of Locally Finite Spaces Computer Agreeable Topology and Algorithms for Computer Imagery , 2008 .

[26]  Atsushi Imiya,et al.  Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000] , 2001 .

[27]  V. A. Kovalevsky,et al.  Finite Topology and Image Analysis , 1992 .

[28]  Azriel Rosenfeld,et al.  Local and global topology preservation in locally finite sets of tiles , 2001, Inf. Sci..

[29]  Petra Wiederhold,et al.  The Alexandroff Dimension of Digital Quotients of Euclidean Spaces , 2002, Discret. Comput. Geom..