Abstract
The conjugate classification, a new classification of the kernel form which defines the spatial structure of binary data on the hexagonal grid, is presented in the paper. The seven points of the kernel form on the hexagonal grid have a state set of 128 states and the conjugate classification arranges the state set in a five level hierarchy. The state set is divided into 2 conjugate sets, 14 groups, 22 clusters, 28 classes and 128 states in the five levels respectively. Using information theory, the conjugate classification is proved to be an optimal structure, in the sense that it has the minimum number of variables and the shortest whole bit length for each variable on each level. When using the new representation, it is necessary to process both conjugate sets in same pass as both sets have equivalent importance. The new classification is potentially useful for cellular automata and mathematical morphology operations in different practical application areas.
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References
E. S. Deutsch (1970). “On Parallel Operations on Hexagonal Arrays” IEEE Trans. Comput. C-19, pp982–983.
E. S. Deutsch (1972). “Thinning Algorithms on Rectangular, Hexagonal and Triangular Arrays” CACM Vol. 15 No. 9, pp827–837.
E. Dubois (1985). “The Sampling and Reconstruction of Time-Varying Imagery with Application in Video System” Proceedings of the IEEE Vol. 73, No. 4, pp502–522.
M. Gardner (1971). “On Cellular Automata, Self-reproduction, the Garden of Eden and the Game ‘life’,” Scientific American 224(2), pp112–117.
M. J. E. Golay (1969). “Hexagonal Parallel Pattern Transformations” IEEE Trans. Comput. Vol. 18, pp733–740.
S. B. Gray (1971). “Local Properties of Binary Images in Two Dimensions.” IEEE Trans. Comput. C-20, 551–561.
R. W. Hamming (1980). “Coding and Information Theory” Prentice-Hall.
B. K. P. Horn (1986). “Robot Vision” Chapter 4: ‘Binary Images: Topological Properties’ The MIT Press.
M. Ingram and K. Preston, Jr. (1970). “Automatic Analysis of Blood Cells” Scientific American Vol. 223, No. 11, pp72–82.
T. Y. Kong and A. Rosenfeld (1989). “Digital topology: introduction and survey” Computer Vision, Graphics and Image Processing 48, 357–393.
T. L. Kunii and Y. Takai (1989). “Cellular Self-Reproducing Automata As a Parallel Processing Model for Botanical Colony Growth Pattern Simulation” New Advances in Computer Graphics: Proceedings of CG International’89, R. A. Earnshaw and B. Wyvill (Eds.), pp7–22, Springer-Verlag Tokyo.
E. Luczak and A. Rosenfeld (1976). “Distance on a Hexagonal Grid” IEEE Trans. Comput. C-25, pp532–533.
A. L. Loeb (1971). “Color and Symmetry” John Wiley k Sons, Inc.
R. M. Mersereau (1979). “The Processing of Hexagonally Sampled Two-Dimensional Signals” Proceedings of IEEE Vol. 67, No. 6, pp930–949.
K. Preston, Jr. (1971). “Feature Extraction By Golay Hexagonal Pattern Transform” IEEE Trans. Comput. C-20, pp1007–1014.
K. Preston, Jr. and M. J. B. Duff (1984). “Modern Cellular Automata: Theory and Application” Plenum Press, New York.
J. Serra (1982). “Image Analysis and Mathematical Morphology” Academic Press London.
H. Steinhaus (1960). “Mathematical Snapshots” New York Oxford University Press.
R. L. Stevenson and G. R. Arce (1985). “Binary Display of Hexagonally Sampled Binary Images” Journal of the Optical Society of America A, Vol. 2 No.6, ppl009–1013.
D. K. W. Walters (1986). “A Computer Vision Model Based on Psychophysical Experiments” Pattern Recognition By Humans and Machines Vol. 2, Visual Perception, E. C. Schwab and H. C. Nusbaum(eds), Academic Press, pp87–120.
A. F. Wells (1977). “Three-dimensional Nets and Polyhedra” John Wiley and Sons, New York.
H. Weyl (1952). “Symmetry” Princeton University Press.
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© 1992 Springer-Verlag Tokyo
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Zheng, Z., Maeder, A.J. (1992). The Conjugate Classification of the Kernel Form of the Hexagonal Grid. In: Kunii, T.L., Shinagawa, Y. (eds) Modern Geometric Computing for Visualization. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68207-3_5
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DOI: https://doi.org/10.1007/978-4-431-68207-3_5
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-68209-7
Online ISBN: 978-4-431-68207-3
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