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Digital Planarity of Rectangular Surface Segments
June 1994 (vol. 16 no. 6)
pp. 647-652

We generalize the concept of evenness which has been developed for digital straight lines. Evenness is a necessary and sufficient condition for a digital arc segment to be a digital straight line segment. We prove that evenness is also a necessary and sufficient condition for a rectangular surface segment to be a digital plane segment. This is not true for surface segments of arbitrary shape. To clarify the relation between shape and evenness we introduce the notion of a regular shape and of an arbitrarily extendable even set.

[1] S. H. Y. Hung, "On the straightness of digital arcs,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-7, pp. 203-215, 1985.
[2] C. E. Kim, "On cellular straight line segments,"Comput. Vision Graphics Image Processing, vol. 18, pp. 369-381, 1982.
[3] C. E. Kim, "Three-dimensional digital planes,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 639-645, 1984.
[4] A. Rosenfeld, "Digital straight line segments,"IEEE Trans. Comput., vol. C-23, pp. 1264-1269, 1974.
[5] J. Stoer and C. Witzgall,Convexity and Optimization in Finite Dimensions I. Berlin: Springer, 1970.
[6] P. Veelaert, "On the flatness of digital hyperplanes,"J. Math. Imaging and Vision, vol. 3, pp. 205-221, 1993.
[7] P. Veelaert, "Geometrical properties of three-dimensional digital planes," Tech. Rep. DG92-03, Univ. of Ghent, Electronics Lab., 1992.
[8] L. Dorst and A. smeulders, "Discrete representation of straight lines,"IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 450-462, 1984.
[9] M. E. Dyer, "Linear time algorithms for two- and three-variable linear programs,"SIAM J. Comput., vol. 13, pp. 31-45, 1984.
[10] C. E. Kim and I. Stojmenovic´, "On the recognition of digital planes in three-dimensional space,"Pattern Recognit. Lett., vol. 12, pp. 665-669, 1991.
[11] I. Stojmenovic´and R. Tosic´, Digitization schemes and the recognition of digital straight lines, hyperplanes, and flats in arbitrary dimensions," inVision Geometry(Comtemporary Mathematics Series Vol. 119), R. A. Melter, A. Rosenfeld, and P. Bhattacharya, Eds. New York: American Mathematical Society, 1991, pp. 197-212.

Index Terms:
computational geometry; rectangular surface segments; evenness; digital straight lines; necessary and sufficient condition; digital arc segment; digital plane segment; arbitrarily extendable even set
P. Veelaert, "Digital Planarity of Rectangular Surface Segments," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 16, no. 6, pp. 647-652, June 1994, doi:10.1109/34.295909
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