Discrete Jordan curve theorem

Li Chen

doi:10.1117/12.364115

23 September 1999 Discrete Jordan curve theorem

Li Chen

Proceedings Volume 3811, Vision Geometry VIII; (1999) https://doi.org/10.1117/12.364115
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1999, Denver, CO, United States

Abstract

According to a general definition of discrete curves, surfaces, and manifolds (Li Chen, 'Generalized discrete object tracking algorithms and implementations, ' In Melter, Wu, and Latecki ed, Vision Geometry VI, SPIE Vol. 3168, pp 184 - 195, 1997.). This paper focuses on the Jordan curve theorem in 2D discrete spaces. The Jordan curve theorem says that a (simply) closed curve separates a simply connected surface into two components. Based on the definition of discrete surfaces, we give three reasonable definitions of simply connected spaces. Theoretically, these three definition shall be equivalent. We have proved the Jordan curve theorem under the third definition of simply connected spaces. The Jordan theorem shows the relationship among an object, its boundary, and its outside area. In continuous space, the boundary of an mD manifold is an (m - 1)D manifold. The similar result does apply to regular discrete manifolds. The concept of a new regular nD-cell is developed based on the regular surface point in 2D, and well-composed objects in 2D and 3D given by Latecki (L. Latecki, '3D well-composed pictures,' In Melter, Wu, and Latecki ed, Vision Geometry IV, SPIE Vol 2573, pp 196 - 203, 1995.).

Citation Download Citation

Li Chen "Discrete Jordan curve theorem", Proc. SPIE 3811, Vision Geometry VIII, (23 September 1999); https://doi.org/10.1117/12.364115

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