Abstract
Euler Number is one of the most important characteristics in topology. In two-dimension digital images, the Euler characteristic is locally computable. The form of Euler Number formula is different under 4-connected and 8-connected conditions. Based on the definition of the Foreground Segment and Neighbor Number, a formula of the Euler Number computing is proposed and is proved in this paper. It is a new idea to locally compute Euler Number of 2D image.
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References
Sonka M, Hlavac V, Boyle R. Image Processing, Analysis, and Machine Vision. 2nd ed. Beijing: PPTPH and Thomson Learning, 2002. 256–259
Dyer C R. Computing the Euler Number of an image from its Quadtree. Comput Graphics Image Process, 1980, 13(3): 270–276
Rosenfeld A, Kak A C. Digital Picture Processing. New York: Academic Press, 1976. 349
Rosenfeld A. Picture Languages—Formal Models for Picture Recognition. New York: Academic Press, 1979. 25–26
Pratt W K. Digital Image Processing. 2nd ed. New York: John Wiley & Sons, 1991. 351
Gray S B. Local properties of binary images in two dimensions. IEEE Trans Comput, 1971, C-20(5): 551–561
Rosenfeld A, Kak A C. Digital Picture Processing. 2nd ed. New York: Academic Press, 1982. 248
Kong T Y, Rosenfeld A. If we use 4-or 8-connectedness for both the objects and the background, the Euler characteristic is not locally computable. Pattern Recogn Lett, 1990, 11: 231–232
Lin X Z, Sha Y, Ji J W, et al. Image Euler Number calculating for intelligent counting. In: Proceedings of the Seventh International Conference on Electronic Measurement & Instruments (ICEMI’2005). Beijing: International Academic Publishers/World Publishing Corporation, 2005, 8: 642–645
Chen M H, Yan P F. A fast algorithm to calculate the Euler Number for binary images. Pattern Recogn Lett, 1988, 8(5): 295–297
Chiavetta F, Gesu V. Parallel computation of the Euler Number via connectivity graph. Pattern Recogn Lett, 1993, 14: 849–859
Diaz-de-Leon S J L, Sossa-Azuela J H. On the computation of the Euler Number of a binary object. Pattern Recogn, 1996, 29(3): 471–476
Nagel W, Ohser J, Pischang K. An integral-geometric approach for the Euler-Poincare characteristic of spatial images. J Microsc, 2000, 189: 54–62
Barth E, Ferraro M, Zetzsche C. Global topological properties of images derived from local curvature features, In Visual Form 2001, Lecture Notes in Computer Science. Berlin: Springer-Verlag, 2001. 285–294
Ohser J, Nagel W, Schladitz K. The Euler Number of discretized sets—surprising results in three dimensions. Image Analysis Stereology, 2003, 22(1): 11–19
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Lin, X., Sha, Y., Ji, J. et al. A proof of image Euler Number formula. SCI CHINA SER F 49, 364–371 (2006). https://doi.org/10.1007/s11432-006-0364-8
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DOI: https://doi.org/10.1007/s11432-006-0364-8