The Fibonacci Numeral System for Computer Vision
One of the most important challenges when creating efficient systems of technical vision is the development of efficient methods for enhancing the speed and noise-resistance properties of the digital devices involved in the system. The devices composed of counters and decoders occupy a special niche among the system’s digital tools. One of the most common ways of creating noise-proof devices is providing special coding tricks dealing with their informational redundancy. Various frameworks make that possible, but nowadays, an acute interest is attracted to noise-proof numeral systems, among which the Fibonacci system is the most famous. The latter helps generate the so-called Fibonacci codes, which can be effectively applied to the computer vision systems; in particular when developing counting devices based on Fibonacci counters, as well as the corresponding decoders. However, the Fibonacci counters usually pass from the minimal form of representation of Fibonacci numbers to their maximal form by recurring to the special operations catamorphisms and anamorphisms (or “folds” and “unfolds”). The latter makes the counters quite complicated and time-consuming. In this chapter, we propose a new version of the Fibonacci counter that relies only on the minimal representation form of Fibonacci numerals and thus leads to the counter’s faster calculation speed and a higher level of the noise-resistance. Based on the above-mentioned features, we also present the appropriate fast algorithm implementing the noise-proof computation and the corresponding fractal decoder. The first part of the chapter provides the estimates of the new method’s noise-immunity, as well as that of its components. The second problem studied in the chapter concerns the efficiency of the existing algorithm of Fibonacci representation in the minimal form. Based on this examination, we propose a modernization of the existing algorithm aiming at increasing its calculation speed. The third object of the chapter is the comparative analysis of the Fibonacci decoders and the development of the fractal decoder of the latter.
KeywordsFibonacci numbers Minimal presentation form Digital devices Noise-immunity Counting velocity
This research was partially supported by the CONACYT grant CB-2013-01-221676 (Mexico). The authors would also like to express their profound gratitude to Prof. Dr. Anatoliy Ya. Beletsky and Prof. Dr. Neale R. Smith Cornejo, whose critical remarks and suggestions have helped us a lot in the revision of the chapter as well as in improving its contents and presentation.
- 1.Borisenko, A.A., Solovey, V.A., Uskov, M.K., Chinaev, P.I.: A robotic-technical combined device for the automatic control of surface defects with an aid of computer vision. Probl. Mach. Build. Autom. 2, 31–37 (1991) (in Russian)Google Scholar
- 2.Borisenko, A.A., Solovey, V.A.: A method for the automatic control of surface defects and its implementation. USSR Author Rights Certificate No. 1715047 (1991) (in Russian)Google Scholar
- 3.Borisenko, A.A., Kalashnikov, V.V., Protasova, T.A., Kalashnykova, N.I.: A new approach to the classification of positional numeral systems. In: Neves-Silva, R., Tshirintzis, G.A., Uskov, V., Howlett, R.J., Jain, L.C. (eds.) Frontiers in Artificial Intelligence and Applications (FAIA), vol. 262, pp. 44–450 (2014)Google Scholar
- 4.Stakhov, A.P.: Introduction to the Algorithmic Measurements Theory. Soviet Radio, Moscow (1997) (in Russian)Google Scholar
- 5.Vorobyov, N.N.: Fibonacci numbers. Nauka, Moscow (1969) (in Russian)Google Scholar
- 6.Goryachev, A.E., Degtyar, S.A.: A method for generation of permutations based on factorial numbers and complementary arrays. Trans. Sumy State Univ. Tech. Sci. 4, 86–92 (2012) (in Russian)Google Scholar
- 7.Stakhov, A.P.: Fibonacci and Golden Proportion Codes as an Alternative to the Binary Numeral System. Part 1. Academic Publishing, Germany (2012)Google Scholar
- 9.Azarov, O.D., Chernyak, O.I., Murashenko, O.G.: A method of developing fast Fibonacci counters. Probl. Inf. Control 46, 5–8 (2014) (in Ukrainian)Google Scholar
- 10.Borysenko, O.A., Matsenko, S.M., Polkovnikov, S.I.: A noise-proof Fibonacci counter. Trans. Natl. Technol. Univ. Kharkiv 18, 77–81 (2013) (in Russian)Google Scholar
- 11.Borysenko, O.A., Stakhov, A.P., Matsenko, S.M.: A Fibonacci decoder. In: 5th International Conference on Information Technologies and Computer Engineering, Vasyl Stefanik Carpathian National University, Ivano-Frankivsk–Vynnytsya, pp. 279–281 (2015) (in Ukrainian)Google Scholar