ON THE INFLUENCE OF NOISE ON THE RECOGNITION OF THREEFOLD ROTATIONAL SYMMETRY IN HEXAGONAL IMAGES
Abstract
The article presents an algebraic approach to the representation and processing of digital images defined on hexagonal lattices. The described approach is based on the representation of images as functions on finite fields of “Eisenstein's integers”. As it turns out, the elements of such fields naturally correspond to the pixels of hexagonal images of certain sizes. The exponential and logarithmic transformations in the Eisenstein fields are described. A method for detecting the centers of threefold rotational symmetry in grayscale images is presented and the corresponding normalized measure of symmetry is introduced. The main purpose of the work is to study the effect of noise on the image on the quality of the symmetry assessment using the introduced measure. The noise factor must be taken into account, since a decrease in the measure can be caused not only by the incomplete symmetry of the real object, but also by distortions due to noise, which is almost always the case. Obviously, this difference will be proportional to the level of the noise component. Analytical estimates of the effect of noise on the criterion for detecting symmetry are obtained in this work. If images are subject to random noise, then the measure of symmetry of local image areas will be a random variable, the distribution law of which is determined by the distribution laws of noise components. At the same time, the standard for image processing assumption is made in the work about the model of normal and independent noise level of the brightness function. The peculiarity of the introduced threefold rotational symmetry measure does not allow directly applying standard methods to obtain probabilistic estimates. For this purpose, an assessment of the cumulative probability distribution function was carried out, on the basis of which an expression was obtained for the probabilities of deviation of the symmetry measure from the true value by a given value. By virtue of the a priori assumptions made, the obtained estimate should be considered as rather "cautious" and it can be expected that in reality the spread of the measure caused by noise in the image will be significantly less than the theoretically established boundaries.
References
Int. J. Robotics Res., 1995, 14 (5), pp. 407-424.
2. Martinet A., Soler C., Holzschuch N., Sillion F. Accurate Detection of Symmetries in
3D Shapes, ACM Trans. Graph., 2006, 25 (2), pp. 439-464.
3. Middleton L., Sivaswamy J. Hexagonal Image Processing: A Practical Approach. Springer,
2005.
4. Xiangjian He,Wenjing Jia, Namho Hur, QiangWu, Jinwoong Kim. Image Translation and Rotation
on Hexagonal Structure, In: 6th IEEE Intern. Conf. on Computer and Information Technology
(CIT'06). Seoul, 141, 2006.
5. Chertok M., Keller Y. Spectral Symmetry Analysis, IEEE Trans. on Pattern Analysis and Machine
Intelligence, 2010, 32 (7), pp. 1227-1238.
6. Derrode S., Ghorbel F. Shape Analysis and Symmetry Detection in Gray-level Objects Using
the Analytical Fourier-Mellin Representation, Signal Processing, 2004, 84 (1), pp. 25-39.
7. Karkishchenko A.N., Mnukhin V.B. Threefold Symmetry Detection in Hexagonal Images
Based on Finite Eisenstein Fields, Analysis of Images, Social Networks, and Texts. 5th International
Conference, AIST’2016. Selected Papers. Communications in Computer and Information
Science 661, Springer, 2017, pp. 281-292.
8. Karkishchenko A.N., Mnukhin V.B. Raspoznavanie simmetrii izobrazheniy v chastotnoy
oblasti [Symmetry Recognition in the Frequency Domain], Tr. 9-y Mezhdunarodnoy
konferentsii «Intellektualizatsiya obrabotki informatsii – 2012», TORUS Press, Moscow,
2012 [In 9th International Conference on Intelligent Information Processing, TORUS Press,
Moscow, 2012], pp. 426-429.
9. Campello de Souza R.M., Farrell R.G. Finite Field Transforms and Symmetry Groups, Discrete
Mathematics, 1985, 56, pp. 111-116.
10. Mnukhin V.B. Transformations of Digital Images on Complex Discrete Tori, Pattern Recognition
and Image Analysis: Advances in Mathematical Theory and Applications, 2014, 24 (4),
pp. 552-560.
11. Karkishchenko A.N., Mnukhin V.B. Primenenie modulyarnykh logarifmov na kompleksnykh
diskretnykh torakh v zadachakh obrabotki tsifrovykh izobrazheniy [Applications of Modular
Logarithms on Complex Discrete Tori in Digital Image Processing], Vestnik Rostovskogo
gosudarstvennogo universiteta putey soobshcheniya [Bulletin of the Rostov State University of
Railway Transport]. Issue 3. Rostov-on-Don: RGUPS, 2013, pp. 137-142.
12. Mnukhin V.B. Fourier-Mellin Transform on a Complex Discrete Torus // In: 11th Int. Conf.
"Pattern Recognition and Image Analysis: New Information Technologies" (PRIA-11-2013),
September 23-28 2013. Samara, Russia, 2013. – P. 102-105.
13. Her I. Geometric Transforms on the Hexagonal Grid, IEEE Transactions on Image Processing,
1995, 4 (9), pp. 1213-1222.
14. Creutzburg R., Labunets V.G. The Early Papers on Number-theoretic Transforms. Available
at: https://www.researchgate.net/publication/229043248.
15. Labunets V.G. Teoretiko-chislovye preobrazovaniya nad kvadratichnymi polyami [Number
Theoretic Transforms over Quadratic Fields], Slozhnye sistemy upravleniya [Complex Control
Systems]. Kiev: Institut kibernetiki USSR, 1982, pp. 30-37.
16. Varichenko L.V., Labunets V.G., Rakov M.A. Abstraktnye algebraicheskie sistemy i tsifrovaya
obrabotka signalov [Abstract Algebraic Systems and Digital Signal Processing]. Kiev:
Naukova Dumka, 1986.
17. Baker H.G. Complex Gaussian Integers for Gaussian Graphics, ACM Sigplan Notices, 1993,
28 (11), pp. 22-27.
18. Bandeira J., Campello de Souza R.M. New Trigonometric Transforms Over Prime Finite
Fields for Image Filtering // In: VI International Telecommunications Symposium (ITS2006),
Fortaleza-Ce, Brazil, 2006, pp. 628-633.
19. Campello de Souza R.M., de Oliveira H.M., Silva D. The Z Transform over Finite Fields,
ArXiv preprint 1502.03371 published online February 11, 2015.
20. Hundt R., Schön J.C., Hannemann A., Jansen M. Determination of Symmetries and Idealized
Cell Parameters for Simulated Structures, Journal of Applied Crystallography, 1999, 32,
pp. 413-416.
21. Spek A.L. Structure Validation in Chemical Crystallography, Acta Crystallographica. D65,
2009, pp. 148-155.
22. Zeyun Yu, Bajaj C. Automatic Ultrastructure Segmentation of Reconstructed CryoEM Maps of
Icosahedral Viruses, IEEE Transactions on Image Processing, 2005, 14 (9), pp. 1324-1337.
23. Seiichi Kondo, Mark Lutwyche, Yasuo Wada Observation of Threefold Symmetry Images due
to a Point Defect on a Graphite Surface Using Scanning Tunneling Microscope (STM), Japanese
Journal of Applied Physics, 1994, 33 (9B), pp. 1342-1344.
24. Ireland K., Rosen M. A Classical Introduction to Modern Number Theory. Springer-Verlag, 1982.
25. Dummit D.S., Foote R.M. Abstract Algebra. John Wiley&Sons, 2004.
26. Karkishchenko A.N., Mnukhin V.B. Topologicheskaya fil'tratsiya dlya raspoznavaniya i analiza
simmetrii tsifrovykh izobrazheniy [Topological Filtration for Digital Images Recognition and
Symmetry Analysis], Mashinnoe obuchenie i analiz dannykh [Journal of Machine Learning
and Data Analysis], 2014, 1 (8), pp. 966-987.
27. Markus M., Mink Kh. Obzor po teorii matrits i matrichnykh neravenstv [Overview on the theory
of matrices and matrix inequalities]. Moscow: Nauka, 1972, 232 p.
28. Kibzun A.I., Goryainova E.R., Naumov A.V. Teoriya veroyatnostey i matematicheskaya statistika.
Bazovyy kurs s primerami i zadachami [Theory of Probability and Mathematical Statistics.
Basic course with examples and tasks]. Moscow: Fizmatlit, 2013, 232 p.
29. Venttsel' E.S., Ovcharov L.A. Teoriya veroyatnostey [Theory of Probability]. Moscow: Nauka,
1969, 368 p.
30. Karkishchenko A.N., Gorban' A.S. K opredeleniyu mer skhodstva polutonovykh izobrazheniy
[On the definition of measures of similarity of halftone images], Izvestiya YuFU.
Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2008, No. 4 (81), pp. 98-103.
