О ВЛИЯНИИ ЗАШУМЛЕНИЯ НА РАСПОЗНАВАНИЕ СИММЕТРИИ 3-ГО ПОРЯДКА В ГЕКСАГОНАЛЬНЫХ ИЗОБРАЖЕНИЯХ

Abstract

The article presents an algebraic approach to the representation and processing of digitalimages defined on hexagonal lattices. The described approach is based on the representation of imagesas functions on finite fields of “Eisenstein's integers”. As it turns out, the elements of such fieldsnaturally correspond to the pixels of hexagonal images of certain sizes. The exponential and logarithmictransformations in the Eisenstein fields are described. A method for detecting the centers ofthreefold rotational symmetry in grayscale images is presented and the corresponding normalizedmeasure of symmetry is introduced. The main purpose of the work is to study the effect of noise on theimage on the quality of the symmetry assessment using the introduced measure. The noise factor mustbe taken into account, since a decrease in the measure can be caused not only by the incompletesymmetry 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 estimatesof the effect of noise on the criterion for detecting symmetry are obtained in this work. If imagesare subject to random noise, then the measure of symmetry of local image areas will be a randomvariable, the distribution law of which is determined by the distribution laws of noise components. Atthe same time, the standard for image processing assumption is made in the work about the model ofnormal and independent noise level of the brightness function. The peculiarity of the introducedthreefold rotational symmetry measure does not allow directly applying standard methods to obtainprobabilistic estimates. For this purpose, an assessment of the cumulative probability distributionfunction was carried out, on the basis of which an expression was obtained for the probabilities ofdeviation of the symmetry measure from the true value by a given value. By virtue of the a prioriassumptions made, the obtained estimate should be considered as rather "cautious" and it can beexpected that in reality the spread of the measure caused by noise in the image will be significantlyless than the theoretically established boundaries.