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.