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• ON THE INFLUENCE OF NOISE ON THE RECOGNITION OF THREEFOLD ROTATIONAL SYMMETRY IN HEXAGONAL IMAGES

  A.N. Karkishchenko, V.B. Mnukhin

  2021-01-19

  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.

• ERROR ESTIMATION FOR MULTIPLE COMPARISON OF NOISY IMAGES

  A.N. Karkishchenko, V. B. Mnukhin

  2021-07-18

  Abstract ▼

  The aim of this work is to study the effect of noise on the image on the quality of comparison of a
  finite set of images of the same shape and size. This task inevitably arises when analyzing scenes, detecting
  individual objects, detecting symmetry, etc. The noise factor must be taken into account, since the
  difference between digital objects can be caused not only by the mismatch of the compared images of
  real objects, but also by distortions due to noise, which is almost always takes place. This differenceturns out to be proportional to the level of the noise component. The main result of this article is an
  analytical estimate for the probability of a given level of error, which may arise in the multiple comparison
  of a finite set of commensurate digital images. This estimate is based on a low-level comparison,
  which is a pixel-by-pixel calculation of image differences using the Euclidean metric. In this case, a
  standard assumption is made about the independent normal noise of image intensities with zero mathematical
  expectation and a priori established standard deviation in each pixel. The evidence presented in
  the article allows us to assert that the obtained estimate should be regarded as sufficiently "cautious"
  and it can be expected that in reality the scatter of the measure caused by noise in the image will be
  significantly less than the theoretically found boundary. The estimates obtained in this work are also
  useful for detecting various types of symmetry in images, which, as a rule, lead to the need to calculate
  the difference of an arbitrary number of commensurate digital areas. In addition, they can be used as
  theoretically grounded threshold values in tasks requiring a decision on the coincidence or difference of
  images. Such threshold values inevitably appear at various stages of processing noisy images, and the
  question of their specific values, as a rule, remains open; at best, heuristic considerations are proposed
  for their selection.

• METHOD FOR DETECTING FEATURE POINTS OF AN IMAGE USING A SIGN REPRESENTATIONS

  A. N. Karkishchenko, V. B. Mnukhin

  2020-11-22

  Abstract ▼

  The aim of the study is to develop a method for detecting feature points of a digital image
  that is stable with respect to a certain class of brightness transformations. The need for such a
  method is due to the needs of detecting feature points of images in video surveillance systems and
  face recognition, often working in a changing light environment. A feature of the proposed method
  that distinguishes it from a number of well-known approaches to the problem of distinguishing
  characteristic points is the use of the so-called sign representation of images. In contrast to the
  usual defining of a digital image by a discrete brightness function, with a sign representation, the
  image is set in the form of an oriented graph corresponding to the binary relation of the increase
  in brightness on a set of pixels. Thus, the sign representation determines not a single image, but a
  set of images, the brightness functions of which are connected by strictly monotonic brightness
  transformations. It is this property of the sign representation that determines its effectiveness for
  solving the problems caused by the goal set above. A feature of the method under consideration is
  a special approach to the interpretation of the characteristic points of the image. This concept in
  image processing theory is not strictly defined; we can say that the characteristic point is characterized
  by increased "complexity" of the image structure in its vicinity. Since the sign representation
  of the image can be represented in the form of a directed graph, in this paper, to evaluate the
  complexity measure of the local neighborhood of its vertices, it is proposed to use the ranking
  method known in the spectral theory of graphs based on the Perron-Frobenius theorem. Its essence
  lies in the fact that the value of the component of the so-called Perron eigenvector of the
  adjacency matrix of this graph acts as a measure of the complexity of the vertex. To conduct experimental
  studies of the proposed approach, a set of programs was developed, the results of
  which confirm the efficiency of the method and demonstrate that with its help it is possible to obtain
  results close to the expected ones on model examples. The paper also offers a number of recommendations
  on the use of this method.