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Subjective modeling of image shape

The paper presents two approaches to subjective modelling incomplete and uncertain information about possible shape of images. The shape is understood as a set of images of the scene, recorded under all possible conditions (lighting, exposure, etc.). To simulate the likelihood, the measure of likelihood introduced by Yu.P. Pytyev is used. This measure is a function defined on a set of statements and ordering them according to plausibility. The obtained subjective models of shapes allow using the theory of optimal strategies for solving morphological problems.


Introduction
Methods of morphological analysis of images use their structural and brightness description. The first approaches to the description of the morphology of images appeared in the 1970s. These include mathematical morphology described in the work of J. Serre [1] and methods of morphological image analysis published in the works of Yu.P. Pyt'ev [2][3][4][5], while the approaches to the description of the shape of images in these works are fundamentally different.
The methods of morphological image analysis of Yu.P. Pyt'ev were continued by his colleagues under his guidance [6][7][8][9]. These methods have been widely and successfully used to solve a number of problems, such as recognizing objects by the shape of their image, distinguishing differences in scenes by their images, estimating the parameters of objects in the depicted scenes, etc. [6,8,10]. They are also used to analyze the shape of images and signals when solving a number of applied problems [11][12][13][14][15].
In the methods of morphological analysis of image given by Pyt'ev the image shape is defined as an invariant of image transformations that simulate a change in the conditions of its registration. Thus, the shape of the image does not depend on the lighting, the parameters of the recording equipment, etc. and is determined only by the properties of the objects of the depicted scene. In the classical methods of morphological analysis in terms of the named invariant, the image shape is defined, a comparison of the images by shape operation is introduced, and a set of methods for solving a number of image analysis tasks is proposed.
However, in solving these problems, additional information about the scene or its images may be essential. This information, on the one hand, is difficult to formalize in terms of classical morphological analysis, and on the other hand, its consideration can significantly increase the accuracy and adequacy of solutions. To account for this kind of information in solving mathematical modeling problems in [16] Yu.P. Pyt'ev proposes to use fuzzy plausibility and belief measures on a set 2 of statements. This paper is devoted to the application and development of this approach in the morphological analysis of images.

Image shape
In the classical morphological analysis, a canonical example is the shape of a mosaic image . For its definition, one uses a mathematical model of the image of a certain scene S in the form here X is a bounded subset of the two-dimensional plane (field of view), , forming the partition X: are unchanged in all images of the scene obtained under various registration conditions. In terms of these indicators, a mosaic image shape description of the scene is given.
The mathematical definition of the image shape is as follows: the image shape (1) is the set To characterize the closeness of the shape of the image of In view of the foregoing, it is possible to give it a meaning of the "noise"/"signal" ratio.

Subjective model of the image shape with fixed segments of mosaic
The set f V of all possible images of the scene S in the previous example is a formally "completely deterministic" set: for any element there is a definite answer whether ) ( g belongs to the shape f V or not. However, in practice, the researcher can set a rule that allows for any pair of images, indicate which of them is more similar to the image of scene S, or say that they are equally alike. The researcher can model his views by giving a measure of plausibility [16] of the statement that To obtain a single proximity criterion, one should specify the likelihood distribution on the whole set R, assuming that the image Having a plausibility distribution, one can apply the mathematical decision-making technique developed in the theory of subjective modeling described in [16].

General subjective model of image shape
In the previous section, a mathematical model of the image shape was constructed, in which the subjective preferences of the researcher concern only the brightness characteristics of the image, and the geometric structure is assumed to be given. However, in practice, it is often necessary to assume that the scene S can generate images with different segmentation. In this paper, it is proposed to choose a subjective model of the image shape based on the visual analysis of the presented image of the scene.
Let the image ) (x  of the scene S be presented as a mixture of the "ideal" image ) ( f and noise . The mathematical model of noise is not defined a priori, but it is believed that the researcher has sufficient experience and intuition to distinguish the noise image from the image of a certain scene. As for the shape of the image

Conclusion
In this paper, two approaches are proposed for constructing a subjective model of the shape of an image. Both allow taking into account the experience and intuition of the researcher. Models constructed applying these approaches make it possible to use a mathematical apparatus for making optimal decisions, developed in the theory of possibilities, for morphological studies.