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The paper is devoted to Descriptive Image Analysis (DA) — a leading line of the modern mathematical theory of image analysis. DA is a logically organized set of descriptive methods, mathematical objects, and models and representations aimed at analyzing and evaluating the information represented in the form of images, as well as for automating the extraction from images of knowledge and data needed for intelligent decision-making.

The basic idea of DA consists of embedding all processes of analysis (processing, recognition, understanding) of images into an image formalization space and reducing it to (1) construction of models/representations/formalized descriptions of images; (2) construction of models/representations/formalized descriptions of transformations over models and representations of images.

We briefly discuss the basic ideas, methodological principles, mathematical methods, objects, and components of DA and the basic results determining the current state of the art in the field. Image algebras (IA) are considered in the context of a unified language for describing mathematical objects and operations used in image analysis (the standard IA by Ritter and the descriptive IA by Gurevich).

We use the concept of fuzzy similarity to compare the objects of a free image algebra (a set of objects which a group is acting on). In particular, we study those fuzzy similarities that are preserved by the action of the group. Later we consider a deformation mechanism of the image algebra and trackle the problem of comparing deformed images. For that purpose, we characterize those deformation mechanisms that are equivalent to the induced action from a subgroup of the group of deformations. In that case, by using techniques from group representation theory, we extend any fuzzy similarity defined on the image algebra to a fuzzy similarity defined on the whole space of deformed images. Moreover, we prove that the invariance of the similarity with respect to the group action is preserved by this extension.

This chapter deals with a new class of specialized algebras, viz. descriptive image algebras, which is an original mathematical language used to formalize and standardize the processing procedures for image models and transformations over them. Descriptive image algebras are used to describe problems, objects, and transformations involved in extracting information from images. When speaking of descriptive image algebras in this publication, we consider descriptive image algebras with one ring in more detail. This class of algebra falls into the class of universal linear algebra with sigma-associative ring with identity. Classes of descriptive image algebras with two or more rings fall into the class of homogeneous graded algebras. Descriptive image algebras are the principal branch of the mathematical apparatus of descriptive image analysis, which is a logically structured totality of descriptive methods and models designed for image analysis and estimation. Up to now, the following principal results associated with “algebraization” have been obtained within the framework of descriptive image analysis: (1) descriptive image algebras are introduced and defined as a mathematical object, (2) the class of descriptive image algebras with one ring is introduced and studied, (3) descriptive image algebras with one ring are defined with the method and necessary and sufficient conditions for constructing them proposed, (4) the specialized versions of descriptive image algebras with one ring over images, models, and image representations as well as over transformations of image models and images are defined, (5) the set of operations of the standard image algebra that ensure construction of descriptive image algebras with one ring is defined, (6) classes of descriptive image algebras with one ring that generate the classes of image models are defined, and (7) the mathematical and functional/physical interpretation of image analysis and processing operations used as the sets of operations (the elements of the ring) of descriptive image algebras with one ring is studied. The following results of the studies obtained within the descriptive image analysis are studied, viz. (1) image models (representations and formalized descriptions) and (2) the algebraic model for solving the problem of automated ophthalmological diagnostics. The principal publications that deal with image algebras are listed in references.