DESCRIPTIVE IMAGE ALGEBRAS: DETERMINATION OF THE BASE STRUCTURESстатья
Статья опубликована в журнале из списка RSCI Web of Science
Статья опубликована в журнале из перечня ВАК
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 8 января 2017 г.
Местоположение издательства:Road Town, United Kingdom
Первая страница:635
Последняя страница:647
Аннотация:The origins, formulations, and results of the first stage of research devoted to the formation of an algebraic approach for the analysis and understanding of images on the basis of the introduced new class of image algebras (descriptive image algebra) are considered. At present, by image algebras (in a broad sense) a certain mathematical theory is meant whose subjects are the transformation and analysis of images in the continuous and discrete domains. An algebraic formalism should provide the following possibilities: (i) the derivation of algebraic structures that makes it possible to use the methods adopted from other fields of mathematics for processing, analysis, and recognition of images; (ii) the formulation of accurate and compact descriptions of images that are convenient both for the interpretation of the performed operations and for the development of new methods; (iii) the development of a new language for the standard description of transformations of images; (iv) description of operations on images in the form of compact sets of simple transformations. In the general case, such an algebraic formalism should constitute a certain formal system for representation and transformation of images; this system should satisfy the following conditions: (I) each object of transformations is a hierarchical structure formed from elementary objects with the use of operations of an image algebra; (II) points, sets, models, operations, and morphisms can be used as the objects; (III) each transformation is a hierarchical structure constructed from a set of basic transformations with the use of operations of an image algebra. Analysis of requirements imposed on functional capabilities and mathematical characteristics of the sought-for formal system showed that this system should constitute a certain image-algebra class ensuring the possibility of writing any algorithm of the image transformation as a combination of elementary basic operations. Thus, this class of image algebras should cover both the basic models of images as the objects of analysis and recognition and the basic models of the transformation procedures that ensure an efficient synthesis and implementation of the base procedures for formal description, processing, analysis, and recognition of images. A new class of image algebras arises: the descriptive image algebras. Their algebraic specificity is defined by the fact that the elements of the ring include both the image models and the operations on images. As the operations, one can use both the conventional algebraic operations and the specialized operations of processing and transformation of images; the latter operations are to be represented in algebraic form. It is established that the descriptive image algebras offer the prospect of (i) representing the operations of formation and transformation of formal descriptions of images and operations on transformations in the problems of processing, analysis, and recognition of images; (ii) putting the representation of input data in correspondence with the requirements imposed by the transformations used; (iii) synthesis of standardized algorithmic schemes with allowance made for the functional capabilities, the competence domain, mathematical properties of the constituent transformations, and also the properties of input and output data. The previous attempts to algebraize the processing and analysis of images are considered and the sources that have initiated the introduction of a new class of an image algebra, the descriptive image algebra, are reported. The descriptive image algebra is formally defined, and illustrative examples are given. The key purposes of developing the algebraic approach to image analysis are mentioned. The main directions of the continuation of the studies are outlined and expected results are formulated. The most promising directions of studies in the field of analysis and estimation of information represented by images are listed. For each direction, the problems whose solution is either closely related to the development of efficient algebraic formalism, and in particular, descriptive image algebras (conventional algebras, superalgebras, and graded algebras) or those which are impossible without using this formalism, are pointed out.