Algebraic Interpretation of Image Analysis Operations

Abstract

The study is devoted to mathematical and functional/physical interpretation of image analysis and processing operations used as sets of operations (ring elements) in descriptive image algebras (DIA) with one ring. The main result is the determination and characterization of interpretation domains of DIA operations: image algebras that make it possible to operate with both the main image models and main models of transformation procedures that ensure effective synthesis and realization of the basic procedures involved in the formal description, processing, analysis, and recognition of images. The applicability of DIAs in practice is determined by the realizability—the possibility of interpretation—of its operations. Since DIAs represent an algebraic language for the mathematical description of image processing, analysis, and understanding procedures using image transformation operations and their representations and models, the authors consider an algebraic interpretation. These procedures are formulated and implemented in the form of descriptive algorithmic schemes (DAS), which are correct expressions of the DIA language. The latter are constructed from the processing and transformation of images and other mathematical operations included in the corresponding DIA ring. The mathematical and functional properties of DIA operations are of considerable interest for optimizing procedures of processing and analyzing images and constructing specialized DAS libraries. Since not all mathematical operations have a direct physical equivalent, the construction of an efficient DAS for image analysis involves the problem of interpreting operations for DAS content. Research into this problem leads to the selection and study of interpretation domains of DIA operations. The proposed method for studying the interpretability of DIA operations is based on the establishment of correspondence between the content description of the operation function and its mathematical realization. The main types of interpretability are defined and examples given of the interpretability/uninterpretability of operations of a standard image algebra, which is a restriction of the DIA with one ring.