In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky.
The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space.
The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated
Subject of research: single-channel digital image, second-order invariants to movements and stretching.
Purpose of research: to determine a new group of invariants of a single-channel digital image with respect to shifts, rotations and stretching.
Methods and objects of research: the object of research is single-channel images. The developed computational algorithms are based on the complex use of analytical and geometric research methods, the theory of invariants and wavelets.
Main results of research: a computational scheme for determining the group of invariant characteristics with respect to such digital image transformations as shifts, rotations and stretching has been developed.
В настоящее время одной из наиболее актуальных задач цифровой обработки изображений является поиск новых математических подходов к анализу и обработке многоканальных изображений. Актуальность этих исследований обусловлена необходимостью оптимизации имеющихся методов цифровой обработки изображений, повышения эффективности и качества получаемых результатов. В работе предложен новый подход к обработке цифрового RGB-изображения, основанный на теории три-тканей В. Бляшке. Определяются и исследуются инварианты трехканальных изображений относительно максимально широкой «топологической» группы преобразований.
In this paper we investigate method of the digital images topological and semantic analysis based on the Morse theory. The description of the MathLab program module for calculation of statistically significant critical points for single-channel image is also submitted.
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