Algorithm for the Reconstruction of the Discontinuous Structure of a Body by Its Projections along Mutually Perpendicular Lines

Mezhuyev, Vitaliy; Pershyna, Iuliia; Nechuiviter, Olesia; Lytvyn, Oleg M.

doi:10.1145/3185089.3185115

Cited by 4 publications

(2 citation statements)

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“…That is, the development of the theory of approximation of discontinuous functions using discontinuous splines is a relevant task. This work belongs to a series of works by the authors aimed at the study and improvement of mathematical models in computed tomography [16][17][18]. To date, tomography has developed many computational methods, algorithms and software tools aimed at restoring the internal properties of an object.…”

Section: Review Of the Literaturementioning

confidence: 99%

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Restoration of Discontinuous Functions by Discontinuous Interlination Splines

Pershyna

2022

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Context. The problem of development and research of methods for approximation of discontinuous functions by discontinuous interlination splines and its further application to problems of computed tomography. The object of the study was the modeling of objects with a discontinuous internal structure. Objective. The aim of this study is to develop a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiated splines. Method. Modern methods of restoring functions are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be represented not only by function values at points, but also in the form of a set of function traces on planes or straight lines. At the same time, practice shows that among the multidimensional objects that need to be investigated, more problems are described by a discontinuous functions. The paper develops a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiable splines. It is considered that the domain of the definition of the required twodimensional function is divided into rectangular elements. Theorems on interlination and approximation properties of such discontinuous constructions are formulated and proved. The method is developed for approximating discontinuous functions of two variables based on the constructed discontinuous splines. The input data are the traces of an unknown function along a given system of mutually perpendicular straight lines. The proposed method has not only theoretical significance but also practical application in the IT domain, especially in computing tomography, allowing more accurately restore the internal structure of the body. Results. The discontinuous interlination operator from known traces of the function of two variables on a system of mutually perpendicular straight lines is researched. Conclusions. The functions of two variables that are discontinuous at some points or on some lines are better approximated by discontinuous spline interlinants. At the same time, equally high approximation estimates can be obtained. The results obtained have significant advantages over existing methods of interpolation and approximation of discontinuous functions. In further research, the authors plan to develop a theory of discontinuous splines on areas of complex shape bounded by arcs of known curves.

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Section: Review Of the Literaturementioning

confidence: 99%

“…First, we construct a discontinuous bilinear approximation spline on a given rectangular grid, for which we use the formula of the approximation spline [18] in each element of the partition…”

Section: As You Can See the Function ( )mentioning

confidence: 99%