Відновлення внутрішньої структури тривимірного об'єкта з використанням невеликої кількості даних

Abstract

У статті запропоновано метод відновлення внутрішньої структури тривимірного тіла, що використовує чотири томограми. Метод будується за допомогою введення оператора інтерфлетації функцій трьох змінних. Вказаний оператор відновлює (можливо, наближено) функцію трьох змінних в точках між заданими площинами за допомогою її слідів на цих площинах (томограм). Доведені теореми про похибку відновлення функції трьох змінних оператором інтерфлетації. Встановлено клас функцій, які описують внутрішню структуру тіла і які точно можуть бути відновлені за допомогою розробленого методу.

One of the most informative method is a tomography that gives more information about the subject being studied than other known diagnostic methods. Significantly improve the informativeness of data obtained in tomography allows the use of various methods for solving three-dimensional problems of computed tomography, allowing to examine individual parts of the object under study at an arbitrary angle. The article proposed a method of restoration of the internal structure of a three-dimensional body, which uses four tomograms. Experimental data is the equation of four planes. Without limiting generality, planes perpendicular to two axis axes, such as the axis Ox and Oy . Input data is also obtained from a computer tomography tomogram, which lie on given planes. In this paper, the definition of the trace of the function of three variables on a plane and the determination of tomograms in the mathematical sense are given as functions of two variables. In the work, the problem is solved, which is to construct an operator that will restore the investigated object by the specified input data and give a general view of functions that will be restored by a precisely constructed operator. The method is constructed by introducing the operator of interflatation of functions of three variables. The indicated operator restores (possibly, approximately) the function of three variables at points between the given planes using its traces on these planes (tomograms). Theorems on the error of recovery of a function of three variables by the operator of interflatation are proved. The class of functions describing the internal structure of the body and which can be accurately restored using the developed method is established. Subsequently, the authors plan to explore the possibility of applying the results in practice to identify the internal structure of objects using only four tomograms. This method can be used for solving problems in a small-angle computed tomography.