Numerical solvers based on the method of approximate inverse for 2D vector and 2-tensor tomography problems

Abstract

A problem of reconstruction of 2D vector or symmetric 2-tensor fields by their known ray transforms is considered. Two numerical approaches based on the method of approximate inverse are suggested for solving the problem. The first method allows to recover components of a vector or tensor field, and the second reconstructs its potentials in the sense of feature reconstruction, where the observation operator assigns to a field its potential. Numerical simulations show good results of reconstruction of the sought-for fields or their solenoidal or potential parts from its ray transforms.

An Analytical Algorithm for Tensor Tomography From Projections Acquired About Three Axes

“…Our work in this paper focused on developing a filtered back-projection algorithm for reconstructing longitudinal and transverse X-ray projections, in the same way reconstruction algorithms can be developed for Radon projections of second rank tensor fields [36], [40], [44], [53]- [55]. To illustrate the differences between X-ray and Radon projections, let x = (x, y, z) be a point in R 3 and let the components t i j (x) of a second rank symmetric tensor field T (x) be real, rapidly decreasing C ∞ functions defined on R 3 .…”

“…It has also been shown for slice-by-slice vector field tomography in [33], [34], [36] that three perpendicular axes are sufficient for a full recovery. Our approach is to separate the tensor field into solenoidal and irrotational components [37]- [40], [44] so that one set of three directional measurements around three axes reconstructs the solenoidal component of the tensor field; and the reconstructed solenoidal component along with a different set of three directional measurements about the same axes reconstructs the irrotational component.…”

The method of approximate inverse for ray transform operators on two-dimensional symmetric m-tensor fields

“…Первый подход позволяет восстановить тензорное поле покомпонентно, в то время как при использовании второго подхода восстанавливаются потенциалы соленоидальной и потенциальных частей. При m = 1, 2 результат получен в работах [29][30][31]. Доказательство.…”

“…Теоремы, доказанные в работе, представляют фундаментальный интерес и носят методологический характер. Полученные формулы для метода приближенного обращения являются существенным обобщением (на случай тензорных полей произвольной валентности) результатов, полученных для решения задач по восстановлению скалярных (m = 0) [25], векторных (m = 1) [29,30] и симметричных 2-тензорных (m = 2) [31] полей.…”

“…Предлагаемые в данной работе подходы являются обобщением результатов, ранее полученных авторами, для восстановления векторных (m = 1) и симметричных 2-тензорных (m = 2) полей [29][30][31].…”