On solving the slice-by-slice three-dimensional 2-tensor tomography problems using the approximate inverse method

Abstract

A numerical solution of the problem of recovering the solenoidal part of a three-dimensional symmetric 2-tensor field using the incomplete tomographic data is proposed. The initial data of the problem are values of the ray transform for all straight lines, which are parallel to at least one of the planes from a finite set of planes. We consider two sets of planes, the number of planes in which are three and six. The recovery algorithms are based on the approximate inverse method.

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 been shown in other work [42], [55] that three orthogonal chosen directions are sufficient for reconstruction of a tensor field. It has also been shown [33], [34], [36] that three orthogonal axes are sufficient for a full recovery of a vector field from slice-by-slice vector field tomography. In [33] an efficient mollifier methodwas proposed for the threedimensional vector tomograph problem.…”

“…In [33] an efficient mollifier methodwas proposed for the threedimensional vector tomograph problem. The mollifier method originally proposed by Louis in 1990 [56] is seen throughout his group's work [35], [36], [44] and provides an approximate solution to a continuous inverse problem which one might see very similar to the determination of a regularized solution in the implementation of discrete Bayesian reconstruction methods. For the tensor tomography problem there isthe longitudinal projection in addition to a transverse projection [34], [43] needed for every projection angle to solve for the 6 unknown tensor elements, whereas only one longitudinal projection for each angle is required for the vector tomography problem.…”

“…The tensor tomographic problem is an extension of the vector tomographic problem [21]- [36] and draws on much of the work in the reconstruction of vector fields (first-rank tensor fields) [31], [35]; in particular, the decomposition of the tensor field into solenoidal and irrotational components [3], [37]- [40] and the extension of the Fourier projection theorem from scalar and vector fields [32], [35] to tensor fields [37]- [40]. This decomposition provides a formulation to analyze data acquisition schemes and reconstruction algorithms from the mathematical construction of projections that might simplify the data acquisition yet provide accurate and precise reconstruction results.…”

“…components, and three for off-diagonal components. 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.…”