Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fields

Abstract

The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree m in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.

Inversion of generalized Radon transforms acting on 3D vector and symmetric tensor fields

“…Theorem 8. For the fields (32) and (33) with the potentials φ ij ∈ S(R 3 ), i, j = 0, 1, 2, i ⩽ j the following equalities hold:…”

“…Theorem 9. Let the symmetric 2-tensor fields Φ kl , k, l = 0, 1, 2, k ⩽ l be defined by the formulas (32) and (33) with potentials φ kl ∈ S(R 3 ). Then for the field w(x) =…”

“…First of all, this allows one to obtain inversion formulas for operators acting on tensor fields using the well known formulas for inverting operators acting on functions [7,12,26]. The method of potentials for constructing basic fields is also used in the development of approaches and algorithms for numerical solution of 2D tensor fields reconstruction problems, for example, using the least squares method [40,42], the method of approximate inverse [10] and the singular value decomposition method [9,11,33]. In the study of ray transforms acting on 2D symmetric m-tensor fields, more detailed decompositions into orthogonal terms each of which is constructed using only one function were previously obtained [12].…”