Solution of the Integral Geometry Problem for 2-Tensor Fields by the Singular Value Decomposition Method

Abstract

We consider the integral geometry problem of finding a symmetric 2-tensor field in a unit disk provided that the ray transforms of this field are known. We construct singular value decompositions of the operators of longitudinal, transversal, and mixed ray transforms that are the integrals of projections of a field onto the line where they are computed. We essentially use the results on decomposition of tensor fields and their representation in terms of potentials. The singular value decompositions are constructive and can be used for creating an algorithm for recovering a tensor field from its known ray characteristics. Bibliography: 20 titles.

An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data

“…As we have seen in (2), that the TRT depends only on the projection normal to the direction of the ray. Now working in pη, ζ " ξˆη, ξq coordinates, let us consider, xJf pξ, xqζ, ζy, which can be transformed into xJf pξ, xqζ, ζy "…”

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

“…The appearance of the factor (1 − r 2 ) m is not accidental and is associated with the boundary conditions imposed on the potentials. Completeness of the system of polynomials (10) for m = 0 over B is a consequence of the completeness of both the Jacobi and harmonic polynomials [19]. Therefore, the system of potentials (10) is complete in the space H m 0 (B).…”

“…Completeness of the system of polynomials (10) for m = 0 over B is a consequence of the completeness of both the Jacobi and harmonic polynomials [19]. Therefore, the system of potentials (10) is complete in the space H m 0 (B). Note also that Φ 2jm 0n (x) ≡ 0, therefore these polynomials do not belong to the basis system of potentials.…”

“…The SV-decompositions have been constructed for the Radon transform operator [5,19,20,26,38] and for the operators of ray transforms of scalar [25], vector [8,9] and two-tensor [10] fields. In [14] an SV-decomposition for the operator of longitudinal ray transform of fan type acting on two-dimensional solenoidal tensor fields of arbitrary degree m was constructed.…”

“…The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original and image spaces are constructed using harmonic functions and classical orthogonal polynomials, as in [8,10]. We propose a new original way for proving orthogonality of basis fields in the initial space for m ⩾ 2, which allows to avoid time-consuming direct calculations (compare with the proof for m = 2 in [10]).…”