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 constru… Show more
“…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 "…”
We give an explicit plane-by-plane filtered back-projection reconstruction algorithm for the transverse ray transform of symmetric second rank tensor fields on Euclidean 3-space, using data from rotation about three orthogonal axes. We show that in the general case two axis data is insufficient but give an explicit reconstruction procedure for the potential case with two 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 "…”
We give an explicit plane-by-plane filtered back-projection reconstruction algorithm for the transverse ray transform of symmetric second rank tensor fields on Euclidean 3-space, using data from rotation about three orthogonal axes. We show that in the general case two axis data is insufficient but give an explicit reconstruction procedure for the potential case with two axis data.
“…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).…”
Section: Sv-decompositions Of the Ray Transform Operatorsmentioning
confidence: 99%
“…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.…”
Section: Sv-decompositions Of the Ray Transform Operatorsmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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]).…”
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.
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