Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

Abstract

In this article we study the fan-beam Radon transform Dm of symmetrical solenoidal 2D tensor fields of arbitrary rank m in a unit disc D as the operator, acting from the object space L2(D ; Sm) to the data space L2([0, 2π) × [0, 2π)). The orthogonal polynomial basis s (±m) n,k of solenoidal tensor fields on the disc D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm was obtained. The inversion formula for the fan-beam tensor transform Dm follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.

Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms

“…• As previously established in [9], if a vanishes nowhere, then both f and g, and thus V , can be reconstructed stably throughout the domain. In general, this also clarifies why f (or the potential part of V ) can only be recovered on the support of a.…”

On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

“…extended to h ∈ L 2 (D, so(m)) by continuity. The finite dimensional approximation spaces E D ⊂ C( D, so(m)) considered below are defined in terms of the SVD of I 0 , which is well known to consist of Zernike polynomials [29]. Postponing a full definition of E D to (6.7), we mention here that if…”

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

“…As the plane-byplane data is written in terms of scalar, vector and tensor longitudinal ray transforms, and the ranges of these operators can be determined in the plane case as a singular function expansion in a suitable Hilbert space. In fan beam coordinates the singular value decomposition of the ray transform is given by [3]. See also [1] and [2] for a parallel beam formulation.…”