Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography

Abstract

The operators of longitudinal and transverse ray transforms acting on vector fields on the unit disc are considered in the paper. The goal is to construct SVD-decompositions of the operators and invert them approximately by means of truncated decomposition for the parallel scheme of data acquisition. The orthogonal bases in the initial spaces and the image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions inversion formulas are derived and the polynomial approximations for the inverse operators are obtained. Numerical tests for data sets with different noise levels of smooth and discontinuous fields show the validity of the approach for the reconstruction of solenoidal or potential parts of vector fields from their ray transforms.

Solving inverse problems using data-driven models

“…More precisely, (2.1) is weakly ill-posed if decays with polynomial rate as and strongly ill-posed if the decay is exponential: see Engl et al. (2000), Derevtsov, Efimov, Louis and Schuster (2011) and Louis (1989) for further details. As a final note, such classification is not possible when the forward operator is non-linear.…”

Range characterizations and Singular Value Decomposition of the geodesic X-ray transform on disks of constant curvature

“…Of crucial importance for practical purposes is the knowledge of the Singular Value Decomposition (SVD) of the operator I 0 , be it for truncation and regularization purposes [25,1], to understand the structure of 'ghosts' in the case of discrete data [13,14], or to seek lowdimensional ansatzes in the case of incomplete data [15,12]. Several results on the SVD of ray transforms have been obtained, mainly existing in the Euclidean case: on functions in [20,17,18,19,27,25], tensor fields in [10] and for the transverse ray transform in [4]. Other transforms on circularly-symmetric families of curves have extensively been studied, see e.g.…”

Reconstruction of potential part of 3D vector field by using singular value decomposition

“…Thus the tomography of vector fields arises for the description of vector characteristics of currents of fluids, vectors of electromagnetic fields inside the conductor in inhomogeneous media, and many others. [1] We consider here a method of solving the 3D vector tomography problem in the case of parallel scheme of observation. As the problem of scalar tomography consists in the inversion of the Radon transform for a function, the vector tomography problem is the problem of inversion of normal Radon transform operator applying to potential part of 3D vector fields.…”