Singular value decomposition of the longitudinal ray transform of vector fields in a ball in cone beam coordinates

Abstract

An analytical singular value decomposition (SVD) of the longitudinal cone-beam transform of solenoidal vector fields in the ball is proposed and described in detail. We exploit the solenoidal vector fields , and constructed by Derevtsov, Kazantsev and Schuster in (2007 J. Inverse Ill-Posed Problems 15 173–185). Our study technique also corresponds to the work Kazantsev (2015 J. Inverse Ill-Posed Problems 23 173–185) in which the scalar case was considered. The calculations use expansions in orthogonal bipolar spherical harmonics. Also the spherical convolution operator Hilbert type , coupling integrals of vector spherical harmonics and Clebsch–Gordan coefficients are involved in our study. The exact formulas for the corresponding singular values , and are obtained and their asymptotic for n → ∞ is studied.