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

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 H-n+1,m((n)), B-n+1-2k,m((n)) and C-n-2k,m((n)) 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 S, coupling integrals of vector spherical harmonics and Clebsch-Gordan coefficients are involved in our study. The exact formulas for the corresponding singular values sigma(H-n+1,m((n))), sigma(B-n+1-2k,m((n))) and sigma(C-n-2k,m((n))) are obtained and their asymptotic for n -> infinity is studied.