Master thesis presentation Ludvig Lindahl, MPPHS

Abstract: In small-angle X-ray scattering tensor tomograph, a reconstruction of the reciprocal
space map of a sample is performed using small-angle X-ray scattering measure-
ments, providing information about the local reciprocal space in the volume of the
sample. Previously, the technique spherical integral geometric tensor tomography
has been used to solve inverse problems of reconstruction using quasi-Newton meth-
ods. The technique uses real and squared complex spherical harmonics to represent
tensor fields, yielding respectively linear and non-linear formulations of inverse prob-
lems. Local representations have advantages, such as making the problem separable
and allowing for positivity constraints on the reconstructed spherical functions. In
this project, two local representations, one based on spherical pixels and one based
on radial basis functions, were implemented. The studied radial basis functions
were based on Gaussian, Matérn and Wendland functions. In addition, regular-
ization using the l1-norm, the l2-norm and the total variation were implemented.
After selecting hyperparameters for the various reconstruction methods using the
L-curve method and minimization of the residual, comparisons of the local represen-
tation methods and a linear formulation of the inverse problem using a real spherical
harmonic representation, SIGTT were carried out on simulated data. Studying the
correlation between the reconstructed reciprocal space map and the true coefficients,
the Gaussian radial basis function representation consistently attained higher cor-
relation values than the other local representations. Small increases in correlation
were seen under the application of regularization terms. Of the regularization terms
used, the total variation outperformed the l1-norm which outperformed the l2-norm.

Examiner: Paul Erhart
Supervisor: Leonard Nielsen
Opponent: Ebba Grönfors