Local Representations for Small-angle X-ray Scattering Tensor Tomography

Abstract

In small-angle X-ray scattering tensor tomography, a reconstruction of the reciprocal space map of a sample is performed using small-angle X-ray scattering measurements, providing information about the local reciprocal space in the volume of the sample. Previous methods of reconstruction have posed it as an inverse problem in linear and non-linear formulations using spherical harmonic representations of the reciprocal space map, solving it using quasi-Newton methods. Local representations have advantages, such as making the problem separable and allowing for positivity constraints on the reconstructed spherical functions. In this work, two local representations, one based on spherical pixels and one based on radial basis functions (including spherical splines), were implemented. The studied radial basis functions were based on Gaussian, Matérn and Wendland functions. In addition, regularization using the L1-norm, the L1-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 representation methods and an existing method using a real spherical harmonic representation were carried out on simulated data as well as a trabecular bone sample. Studying the correlation between the reconstructed reciprocal space map and the true coefficients, the radial basis function representations were shown to perform comparably to the real spherical harmonics representation. Increases in correlation were seen under the application of regularization terms. The correlation between neighboring voxels of the orientation of features increased under regularization, where total variation regularization was shown to perform similarly to L1-regularization.