Search

Phase contrast tomography is an alternative to classic absorption contrast tomography that leads to higher contrast reconstructions in many applications. We review how phase contrast data can be acquired by using a combination of phase and absorption gratings. Using algebraic reconstruction techniques the object can be reconstructed from the measured data. In order to solve the resulting linear system we propose the Generalized Bidiagonal Tikhonov (GBiT) method, an adaptation of the generalized Arnoldi-Tikhonov method that uses the bidiagonal decomposition of the matrix instead of the Arnoldi decomposition. We also study the effect of the finite difference operator in the model by examining the reconstructions with either a forward difference or a central difference approximation. We validate our conclusions with simulated and experimental data.

Classical iterative methods for tomographic reconstruction include the class of Algebraic Reconstruction Techniques (ART). Convergence of these stationary linear iterative methods is however notably slow. In this paper we propose the use of Krylov solvers for tomographic linear inversion problems. These advanced iterative methods feature fast convergence at the expense of a higher computational cost per iteration, causing them to be generally uncompetitive without the inclusion of a suitable preconditioner. Combining elements from standard multigrid (MG) solvers and the theory of wavelets, a novel wavelet-based multi-level (WMG) preconditioner is introduced, which is shown to significantly speed-up Krylov convergence. The performance of the WMG-preconditioned Krylov method is analyzed through a spectral analysis, and the approach is compared to existing methods like the classical Simultaneous Iterative Reconstruction Technique (SIRT) and unpreconditioned Krylov methods on a 2D tomographic benchmark problem. Numerical experiments are promising, showing the method to be competitive with the classical Algebraic Reconstruction Techniques in terms of convergence speed and overall performance (CPU time) as well as precision of the reconstruction., Comment: Journal of Computational and Applied Mathematics (2014), 26 pages, 13 figures, 3 tables