A recovery algorithm based on the kaczmarz algorithm and admm splitting with application to convex optimization in magnetic particle imaging

Abstract

This work introduces a strategy for the extension of the standard Kaczmarz algorithm, which is popular for solving very large inverse problems, to priors other than the commonly used Tikhonov regularization. The proposed reformulation of the algorithm allows us to include more sophisticated priors while inheriting the row-wise operation structure of the classical Kaczmarz algorithm. The new method is developed with help of the alternating direction method of multipliers. The results show that also with suboptimal alternating direction method of multiplier steps, the proposed algorithm is able to solve convex optimization problems with very high accuracy. Especially, on the relative young preclinical medical imaging modality of magnetic particle imaging, the algorithm demonstrates high convergence rates. When the underlying matrix nearly shows mutually orthogonal rows, which is observed in the field of magnetic particle imaging, very high convergence rates can be expected.