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Magnetic particle imaging (MPI) is a new tomographic imaging method which is able to capture the fast dynamic behavior of magnetic tracer material. From measured induced signals, the unknown magnetic particle concentration is reconstructed using a previously determined system function, which describes the relation between particle position and signal response. After discretization, the system function is represented by a matrix, whose size can prohibit the use of direct solvers for matrix inversion to reconstruct the image. In this paper, we present a new reconstruction approach, which combines efficient compression techniques and iterative reconstruction solvers. The data compression is based on orthogonal transforms, which extract the most relevant information from the system function matrix by thresholding, such that any iterative solver is strongly accelerated. The effect of the compression with respect to memory requirements, computational complexity and image quality is investigated. With the proposed method, it is possible to achieve real-time reconstruction with almost no loss in image quality using measured 4D MPI data.

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The total least squares (TLS) method is a successful approach for linear problems if both the matrix and the right hand side are contaminated by some noise. In a recent paper Sima, Van Huffel and Golub suggested an iterative method for solving regularized TLS problems, where in each iteration step a quadratic eigenproblem has to be solved. In this paper we prove its global convergence, and we present an efficient implementation using an iterative projection method with thick updates.