Accelerating CS in Parallel Imaging Reconstructions Using an Efficient and Effective Circulant Preconditioner

Purpose: Design of a preconditioner for fast and efficient parallel imaging and compressed sensing reconstructions. Theory: Parallel imaging and compressed sensing reconstructions become time consuming when the problem size or the number of coils is large, due to the large linear system of equations that has to be solved in l1 and l2-norm based reconstruction algorithms. Such linear systems can be solved efficiently using effective preconditioning techniques. Methods: In this paper we construct such a preconditioner by approximating the system matrix of the linear system, which comprises the data fidelity and includes total variation and wavelet regularization, by a matrix with the assumption that is a block circulant matrix with circulant blocks. Due to its circulant structure, the preconditioner can be constructed quickly and its inverse can be evaluated fast using only two fast Fourier transformations. We test the performance of the preconditioner for the conjugate gradient method as the linear solver, integrated into the Split Bregman algorithm. Results: The designed circulant preconditioner reduces the number of iterations required in the conjugate gradient method by almost a factor of 5. The speed up results in a total acceleration factor of approximately 2.5 for the entire reconstruction algorithm when implemented in MATLAB, while the initialization time of the preconditioner is negligible. Conclusion: The proposed preconditioner reduces the reconstruction time for parallel imaging and compressed sensing in a Split Bregman implementation and can easily handle large systems since it is Fourier-based, allowing for efficient computations.