High accuracy reconstruction algorithm for CS-MRI using SDMM

We propose a high accuracy algorithm for compressed sensing magnetic resonance imaging (CS-MRI) using a convex optimization technique. Lustig et al. proposed CS-MRI technique based on the minimization of a cost function defined by the sum of the data fidelity term, the 11-norm of sparsifying transform coefficients, and a total variation (TV). This function is not differentiable because of both l₁-norm and TV. Hence, they used approximations of the non-differentiable terms and a nonlinear conjugate gradient algorithm was applied to minimize the approximated cost function. The obtained solution was also an approximated one, thus of low-quality. In this paper, we propose an algorithm that obtains the exact solution based on the simultaneous direction method of multipliers (SDMM), which is one of the convex optimization techniques. A simple application of SDMM to CS-MRI cannot be implemented because the transformation matrix size is proportional to the square of the image size. We solve this problem using eigenvalue decompositions. Simulations using real MR images show that the proposed algorithm outperforms the conventional one regardless of compression ratio and random sensing patterns.