Efficient, Convergent SENSE MRI Reconstruction for Nonperiodic Boundary Conditions via Tridiagonal Solvers
Undersampling is an effective method for reducing scan acquisition time for MRI. Strategies for accelerated MRI, such as parallel MRI and compressed sensing MRI present challenging image reconstruction problems with nondifferentiable cost functions and computationally demanding operations. Variable splitting (VS) can simplify implementation of difficult image reconstruction problems, such as the combination of parallel MRI and compressed sensing, CS-SENSE-MRI. Combined with augmented Lagrangian (AL) and alternating minimization strategies, variable splitting can yield iterative minimization algorithms with simpler auxiliary variable updates. However, arbitrary variable splitting schemes are not guaranteed to converge. Many variable splitting strategies are combined with periodic boundary conditions. The resultant circulant Hessians enable  $\mathcal {O} (n \log {} n)$ computation but may compromise image accuracy at the spatial boundaries. We propose two methods for CS-SENSE-MRI that use regularization with nonperiodic boundary conditions to prevent wrap-around artifacts. Each algorithm computes one of the resulting variable updates efficiently in $\mathcal {O}\left(n\right)$ time using a parallelizable tridiagonal solver. AL-tridiag is a VS method designed to enable efficient computation for nonperiodic boundary conditions. Another proposed algorithm, ADMM-tridiag, uses a similar VS scheme but also ensures convergence to a minimizer of the proposed cost function using the alternating direction method of multipliers (ADMM). AL-tridiag and ADMM-tridiag show speeds competitive with previous VS CS-SENSE-MRI reconstruction algorithm AL-P2. We also apply the tridiagonal VS approach to a simple image inpainting problem.