An Augmented Lagrangian Method for MR Coil Sensitivity Estimation

ԡ ଶ ൅λԡ ܝ ૙ԡ ଶ ൅μν଴ 2 ⁄ԡ ܝ ૙ െ܀ ܝ ૚ െિ ૙ԡ ଶ ൅μνଵ 2 ⁄ԡ ܝ ૚ െܛെિ ૚ԡ ଶ , where η=[િ ૙ ,િ ૚ሿ are Lagrange multipliers, u=[ܛ, ܝ ૙ ,ܝ ૚ሿ , and μ,ν଴,νଵ ൐0 are scalar parameters that control the convergence properties. We use an alternating minimization scheme to solve the AL function resulting in the final estimation algorithm (see below). In step 2 of the algorithm, we avoid inverting a large matrix by approximating the update using a few iterations of the conjugate gradient (CG) method with a circulant pre-conditioner. Step 3 can be efficiently computed as the matrix requiring inversion is diagonal. We choose the AL parameters by first setting νଵ ൌ0 .05 and then selecting ν଴ and μ so that the condition numbers of the matrices in step 2 and step 3 are 120 and 10, respectively. These parameter values have worked well for several distinct datasets. To reduce the convergence time of the iterative methods, we initialize with an estimate based on polynomial fitting. We avoid computing the ratio z/b by posing the initialization problem as ી ൌa rgmin ી ԡ ۻሺܢ െ ۲۾ીሻ ԡ ଶ , where ۾ൌሾ ܘ ૚ ,ܘ ૛ ,…,ܘ ܖሿ is a matrix of 2D-Chebyshev polynomials of the first kind. The least squares solution to this problem is well conditioned due to the orthonormality of the Chebyshev polynomials.