Orthonormal expansion ℓ ₁-minimization for compressed sensing in MRI

Compressed sensing (CS) enables the reconstruction of MR images from highly under-sampled k-space data via a constrained ℓ ₁-minimization problem. However, existing convex optimization techniques to solve such a constrained optimization problem suffer from slow convergence rate when dealing with data of a large size. On the other hand, many iterative thresholding techniques improve the convergence rate but at the cost of accuracy. In this work, we present a new iterative optimization technique to efficiently solve the constrained ℓ ₁ optimization without compromising the accuracy of the solution. The key idea is to expand the sensing matrix into an orthonormal matrix, which casts the ℓ ₁ constrained optimization into an equivalent convex optimization problem that can be exactly solved by the joint application of augmented Lagrange multipliers (ALM) method and alternating direction method (ADM). The proposed algorithm, dubbed as One - ℓ ₁, provides much faster convergence rate without compromising the reconstruction accuracy, when compared with commonly used optimization techniques, such as nonlinear conjugate gradient (NCG) method, as demonstrated with both phantom and in-vivo MR experiments.