Compressed Sensing MRI via Extended Anisotropic and Isotropic Total Variation

Compressed sensing (CS) is a technique to reconstruct images from undersampling data, reducing the scanning time of magnetic resonance imaging (MRI). It utilizes the sparsity of images in some transform domains. Total variation (TV) has been applied to enforce sparsity. However, traditional TV based on the l ₁-norm is not the most direct way to induce sparsity, and it cannot offer a sufficiently sparse representation. Since the l _p -norm (0< p < 1) promotes the sparsity better than that of the l ₁-norm, we propose two extended TV algorithms based on the l _p -norm: anisotropic and isotropic total p-variation (TpV). Then we introduce them to the MRI reconstruction model. We apply the Bregman iteration technique to handle the proposed optimization problem. During the iteration, the p-shrinkage operator is employed to resolve the nonconvex problem caused by the l _p -norm. Experimental results illustrate that our algorithms could offer the higher SNR and lower relative error compared with traditional TV algorithms and high-degree TV (HDTV) algorithm in MRI reconstruction problem.