An Adaptive Algorithm for Compressively Sampled MR Image Reconstruction Using Projections onto $l_{p}$-Ball

Abstract

Compressed sensing (CS) is an emerging technique for magnetic resonance imaging (MRI) reconstruction from randomly under-sampled k-space data. CS utilizes the reconstruction of MR images in the transform domain using any non-linear recovery algorithm. The missing data in the $k$-space are conventionally estimated based on the minimization of the objective function using $l_{1} - l_{2}$ norms. In this paper, we propose a new CS-MRI approach called tangent-vector-based gradient algorithm for the reconstruction of compressively under-sampled MR images. The proposed method utilizes a unit-norm constraint adaptive algorithm for compressively sampled data. This algorithm has a simple design and has shown good convergence behavior. A comparison between the proposed algorithm and conjugate gradient (CG) is discussed. Quantitative analyses in terms of artifact power, normalized mean square error and peak signal-to-noise ratio are provided to illustrate the effectiveness of the proposed algorithm. In essence, the proposed algorithm improves the minimization of the quadratic cost function by imposing a sparsity inducing $l_{p}$-norm constraint. The results show that the proposed algorithm exploits the sparsity in the acquired under-sampled MRI data effectively and exhibits improved reconstruction results both qualitatively and quantitatively as compared to CG.