Comparison of L1-norm surrogate functions used for the recovery of MR images

Applications of compressive sensing (CS) theory involve recovering sparse signals by solving ℓ
1
norm regularized objective functions. Due to discontinuity of ℓ
1
-norm, usage of gradient based algorithm for reconstruction of sparse signals is not possible. Different smooth surrogate functions have been used to approximate ℓ
1
norm. This article presents a performance comparison of two such surrogate functions. For comparison transformed domain MR images are recovered using smooth l
1
-norm approximations. Convergence rate of non linear conjugate gradient algorithm with backtracking line search is investigated for the two surrogate functions. Structural similarity index (SSIM) for different acceleration rates, Gaussian noise levels in MR images and root mean square error (RMSE) for recovered MR images is calculated. Simulation results show the improved convergence rate, low RMSE and better SSIM for different sampling factors as well as for different Gaussian noise levels for hyperbolic tangent as a surrogate function for l
1
-norm.