We use a small positive parameter to change the totalvariation function for unconstrainedMRimage reconstruction to a strictly convex perturbed function. Bregman iteration is applied to solve the modified total-variation MR image (TVMRI) reconstruction problem. A lagged diffusivity fixed-point algorithm is applied to solve the minimization problem in the Bregman iteration. We use the periodic boundary condition and a Fourier transform to accelerate TVMRI reconstruction.RealMR images are used to test the approach in numerical experiments. The experimental results demonstrate that the proposed method is very efficient for TVMRI reconstruction.

The problem of compressive-sensing (CS) L2-L1-TV reconstruc- tion of magnetic resonance (MR) scans from undersampled k-space data has been addressed in numerous studies. However, the regularization parameters in models of CS L2-L1-TV reconstruction are rarely studied. Once the regu- larization parameters are given, the solution for an MR reconstruction model is xed and is less eective in the case of strong noise. To overcome this shortcoming, we present a new alternating formulation to replace the standard L2-L1-TV reconstruction model. A weighted-average alternating minimization method is proposed based on this new formulation and a convergence analysis of the method is carried out. The advantages of and the motivation for the pro- posed alternating formulation are explained. Experimental results demonstrate that the proposed formulation yields better reconstruction results in the case of strong noise and can improve image reconstruction via exible parameter slection.

We use a positive parameter to develop a dierentiable perturbed reconstruction model to solve the L1–L2 magnetic resonance image (MRI) reconstruction problem. We use Bregman iterative formulation to solve the dierentiable perturbed L1–L2 model, and lagged diusivity xed-point iteration to solve the minimization problem in the Bregman iteration. Two Fourier transforms and an inverse Fourier transform are used to accelerate L1–L2 MRI reconstruction. Real MR images are used to test the method in numerical experiments. The results demonstrate that the proposed method is very ecient for L1–L2MRI reconstruction.

Image restoration is a common problem in visual process. In this paper, a modified minimization model is presented, which combines the L1 and L2 fidelity terms with a combined quadratic L2 and TV regularizer just as the regularizer of Cai et al. (2013). The combined regularizer has the priorities of preserving desirable edges and ensuring several kinds of noises can be removed clearly. Split-Bregman algorithm is effi- ciently employed to solve this model and convergence analysis is also discussed. Moreover, we extend the proposed model and algorithm for image restoration involving blurry images and color images. Experimental results show that our proposed model and algorithm have good performance both in visual and ISNR values for different kinds of blurs and noises including mixed noise.

Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with $ {L^\infty }$ initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.