Dynamic Magnetic Resonance Imaging Reconstruction Based on Nonconvex Low-Rank Model

1School of Mathematics and Computing Science and Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China 2Guangxi Experiment Center of Information Science, Guilin, Guangxi 541004, China 3School of Computer Science and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China


Introduction
Image reconstruction is widely applied in the medical field, and most clinical diagnoses depend on computer hardware equipment; thus the improvement on image reconstruction algorithm has great significance.At present, among all kinds of detection methods, magnetic resonance imaging (MRI) and computed tomography (CT) are the most common and important ways for clinical diagnosis.When checking organs by MR, the final image is affected in varying degrees by artifacts, which results in degrading in quality and affects the diagnoses.Therefore, reconstruction algorithms with high quality and fast calculation have become one of the research focuses in this field.
In recent years, in order to ensure image quality and to speed up the pace of reconstruction, there emerge many improved methods, such as multicoil parallel imaging [1], keyhole imaging technology [2,3], unfold method [4], and - SENSE [5].After the theory of compressed sensing (CS) [6,7] was proposed, many scholars applied it to the dynamic MR reconstruction [5,8,9].These methods process the original data by downsampling method instead of traditional full-sampling and reduce the computing time.
Since the development of robust principal component analysis (RPCA) theory [10], breakthroughs have emerged in the field of MR image reconstruction.This theory suggests that images with spatiotemporal correlation can be decomposed into low-rank (LR) matrices and sparse matrices.Many researchers introduced this theory to reconstruct dynamic MR images to achieve better results.In [11], a method was proposed based on accelerated diffusion weighted sequences for image reconstruction by the  +  decomposition.In [12], 3D cardiac MRI was reconstructed by combining  +  decomposition with prior knowledge.In [13], a new  +  model was proposed, which decomposed a series of dynamic MR images with temporal and spatial correlation into LR matrices and sparse matrices.As a result, the dynamic MRI could be divided into the foreground and background to help diagnosis.In order to reconstruct dynamic MR images with better quality, based on the traditional  +  model, a rank-1 and sparse model was proposed in [14].
Recently, a lot of researchers focused on the problem of  0 norm approximation [15][16][17].In particular, in [15], Candés et al. pointed out that a nonconvex problem can be transformed into a convex optimization problem using the  1 norm instead of  0 norm.However, the  1 norm and  0 norm suggested that if the nuclear norm of the matrix is regarded as a  1 norm and the rank of the matrix is regarded as a  0 norm, the error between them can not be ignored in [18].The algorithm in [19] achieved good approaching effect by introducing a nonconvex regularization which was like a   norm.In [20], Quach et al. proposed a nonconvex online RPCA model which enhanced the sparsity by minimization of the   norm and achieved good results in many fields, such as face modeling and online background removal.Inspired by the above discussion, combining RPCA theory with a nonconvex idea, we propose a new MR image reconstruction model.The contributions of this paper are as follows: (1) in order to better approximate  0 norm, we introduce a nonconvex regularization using the   norm instead of the  1 norm.(2) The alternating direction method of multipliers (ADMM) is used to solve the nonconvex problem.In addition, we point out that the solution is convergent.The framework of this proposed method is shown in Figure 1.
The rest of this paper is organized as follows: in Section 2, a review of the  +  model is presented.In Section 3, we firstly introduce the proposed model and then provide the detailed implementation of the proposed model.Finally, we illustrate the convergence of the proposed algorithm.In Section 4, in order to verify the effectiveness of our algorithm, comparisons between our method and the state-of-the-art algorithms are provided.In Section 5, we conclude our works.

Related Works
According to the theory of RPCA [10], we focus on the model min where  is the observed data,  and  are the LR and the sparse matrix, respectively, , , and  ∈  × , and  denotes the regularization parameter.Since (1) is a nonconvex optimization problem, the effective solution of the problem is not easy to obtain [21].
According to the theory of RPCA, Otazo et al. [13] used the nuclear norm of  and the  1 norm of  instead of the rank of  and the  0 norm of  in (1), respectively, and proposed a convex optimization problem ( +  model): where ‖ ⋅ ‖ * is the nuclear norm.‖ ⋅ ‖ 1 denotes  1 norm, which is the sum of absolute values of the elements in matrix .
Let the data of an image correspond to the data of the - space by introducing an encoding operator .And assume  to be the sparse transformation of ; that is, the dynamic component  can be expressed by the known sparse basis .So the minimization problem (2) can be written as min Singular Value Thresholding (SVT) algorithm [15] is introduced to solve the minimization problem (3).

The Proposed Model
3.1.Modeling.Although the above methods have good theoretical guarantee and have made major breakthroughs in dynamic MR image reconstruction,  1 norm can not well approximate  0 norm [15,16,20].In addition, authors of [20,22] pointed out that, among common nonconvex functions, such as -norm [23], SCAD [24], log [15], MCP [25], ETP [26], and Geman [27], -norm is the best approximation to  0 norm.Inspired by this, we propose a new model as follows: where  is a LR matrix,  is a sparse matrix, and ‖ ⋅ ‖  denotes   norm. is an undersampled observed data in the - space and (+) ∈  × denotes the ideal reconstructed MR image, where  ×  denotes the size of the image. is an encoding operator, which makes the data of an image correspond to the data of the - space.
Combining the properties of supergradient for nonsmooth points with Taylor expansion [20], the second term of ( 5) can be approximated as where  is the number of iterations.
Algorithm 1: Dynamic MRI reconstruction algorithm based on nonconvex low-rank model.
Finally, the parameters  and  are updated as follows: where  > 1.The complete process of this algorithm is shown in Algorithm 1.
According to [20], the above solving process is convergent.

Numerical Experimental Comparisons and Analysis
In order to evaluate the performance of the proposed model, we compare our algorithm with the - SENSE [5], XD-GRASP [9], and the LplusS [13].All experiments are run on a PC with Intel Core i5-4690 processor, 8 GB of memory, Win 7 64-bit operating system, and MATLAB 2014a.The quality of the reconstructed image is evaluated by the root mean square error (RMSE), which is defined as where  is the observation data and  and  are the LR matrix and sparse matrix for the reconstructed image, respectively.The smaller the RMSE is, the better the MR image is reconstructed.
The iteration stop indicator for all algorithms of this paper is linked to the relative error (Err), which is defined as The iteration stops when the Err reduction of the consecutive iterations is less than a predefined threshold.For  norm (0 <  < 1), the choice of  heavily influences the convergence rate.Figure 2 shows the convergence rate with  = 0.1, 0.4, and 0.8.The greater  results in the faster convergence rate.A large number of experiments also prove that the smaller  leads to the smaller Err and the more precise solution.Through repeated trials, we set  = 0.2 in all experiments for our algorithm.

Comparisons and Analysis.
In this section, we compare our method with the - SENSE, XD-GRASP, and LplusS algorithm in subjective and objective aspects.Figures 4-9 present the visual comparisons, and the differences between the reconstructions by four algorithms are depicted with white arrows.The comparisons on RMSE, Err, and time consumption are listed in Tables 1 and 2, where the boldfaced numbers are the best results of all.
Figures 4 and 5 are the reconstructed comparisons of the first frame and the fifth frame of dynamic cardiac perfusion, respectively.Figure 4(d) is the best reconstructed image of Figure 4.In Figures 4(a Figures 6 and 7 represent the reconstructions about the dynamic abdomen image with the second frame and the fifth frame, respectively.By comparing, the proposed algorithm is better than the other three algorithms in brightness and resolution.
Figure 8 shows the reconstructed images of the 20th frame of the cardiac cine.The reconstruction effects of all four algorithms have no obvious visual differences, but, through careful observation on the enlarged regions to which the white arrows point, the proposed algorithm performs better.
Figure 9 shows the reconstructed effects of images 1, 2, 4, and 6, respectively.In images 1 and 2, we find that the features of our algorithm are the most obvious among four compared methods, especially in the regions where the white arrows point.In images 4 and 6, the outline and details of the organ of the proposed algorithm are clearer than those of the other reference algorithms, and the result is brighter.
Tables 1 and 2 list the numerical results of the four algorithms.The proposed algorithm ranks the first on the Err and RMSE among the four comparison methods, which denotes that the reconstruction quality is the best.Of all experiments, the time consumption of our algorithm is the second best except for cardiac cine in Table 1, on which the proposed algorithm ranks the first.

Conclusions
In this paper, we propose a nonconvex model for reconstructing high-quality dynamic MR images.In the new model, based on the RPCA theory, the  1 norm is substituted by the   norm to approximate the  0 norm; thus the accuracy of the solution is improved.An alternate iteration method is applied to solve the proposed model.Experimental results show that the proposed algorithm has advantages in terms of visual effect, RSME, and Err over three reference algorithms.

Figure 1 :
Figure 1: Framework of the proposed dynamic MRI reconstruction.

Figure 2 :
Figure 2: The relationship between the function values and the parameter .

Figure 3 :Figure 4 :
Figure 3: The 5th frame of each data set.(a)-(f) are named as image 1 to image 6.

Figure 9 :
Figure 9: Reconstruction comparisons on four images.From top to bottom are images 1, 2, 4, and 6, respectively.From left to right are - SENSE, XD-GRASP, LplusS, and our algorithm, respectively.

Table 1 :
Comparison on object indexes.But, in terms of time consumption, the proposed algorithm is slightly inferior to LplusS, which requires further improvement.

Table 2 :
Comparison on object indexes.