Jing Cheng^{1}, Sen Jia^{1}, Haifeng Wang^{1}, and Dong Liang^{1}

The combination of Parallel imaging (PI) and compressed sensing (CS) allow high quality MR image reconstruction from partial k-space data. However, most CS-PI MRI methods suffer from detail loss with large acceleration and complicated parameter selection. In this work, we describe and evaluate an efficient and robust algorithm to overcome these limitations. The experimental results on in vivo data show that, the proposed method using a first-order primal-dual algorithm can successfully remove undersampling artifacts while keeping the details with little parameter tuning compared with the existing advanced method.

**INTRODUCTION **

**METHOD**

The reconstruction
of parallel MR imaging with TV regularization can be formulated as:$$\begin{cases}\min_{u}‖∇u‖_1 \\s.t. Eu=f \end{cases}(1)$$where $$$u$$$ is the image we need to reconstruct, $$$E$$$ denotes the sensitivity encoding matrix, $$$f$$$ is the data acquired with arbitrary k-space sampling patterns, and $$$∇$$$ is the gradient operator.The
Chambolle-Pock (CP) algorithm is a first-order primal-dual algorithm suitable
for non-smooth convex optimization problems, and has been proved convergence in
mathematics^{2}. It is usually applied to a general form of the primal
minimization:$$\min_{x}{F(Kx)+G(x)} (2)$$and solves it simultaneously
with its dual, which provides a robust convergence check – the duality gap. Therefore,
we can rewrite Eq. (1) as$$\min_u{‖∇u‖_1+\frac{λ}{2} ‖Eu-f‖_2^2 } (3)$$$$$λ$$$ is the regularization parameter, The
whole minimization objective can be written in the form of CP algorithm with
the following assignments:$$F(Kx)=‖∇x‖_1,G(x)=\frac{λ}{2} ‖Ex-f‖_2^2 (4)$$$$x=u,K=∇ (5)$$$$$G(x)$$$ is uniformly
convex, hence we can use the CP algorithm to solve the MR
reconstruction problem as follows:$$\begin{cases}y_{n+1}=prox_σ [F^* ](y_n+σ_n Ku ̅_n)=\frac{y_n+σ_n Ku ̅_n}{max(1,y_n+σ_n Ku ̅_n)}\\u_{n+1}=prox_τ [G](u_n-τ_n K^* y_{n+1})=E^{-1} [\frac{E(u_n-τ_n K^* y_{n+1} )+τ_n λf}{1+τ_n λ}]\\θ_n=1/sqrt{(1+2γτ_n) },τ_{n+1}=θ_n τ_n,σ_{n+1}=σ_n/θ_n \\u ̅_{n+1}=u_{n+1}+θ_n (u_{n+1}-u_n)\end{cases} (6)$$where $$$τ_0=σ_0=1/L, L$$$ is the $$$l_2-norm $$$ of the matrix $$$K$$$, $$$γ$$$ is the parameter related to function $$$G$$$, $$$F^*$$$is the convex conjugate of the function $$$F$$$ which can be computed by the Legendre transform^{3}, and the proximal mapping of the convex function $$$H(x)$$$ is obtained by the following minimization:$$prox_σ [H](x)=arg \min_{z}{\left\{H(z)+\frac{‖z-x‖_2^2}{2σ}\right\}} (7)$$The proximal mapping has a closed-form representation if the function $$$H$$$ is simple.There are no other
free parameters except the regularization parameter in Eq. (6). This is an important feature for our
purpose of CS-pMRI reconstruction.

**RESULTS**

**CONCLUSION**

1. Liang D, Liu B, Wang J, Ying L. Accelerating SENSE using compressed sensing. Magn Reson Med 2009; 62: 1574-1584.

2. Chambolle A, Pock T. A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imag. Vis. 2011; vol. 40:1–26.

3. Rockafellar, RT. Convex analysis. Princeton Univ. Press; 1970.

4. Huang F, Yin W, Lin W, Ye X, Guo W, Reykowski A. A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self-feeding sparse SENSE. Magn Reson Med 2010; 6(4):1078-1088.

Figure 1.
Comparison of reconstruction methods for R=4. Both SFSS and CP can reduce the
level of aliasing artifacts, whereas the SFSS method shows noiser than CP.

Figure 2.
Comparison of reconstruction methods for R=6. The second row are the respective
zoom-in images indicated by the red rectangular in the reference image. The CP
method contains more features than SFSS, especially at the location indicated
by the red arrows.

Figure
3. Reconstructions with different values of λ
. The number on the left top of each
subfigure is the value of λ. The
brain image on the last row is the reference form full data and the plots are
the root mean square error (RMSE) and the mean structural similarity index (mSSIM)
of the reconstruction versus the values of λ. The CP method has
wider range of proper parameter.