012).3 45%1%5)1-5) *'%-)651 7,1(%-.'2 -) 851).-& !#()5)& 92$1-)1 … · 2019-11-16 · (TV) norm and L1 norm or three regularizations consisting of total variation, L1 norm and wavelet

CHINESE J. BIOMED. ENG. VOL.23 NO.3, SEP. 2014CHINESE J. BIOMED. ENG. VOL.23 NO.3, SEP. 2014

Research on Split Augmented LargrangianShrinkage Algorithm in Magnetic Resonance Imaging

Based on Compressed Sensing

ZHENG Qing-bin, DONG En-qing, YANG Pei, LIU Wei, JIA Da-yu, SUN Hua-kui

School of Mechanical, Electrical and Information Engineering, Shandong University, Weihai 264209, China

Abstract. This paper aims to meet the requirements of reducing the scanning time of

magnetic resonance imaging (MRI), accelerating MRI and reconstructing a high quality

image from less acquisition data as much as possible. MRI method based on compressed

sensing (CS) with multiple regularizations (two regularizations including total variation

(TV) norm and L1 norm or three regularizations consisting of total variation, L1 norm and

wavelet tree structure) is proposed in this paper, which is implemented by applying split

augmented lagrangian shrinkage algorithm (SALSA). To solve magnetic resonance image

reconstruction problems with linear combinations of total variation and L1 norm, we

utilized composite split denoising (CSD) to split the original complex problem into TV

norm and L1 norm regularization subproblems which were simple and easy to be solved

respectively in this paper. The reconstructed image was obtained from the weighted

average of solutions from two subproblems in an iterative framework. Because each of the

splitted subproblems can be regarded as MRI model based on CS with single

regularization, and for solving the kind of model, split augmented lagrange algorithm has

advantage over existing fast algorithm such as fast iterative shrinkage thresholding(FIST)

and two step iterative shrinkage thresholding (TwIST) in convergence speed. Therefore,

we proposed to adopt SALSA to solve the subproblems. Moreover, in order to solve

magnetic resonance image reconstruction problems with linear combinations of total

variation, L1 norm and wavelet tree structure, we can split the original problem into three

subproblems in the same manner, which can be processed by existing iteration scheme.

A great deal of experimental results show that the proposed methods can effectively

reconstruct the original image. Compared with existing algorithms such as TVCMRI,

RecPF, CSA, FCSA and WaTMRI, the proposed methods have greatly improved the

quality of the reconstructed images and have better visual effect.

Key words: magnetic resonance imaging (MRI); compressed sensing (CS); split

augmented lagrangian; total variation(TV) norm; L1 norm

CLC number: TP 391 Document code:A Article ID: 1004-0552(2014)03-0108-13Grant sponsor: Natural Science Foundation of China; grant number: 81371635; grant sponsor: Research Fund for the Doctoral Program of Higher Education ofChina;grant number:20120131110062; grant sponsor:Shandong Province Science and Technology Development Plan;grant number:2013GGX10104Corresponding author: DONG En-qing. E-mail:[email protected] 10 June 2014; revised 16 September 2014

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Chinese Journal of Biomedical Engineering (English Edition) Volume 23 Number 3, September 2014Chinese Journal of Biomedical Engineering (English Edition) Volume 23 Number 3, September 2014

INTRODUCTION

Magnetic resonance (MR) imaging has been widely used in medical diagnosis because of its non-radiation injury, non -contrast agent, multi -parameter control and excellent depiction of soft tissuechanges. Recent development in compressed censing theory[1-2] show that it is possible to accuratelyreconstruct MR images from highly under -sampled K -space data, and thus significantly reducingscanning duration.

Firstly, Lustig et al.[3] proposed the pioneering work for MR images reconstruction. Their methodcan effectively reconstruct MR images with only 20% sampling. The improved results were obtained byan objective function with both a wavelet transform and a discrete gradient, which is formulated asEquation(1):

x赞=argminx

12 Rx-b

2

2 +α x TV+β Φx 1� � (1)

where α and β are two positive parameters, x is a MR image, R is partial Fourier transform, b is the

under-sampled measurements of K-space data of x, i.e., and x赞 is the estimated value of x. Φ is awavelet transform matrix. Equation (1) is based on the fact that MR images of organs can be sparselyrepresented by wavelet basis ( 准x 1, L1) and should have small total variation ( x TV, TV).Since both L1 and TV norm regularization terms are non-smooth, Equation (1) is very difficult to besolved directly.

To solve the problem of Equation(1), Conjugate Gradient [3] (CG) method was proposed by Lustig.However, it is very slow and impractical for real MR images. Then, Ma[4] proposed an operator-splittingalgorithm (compressed MRI reconstruction based on total variation, TVCMRI) to attack the MR imagereconstruction problem. But more practices show that the effectiveness of the algorithm is poor. Toimprove the quality of the reconstructed image, Yang [5] proposed a variable splitting method(reconstruction from partial Fourier data, RecPF). RecPF introduces an auxiliary variable to convert theoriginal unconstrained convex optimization problem into a constrained convex optimization problem.The converted problem can be solved by a classical alternating method, so a good reconstructed imagecan be obtained effectively. Recently, Huang [6] proposed composite splitting algorithm (CSA) and fastcomposite splitting algorithm (FCSA) which solve Equation (1) by using fast iterative shrinkage-thresholding algorithm (FISTA)[7]. CSA and FCSA are the fastest MR image reconstruction method so far.

In addition, based on CS theory, Huang et al. [8] proposed a new MR imaging model which isdenoted as Equation (2) and has three regularizations items.

x=argminx

12 Rx-b

2

2 +α x｜TV+β ｜准x 1+g∈GΣ 准xg 2Σ Σ� � (2)

whereg∈GΣ 准xg 2 utilizes the benefit of wavelet tree structure. G denotes a set of all parent-child

groups and g is an element of G. The model reasonably represents the relationship among waveletsparsity, gradient sparsity and tree sparsity seamlessly. Huang [8] proposed a new algorithm called

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WaTMRI (Wavelet Tree Sparsity MRI) to solve the Equation (2).However, the existing algorithms have much space for the improvement in image quality and

imaging speed. It is still an open problem in compressed MR image reconstruction for us to developefficient algorithms to solve Equation (1) and Equation (2) with nearly optimal reconstruction accuracy.

Composite CSD[6] can split the complex Equation (1) into two easy subproblems. And SALSA wasproposed in literature [9], which was more faster than FIST and TwIST [10] to solve the MR imagereconstruction problem with wavelet sparsity or gradient sparsity.

Therefore, we proposed two algorithms to solve the two problems in this paper. One of them is tosolve the Equation (1). We split the Equation (1) into TV norm and L1 norm regularization subproblemsrespectively, which are simple and easy to be solved by SALSA. And the reconstructed image isobtained from the weighted average of solutions from two subproblems in a framework. We named thealgorithm as SALSA-2R-CSMRI (apply SALSA to solve CS-MRI with two Regularization).

Another algorithm is to attack Equation (2). We split the original problem into three subproblems(L1 norm, TV norm and wavelet tree structure regularization). We utilized SALSA to solve TV normand L1 norm regularization subproblems. At the same time, wavelet tree structure can be effectivelysolved by existing iterative techniques. We named the algorithm as SALSA-3R-CSMRI (apply SALSAto solve CS-MRI with three Regularizations).

SALSA

In this section, for better illustrating our proposed methods, we need briefly review SALSA.SALSA considers minimizing the following problem of Equation (3):

x赞=argminx

12 Rx-b

2

2 +γφ(x� �) (3)

where φ (x) is a regular function, R is the under-sampled measurement, γ is a positive parameter.Normally, φ(x)= Φx 1 or φ(x)= x TV,where Φ is a wavelet transform matrix.

This non-constrained optimization formulation can be equivalently transformed into the followingconstrained optimization problem as Equation (4).

minx∈Rn,y∈Rn

12 Rx-b

2

2 +γφ(y� �) , subject to y=x (4)

Table 1 is the computational process of SALSA for solving the Equation(4). Thus,SALSA can solvethe non-constrained optimization Equation (3). The efficiency of SALSA highly depends on being ableto quickly solving its step ② and step ③.

The solving process of step ② is as follows:

Set F(x)= 12 Rx-b

2

2 +μ2 x-yk-dk and then

坠F(x)坠x =RT(Rx-b)+μ(x-yk-dk)

=(RTR+μ)x-(RHb+μ(yk+dk)),

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Let 坠F(x)坠x =0, we can get

xk+1=(RTR+μ)-1(RHb+μ(yk+dk)).The solving process of step ③ is as follows:For φ(y)= y 1, then

fsoft (y, μ)=argminyγ y 1+ μ

2 xk+1-y-dk2

2 ,

where fsoft (y, μ) is a soft threshold,

yk+1=sign(xk+1-dk)·max xk+1-dk - γμ ,� �0 .

For φ(y)= y TV, we can use Chambolle[11] to compute with computational cost O(n).Table 1 SALSA

THE PROPOSED IMAGE RECONSTRUCTION ALGORITHM

SALSA-2R-CSMRIFrom the above, the SALSA can rapidly solve the L1 regularization problem and the TV

regularization problem respectively. However, the SALSA cannot efficiently solve the compositeEquation (1) with L1 and TV regularizations, this reason is that there is no efficient algorithm to solvethe following Equation (5).

yk+1∈argminyα y TV+β Φy 1+ μ2 xk+1-y-dk

2

2 (5)

To solve Equation (1), the key problem is to develop an efficient algorithm to solve Equation (5).In the following section, we will give a scheme based on composite splitting techniques to solveEquation (5).

Before giving the new proposed algorithm, we need introduce CSD algorithm. CSD is a specialcase [12], which is an efficient algorithm for solving the following composite denoising problem asEquation (6).

① Set k=0 , choose μ>0 , y0 and d0 ;

② xk+1∈argminx

12

Rx-b2

2 + μ2

x-yk-dk2

2 ;

③ yk+1∈argminyγφ(y)+ μ

2xk+1-y-dk

2

2 ;

④ dk+1 =dk-(xk+1-yk+1);

⑤ k←k+1 ;

⑥ Until stopping criterion is satisfied.

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x赞=arg minx∈Rn

12 x-xg

2

2 +m

i=1Σgi (Bi xΣ Σ) (6)

where xg is a noisy signal, gi (.) is a regularization, Bi is a transform matrix and m is the number ofthe regularizations.

The main ideas of CSD are：① splitting variable into multiple variables (xi)i=1,…,m；② performingoperator splitting over each independently；③ obtaining the solution by linear combination of (xi)i=1,…, m.Table 2 describes the procedure of the algorithm in details.

Table 2 CSD

With CSD and SALSA, we proposed a new algorithm (SALSA -2R -CSMRI) for MR imagereconstruction as Equation (1). In practice, we find that few iteration numbers in CSD is enough forSALSA-2R-CSMRI to obtain good reconstruction results. Especially, the iteration number is set as 1in our algorithm. Numerous experimental results in the next section will show that this setting is goodenough for real MR image reconstruction.

Table 3 outlines the proposed SALSA-2R-CSMRI. SALSA-2R-CSMRI efficiently solves the MRimage reconstruction as Equation (1) and obtains better reconstruction results in the reconstructionaccuracy. Compared with all previous methods for compressed MR image reconstruction, experimentalresults will demonstrate its superior performance.

① Initialization {ω0,i}i=1,…, m = xg , k=1

② Compute

xk，i=argminx∈Rn

12

x-ωk-1,i2

2 +gi(Bix)

where i=1,…,m;

③ xk= 1m

m

i=1Σxk, i ;

④ Update ωj,i ,ωk,i =ωk-1,i+xk-xk-1,i

where i=1,…,m;

⑤ Update k=k+1 ;


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① Set {ω1,i}i=1,2 =RTb, y1,1, y1,2, d1,1, d1,2, k=1,α and β;② Compute

xk+1，1 =argminx

12

Rx-ωk,12

2 + μ2

x-yk,1-dk,12

2

yk+1，1 =argminy

2α y TV + μ2

xk+1,1-y-dk,22

2

xk+1，2 =argminx

12

Rx-ωk,22

2 + μ2

Φx-yk,2-dk,12

2

yk+1，2 =argminy

2β y 1+μ2

Φxk+1，2-y-dk,22

2

③ The k iteration of the reconstructed image,

xk+1=12

(xk+1，1+xk+1，2) ;

Table 3 SALSA-2R-CSMRI

SALSA-3R-CSMRIIn this section, we will give an algorithm (SALSA-3R-CSMRI) to solve Equation (2). Since Equation(2)

is an overlapping convex optimization equation, it is very hard to solve the equation directly.

Firstly, we introduce a variable z to constrain x with overlapping structure, where, z= zT

gi ,…, zT

gs� �T,

�n =Σsi=1 ni>d, z∈R�n , ni is the number of the elements of the group i , s is the total number of groups

and d are coefficients of image x in wavelet bases Φ. gi indicates the group i.Let zgi =(Φx)gi, then z=AΦx, where A∈R�n ×d, is an operator matrix in which elements are 0 or 1.

Then, Equation (2) can be written as Equation (7):

(x赞,z赞)=argminx,z

12 Rx-b

2

2+α x TV+β Φx 1+s

i=1Σ zgi 2∈ ∈+λ2 z-AΦx

2

22 2 (7)

Obviously, Equation (7), which is a non-overlapping structure about (Φx)gi, is optimization problemon variable x and z. We can solve Equation (7) by alternating minimization with respect to x and z. So wecan decompose the original optimization Equation (7) into x-subproblem and z-subproblem.

1) z-subproblemWhen given x, we can get an optimization equation from Equation (7) on z.

④ Update dk+1，1 , dk+1，2 , ωk+1，1 and ωk+1，2 ,

dk+1，1 =dk，1-(xk+1-yk+1，1) ;

dk+1，2 =dk，2-(Φxk+1-yk+1，2) ;

ωk+1，1 =ωk，1 +xk+1-xk+1，1;

ωk+1，2 =ωk，2 +xk+1-xk+1，2;

⑤ k←k+1;


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arg minz

βs

i=1Σ zgi 2Σ Σ x-xg

2

2 +λ2 z-AΦx

2

22 2 (8)

Obviously, Equation (8) is equivalent to Equation (9).

arg minz

s

i=1Σ β zgi 2+

λ2 z-(AΦx)gi

2

22 Σ2 2 (9)

Then, z-subproblem can be written as Equation (10):

arg minzgi

β zgi 2+λ2 zgi-(AΦx)gi

2

22 2, i=1,2,…,s (10)

We can get an approximate form solution by soft thresholding for each as Equation (11):

zgi=max ri 2 -βλ ,2 Σ0 · ri

ri 2

, i=1,2,…,s (11)

where ri=(AΦx)gi.2) x-subproblemFor the following x-subproblem as Equation (12),

arg minx

12 Rx-b

2

2+α x TV+β φx 1+λ2 z-AΦx2

22 2 (12)

Let f(x)= 12 Rx-b

2

2+λ2 z-AΦx

2

2 , where f(x) is a convex and smooth function. α x TV and

β φx 1 are convex but non-smooth. Then,this x-subproblem can be solved efficiently by SALSA-2R-CSMRI.

Now, we can summarize SALSA-2R-CSMRI as shown in Table 4.Table 4 SALSA-3R-CSMRI

①Set y1 ,1, y1,2, d1,1, d1,2, α, β, λ and {ω1,i}i=1,2 =x1 ;

②zgi=max ri 2 -βλ ,2 Σ0 · ri

ri 2

, i=1,2,…,s;

③ z= zT

gi ,…, zT

gs2 2T;

④ xk+1，1 =argminx

12

Rx-ωk,12

2 +λ2

z-AΦx2

2+ μ2

x-yk,1-dk,12

2 ;

yk+1，1 =argminy

2α y TV + μ2

xk+1,1-y-dk,12

2 ;

xk+1，2 =argminx

12

Rx-ωk,22

2 +λ2

z-AΦx2

2+ μ2

φx-yk,2-dk,22

2 ;

yk+1，2 =argminy

2β y 1+μ2

Φxk+1，2-y-dk,22

2 ;

⑤ xk+1=12

(xk+1，1+xk+1，2) ;

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EXPERIMENTS ON SALSA-2R-CSMRI

Experimental setupIn order to verify the effectiveness of SALSA-2R-CSMRI, we compare the SALSA-2R-CSMRI

with TVCMRI[4], RecPF[5], CSA and FCSA[6]. CSA and FCSA are the fastest MR reconstruction methods.And we download the source codes from their websites for existing algorithms[4-6].

Test original MR images are 2D Brain and Cardiac MR images respectively as shown in Fig.1.The images are the same size of 256×256. The sample ratio is set to be approximately 20%.

All experiments were performed using Matlab 7.10.0 for Windows 7. For fair comparisons, wecarefully abided by the following experiment setup. The observation measurement b is synthesized as b=Rx+N, where N is the Gaussian white noise with standard deviation δ=0.001. The regularization parameterα and β are set as 0.001 and 0.035, respectively. R and b are given as inputs, and x is the unknown target.

(a) Brain (b) CardiacFig.1 Original MR images

All methods run 50 iterations except that the SALSA-2R-CSMRI runs only 40 iterations due toits each iteration has high convergence.Experimental resultsVisual comparisons

Fig.2 and Fig.3 show the visual comparisons of the reconstructed results on brain and cardiac bydifferent methods. As shown in Fig.2 and Fig.3, although TVCMRI can reconstruct the elementarycontour of original images, the reconstructed images are blurry. RecPF is slightly better than theTVCMRI in the reconstruction image quality. CSA and FCSA are better than TVCMRI and RecPF invisual effect, but FCSA is better than CSA. Comparing with the above methods, SALSA-2R-CSMRIalways obtains the best visual effect and clear image details on all MR images.

⑥ dk+1，1 =dk，1-(xk+1-yk+1，1) ;

dk+1，2 =dk，2-(Φxk+1-yk+1，2) ;

ωk+1，1 =ωk，1 +xk+1-xk+1，1;

ωk+1，2 =ωk，2 +xk+1-xk+1，2;

⑦ k←k+1 ;

⑧ Until stopping criterion is satisfied.

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（a）Original （b）TVCMRI （c）RecPF

（d） CSA （e） FCSA （f）SALSA-2R-CSMRIFig.2 Visual comparisons of brain MR image reconstruction

（a）Original （b）TVCMRI （c）RecPF

（d） CSA （e） FCSA （f）SALSA-2R-CSMRIFig.3 Visual comparisons of cardiac MR image reconstruction

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Table 5 shows signal to noise ratio (SNR) of brain MR images reconstructed by different methods,which are derived from the average of 100 experiments.

SNR is defined as Equation (12)：

SNR=10×log10xtrue 2

2

x-xtrue 2

2

22222222222

22222222222

(12)

where xtrue is original image, is reconstructed image. From SNR formula, we know that the closer to theoriginal image the reconstructed image is, the higher SNR is. Therefore, it is reasonable for us to useSNR as objective evaluation for reconstruction algorithms.

Table 5 SNR of reconstructed images

As shown in Table 5, SALSA-2R-CSMR can achieve the highest SNR. Obviously, thereconstructed image by SALSA-2R-CSMR is closer to original image than others.CPU time and SNR

Fig.4 and Fig.5 show the performance comparisons between different methods in terms of the CPUtime over SNR, the results are the average of 100 runs for each method respectively.

SALSA-2R-CSMRI is lower than FCSA in reconstructed image SNR before 0.2 s. However, thereconstructed image SNR of FCSA soon flattens out after 0.2 s, which shows the algorithm hasconverged. But SALSA-2R-CSMRI is still increasing and 2.3 db higher than FCSA when SALSA-2R-CSMRI reaches convergence. Since FCSA is an improved version of CSA, FCSA is significantlybetter than CSA. TVCMRI and RecPF are always lower than other three algorithms. And TVCMRI isthe worst among these algorithms.

Algorithms TVCMRI RecPF CSA FCSA SALSA-2R-CSMRI

SNR(db) 12.15 12.69 13.56 13.91 16.22

Fig.4 Performance comparison (CPU timevs. SNR) on brain MR images

Fig.5 Performance comparison (CPU timevs. SNR) on cardiac MR images

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EXPERIMENTS ON SALSA-3R-CSMRI

Experimental setupIn order to verify the effectiveness of the SALSA-3R-CSMRI, we compared the SALSA-3R-

CSMRI with WaTMRI[8]. All experimental data are real MR images. Fig.6 shows an original brain imageand a shoulder image, they have the same size of 256 ×256. In the process of iterative algorithm,WaTMRI run 50 iterations. However, because the SALSA-3R-CSMRI has higher convergence in eachiteration, it need run only 25 iterations. For fair comparison, α, β and λ are set as 0.001, 0.035 and0.2×β, respectively.Visual comparison

Fig.7 and Fig.8 show the visual comparisons of the reconstructed results by WaTMRI and ALSA-3R-CSMRI on brain MR image and cardiac MR image. SALSA-3R-CSMR always has better visualeffect than WaTMRI, and the reconstructed image of SALSA -3R -CSMR is closer to the originalimage.CPU time and SNR

Fig.9 and Fig.10 show the statistical curves of WaTMRI and SALSA-3R-CSMRI in terms of theCPU time over SNR, which are obtained from the average of 100 runs for each method. The horizontalaxis represents iterative time and the vertical axis represents SNR of reconstructed images. TheSALSA-3R-CSMRI always obtains the best reconstruction results on all MR images in SNR and CPU time.

(a) Brain (b) CardiacFig.6 Original MR images

（a）Original （b）WaTMRI （c）SALSA-3R-CSMRIFig.7 Visual comparison on brain MR images

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（a）Original （b）WaTMRI （c）SALSA-3R-CSMRIFig.8 Visual comparison on cardiac MR images

Fig.9 Performance comparison (CPU time vs. SNR) on brain MR images

Fig.10 Peformance comparison (CPU time vs. SNR) on cardiac MR images

CONCLUSION

In this paper, we proposed SALSA-2R-CSMRI to solve compressed MR image recover regularizedby the linear combination of L1 norm and TV norm.

SALSA-2R -CSMRI decomposes a hard composite regularization problem into two simplersubproblems and efficiently solves them by SALSA. At the same time, we also proposed a new algorithmcalled SALSA-3R-MRI to effectively solve MR imaging based on CS with L1 norm, TV norm andwavelet tree structure norm. A great deal of real MR images experiments demonstrate the superiorperformance of two proposed algorithms for lower sampling MR image reconstruction. In the future work,

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[3] ZHAO Namula, WANG Mei, LI Xueen. Biological macrofeatures and criterion thereof for fracture immobilization in

Chinese Mongolian traditional osteopathy[J]. Joumal of Meddical Biomechanics, 2011, 26(2):189-192.

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Documents

012).3 45%1%5)1-5) *'%-)651 7,1(%-.'2 -) 851).-& !#()5)& 92$1-)1 … · 2019-11-16 · (TV) norm and L1 norm or three regularizations consisting of total variation, L1 norm and wavelet