# An AtanTV Nonconvex Regularization Model for MRI Reconstruction

Authors: Zhijun Luo; Zhibin Zhu; Benxin Zhang
Journal: Journal of Sensors
 (2022)

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## Abstract

Recently, the nonconvex regularization model has attracted much research attention in magnetic resonance imaging (MRI). In comparison with the traditional total variation (TV) regularization method, a nonconvex TV regularizer can effectively improve the fitting performance and prevent bias problems. In this work, we utilize the arctangent function as nonconvex TV regularize term (AtanTV) for MRI reconstruction. With appropriate parameters, the AtanTV model can avoid the systematic underestimation characteristic and maintain the objective function still convex. To address the AtanTV model, we propose an efficient alternating direction method of multipliers (ADMM) method to solve it. Experimental results indicate the superior performance (PSNR, RE, SSIM, etc.) of the proposed MRI reconstruction method.


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## Body

## 1. Introduction

Since the emergence of magnetic resonance imaging (MRI) in the twentieth century [1], it has been widely used in the field of medicine because of its excellent properties such as nonradiation and nonionization. In the traditional MRI field, k-space undersampling follows the Nyquist theory, which leads to long data acquisition time (or slow speed). Therefore, the application of MRI technology in some medical clinical fields is limited. In recent years, compressed sensing theory ([2, 3]) has shown great potential in processing sparse signal reconstruction. The compressed sensing (CS) in MRI demonstrates that it can reconstruct high-quality MRI images from less under sampled k-space measurements [4, 5]. Compression sensing has become the focus of MRI research since the invention of the pioneering work CS-MRI was proposed [6–8]. The researchers commonly formulate the under-sampled model for CS-MRI as
(1)y=Ax+ε,where y represents the observed MR data, x is the original MRI image, and ε is the noise. Generally, the observed MR data y’s scale is smaller than x. The matrix A=RF; R and F represent undersampling template (Cartesian or Radial sampling, etc.) and sparse transform (Wavelet or Fourier transform, etc.), respectively. MR image reconstruction aims to recover x from y. As a general rule, Equation (1) is an under-determined system, and the solution of the equation system is not unique. To stabilize the recovery of the data x, a common method to approximately solve Equation (1) is to adopt the total variation (TV) [9] form as a regularization term, and the model can express as
(2)minxλxTV+12y−Ax22,where λ>0 is the regularization parameter and xTV=Dx1 is called the total variation norm of x (D is first-order difference matrix). If the matrix A is under some proper conditions, the model (2) can be regarded as a convex optimization problem with respect to ℓ1-norm regularization. Unfortunately, there are two distinct drawbacks to this type of ℓ1-norm regularization problem [10, 11]: (1) it would underestimate high-amplitude values, and (2) it cannot recover a signal with fewer measurements. Hence, ℓ1-norm (or traditional TV) is not a good approximation of ℓ0-norm. To improve the performance of the model, many improved TV regularization works are proposed, such as generalized TV [12–15], hybrid TV [16, 17], adaptive TV [18, 19], and high-order TV [20, 21].In recent years, nonconvex regularization techniques have attracted much attention. Compared withℓ1-norm (or traditional TV) regularization, nonconvex regularizations can usually get better and more accurate estimates [22–29]. Specifically, it is verified in the studies [10, 23] that ℓp-norm regularization is significantly better than ℓ1-regularization in recovery performance. Xu [27] propose a L1/2 regularization model for CS model. For MRI, Yuan [28] considered the CS-MRI model based on L1/2 regularization. Selesnick [11, 29] introduced minmax-concave (MC) penalty in the sparse model and defined nonconvex TV regularization. The main characteristic of this nonconvex regularization model is that it can maintain convexity of the cost function. Therefore, some excellent algorithms in convex optimization can solve it. On this basis, some researchers separately considered the MCTV model of MRI, and their simulation experiments show the effectiveness of this type of method [30–32].Finding the “best” penalty function is an open problem, and its exploration is far from over. In this paper, we consider the arctangent penalty function (Atan) which was originally proposed by Selesnick [33]. As an alternative to ℓ1-norm, the Atan penalty function shows good sparsity and follows a “convex nonconvex” strategy similar to MC penalty [29]. However, they only consider its application in one-dimensional TV denoising [34]. Inspired by [29, 30], we introduce the Atan penalty function into traditional TV, define AtanTV, and apply it to MRI reconstruction.In summary, the main contributions are as follows:(i)
For MRI reconstruction model, we first use a multivariate generalization Atan penalty function to define AtanTV regularization and apply it to construct a more accurate nonconvex reconstruction model. The core of the nonconvex model is AtanTV as a regular term, so it is closer toℓ0-norm than ℓ1-norm(ii)
By selecting the parameters appropriately, the proposed model can preserve the convexity of the objective function. Therefore, some fast and effective convex optimization algorithms [35, 36] can solve the model. In this work, we consider alternating direction method of multipliers (ADMM) algorithms [37–42], which is considered to be one of the most successful methods for solving convex/nonconvex optimization problems [43](iii)
In the simulation experiment, different sampling masks, sampling rates, and datasets are tested, which shows that the performance of the proposed method (AtanTV) is better than that of traditional TV as well as MCTV method.The remainder of this work is organized as follows. In Section2, we will construct a nonconvex model of MRI reconstruction by the Atan penalty function. The ADMM algorithm of the proposed model is described in Section 3. Section 4 demonstrates simulation experiment results and Section 5 gives some conclusions.


## 2. AtanTV MRI Model

In this section, we first reviewed the definition of arctangent penalty function (Atan) and demonstrated its properties and then use the penalty function to define AtanTV regularization. At the end of the section, we structured a nonconvex AtanTV model for MRI reconstruction.


### 2.1. Some Propositions of the Atan Penalty Function

Definition 1 (see [33]).
The arctangent penalty function can be express as(3)ϕatanx;a=x,ifa=0,2a3arctan1+2ax3−π6,ifa>0..Figure1 intuitively shows that the Atan function is a good alternative to ℓ1-norm, and it is even more effective than MC function through parameter settings. Combined with the definition of Atan function, we propose a function ψax related to it as the following:
(4)ψax≔x−ϕatanx,a.Figure 1
Curves of function ofx, MC, Atan (a=0.5,2).The graph ofψaxa=0.5 is illustrated in Figure 2. We can find that the function ψax has some good properties such as ψax is a differentiable function, convex, and satisfies the following inequality 0<ψax≤x. Hence, we rewrite the Atan function as the following:
(5)ϕatanx;a=x−ψax.Figure 2
A DC decomposition of the Atan function.Combined with proposition 1 of [34], we have the following results:Proposition 2.
Let0≤a≤1/λ,
(1)
The function(6)θx=λϕatanx;a+12x−t2,is strictly convex.
(2)
The threshold function(7)proxϕt;λ,a=argminxλϕatanx;a+12x−t2,
whereλ>0 is called threshold penalty parameter. The threshold function is a continuous nonlinear function.Figure3 illustrates the threshold function proxϕt;1,0.5. Note that Equation (7) is also called the proximity operator of ϕ [44], which is a fundamental tool in efficient sparse model [45, 46].Figure 3
Atan proximity operator.


### 2.2. AtanTV MRI Model

To present our AtanTV MRI reconstruction model, we first consider the multivariate generalization form of the functionψax as follows:
(8)Ψau=∑i=1ℕψaui,u∈ℝℕ,where ψaui is given by Equation (4). Then, let
(9)Φau=u1−Ψau,∀u∈ℝℕ.Considering the substitution ofDx for u in the above equation, we get the definition of AtanTV.Definition 3 (AtanTV) The AtanTV regularizer:xAtanTV:ℝℕ⟶ℝ.
(10)xAtanTV≔ΦaDx=Dx1−ΨaDx,where D is the so-called first-order difference operator/matrix.Then, the MRI reconstruction model can be formulated as(11)minxλxAtanTV+12y−Ax22,where xAtanTV is given by Equation (10) and λ>0 is a parameter. For the Atan penalty function is a nonconvex function, the model Equation (11) is nonconvex.Lemma 4.
LetE:Rn⟶R be the function
(12)Ex≔λxAtanTV+12y−Ax22,if the parameter a>0 such that λaDTD≺¯ATA, then Ex is convex.Proof.
Firstly, from Equations (10)–(12), we have
(13)Ex=λxAtanTV+12y−Ax22=λΦaDx+12y22+Ax22−2yTAx=λDx1+12y22−yTAx+12Ax22−λΨaDx.
Then, the functionEx is convex if Hx is convex, where Hx=1/2Ax22−λΨaDx. Now, we prove that function Hx is convex and rewrite Hx as
(14)Hx=H1x+H2x=12Ax22−λaDx22⏟H1x+λa2Dx22−ΨaDx⏟H2x.
SinceλaDTD≺¯ATA is given, H1x is convex. On the other hand, from the proposition 1 in [34], we can obtain that H2x is convex. Hence, the conclusion holds.


## 2.1. Some Propositions of the Atan Penalty Function

Definition 1 (see [33]).
The arctangent penalty function can be express as(3)ϕatanx;a=x,ifa=0,2a3arctan1+2ax3−π6,ifa>0..Figure1 intuitively shows that the Atan function is a good alternative to ℓ1-norm, and it is even more effective than MC function through parameter settings. Combined with the definition of Atan function, we propose a function ψax related to it as the following:
(4)ψax≔x−ϕatanx,a.Figure 1
Curves of function ofx, MC, Atan (a=0.5,2).The graph ofψaxa=0.5 is illustrated in Figure 2. We can find that the function ψax has some good properties such as ψax is a differentiable function, convex, and satisfies the following inequality 0<ψax≤x. Hence, we rewrite the Atan function as the following:
(5)ϕatanx;a=x−ψax.Figure 2
A DC decomposition of the Atan function.Combined with proposition 1 of [34], we have the following results:Proposition 2.
Let0≤a≤1/λ,
(1)
The function(6)θx=λϕatanx;a+12x−t2,is strictly convex.
(2)
The threshold function(7)proxϕt;λ,a=argminxλϕatanx;a+12x−t2,
whereλ>0 is called threshold penalty parameter. The threshold function is a continuous nonlinear function.Figure3 illustrates the threshold function proxϕt;1,0.5. Note that Equation (7) is also called the proximity operator of ϕ [44], which is a fundamental tool in efficient sparse model [45, 46].Figure 3
Atan proximity operator.


## 2.2. AtanTV MRI Model

To present our AtanTV MRI reconstruction model, we first consider the multivariate generalization form of the functionψax as follows:
(8)Ψau=∑i=1ℕψaui,u∈ℝℕ,where ψaui is given by Equation (4). Then, let
(9)Φau=u1−Ψau,∀u∈ℝℕ.Considering the substitution ofDx for u in the above equation, we get the definition of AtanTV.Definition 3 (AtanTV) The AtanTV regularizer:xAtanTV:ℝℕ⟶ℝ.
(10)xAtanTV≔ΦaDx=Dx1−ΨaDx,where D is the so-called first-order difference operator/matrix.Then, the MRI reconstruction model can be formulated as(11)minxλxAtanTV+12y−Ax22,where xAtanTV is given by Equation (10) and λ>0 is a parameter. For the Atan penalty function is a nonconvex function, the model Equation (11) is nonconvex.Lemma 4.
LetE:Rn⟶R be the function
(12)Ex≔λxAtanTV+12y−Ax22,if the parameter a>0 such that λaDTD≺¯ATA, then Ex is convex.Proof.
Firstly, from Equations (10)–(12), we have
(13)Ex=λxAtanTV+12y−Ax22=λΦaDx+12y22+Ax22−2yTAx=λDx1+12y22−yTAx+12Ax22−λΨaDx.
Then, the functionEx is convex if Hx is convex, where Hx=1/2Ax22−λΨaDx. Now, we prove that function Hx is convex and rewrite Hx as
(14)Hx=H1x+H2x=12Ax22−λaDx22⏟H1x+λa2Dx22−ΨaDx⏟H2x.
SinceλaDTD≺¯ATA is given, H1x is convex. On the other hand, from the proposition 1 in [34], we can obtain that H2x is convex. Hence, the conclusion holds.


## 3. ADMM Algorithm for AtanTV Model

In this section, we use the ADMM algorithm to solve the proposed AtanTV MRI reconstruction model. Firstly, Equation (11) can be rewritten as
(15)minxλΦaDx+12y−Ax22,where Φa· is given by Eq. (11). To solve this problem, firstly, we introduce u=Dx, and then, the unconstrained optimization problem (15) can be reformulated as
(16)minx,uλΦau+12y−Ax22,s.tu=Dx.The corresponding augmented Lagrangian function can be written as(17)Lx,u,τ=λΦau+12y−Ax22−τTu−Dx+β2u−Dx22,where τ is the Lagrange multiplier and β>0 is a penalty parameter. Given an initial point u0,τ0, the standard ADMM algorithm generates the new iterate by the following scheme:
(18)xk+1=argminLx,uk,τk=argminx12y−Ax22+τkTDx+β2zk−Dx22,uk+1=argminLxk+1,u,τk=argminzλΦau−τkTu+β2u−Dxk+122,τk+1=τk−βDxk+1−uk+1.Forx-update step, using the first-order optimality condition, we obtain that the xk+1 is
(19)xk+1=βDTD+ATA−1βDTuk+ATy−DTτk.In the field of MRI,DTD is a cyclic matrix that can be diagonalized by Fourier transform. The matrix A=RF, where F is a Fourier matrix with the property of F=F−1. Therefore, the optimal solution of xk+1 may obtain by twice fast Fourier transform.Foru-update step
(20)uk+1=argminuLxk+1,u,τk=argminuΦau+β2u−Dxk+1+τkβ22,let tk=Dxk+1+τk/β, follows Eq. (7); the new iteration point of uk+1 is determined by the following expression:
(21)uk+1=proxΦa,βtk,where proxΦ· is the proximal operator of the Atan penalty function.Based on the above analysis, we now give a specific algorithm to solve Equation (16). We call the Algorithm 1 AtanTV method because the AtanTV is introduced in the Eq. (16).Algorithm 1: (AtanTV).
Step 0 Initialization:Inputa≥0,β>0,λ>0, ε>0. Let k≔0, given u,τ≔u0,τ0,Step 1 A new iteration point wk+1=xk+1,uk+1,τk+1 is generated by the following formulaxk+1=ATA+βDTD−1H^k,uk+1=proxΦa,βtk,τk+1=τk−βDxk+1−zk+1,where H^k=βDTuk+ATy−DTτk.Step 2 If wk−wk+122≤ε, STOP; otherwise, let k≕k+1. Go back to step 1.


## 4. Experimental Results

In this section, we verified the effectiveness of the proposed AtanTV method by some simulation experiments and made some numerical comparisons with TV [37] and MCTV [30, 31]. The programming software was MATLAB R2015a. All experiments were carried out on a standard PC with 8.0 GB RAM and Intel Core CPU 2.2 GHz.The experimental data are shown in Figure4. These data are commonly used undersampling templates and MRI image data in the field of MRI research, which are easy to obtain on the Internet. The first row in Figure 4 shows three common sampling templates, radial sampling, random sampling, and Cartesian sampling from left to right. The second row of Figure 4 is three typical MRI data. The size of sampling masks and MRI data are 256×256.Figure 4
Different sampling masks and MR datasets.In order to evaluate the image performance of the experimental results, we selected two indexes commonly used in image processing research, namely, the relative error (RE) and the peak signal-to-noise ratio (PSNR). Their calculation formula is as follows:(22)PSNR=10lgUmax2xk−x2,RE=20x−xk2x2,where Umax is the possible maximum value of all pixels in the image (for example, for an 8-bit image Umax=255 [47]), x and xk denote the original image and restored image, respectively. Generally speaking, a larger PSNR value and a smaller RE value represent better image restoration quality. The structural similarity (SSIM) is an index to measure the similarity of two images. Different from PSNR index, SSIM is a perception index based on the visible structure in the image. Its value range is between 0 and 1, the closer it is to 1, the better the result.
(23)SSIMx,xk=2x¯·x¯k+C12σxxk+C2x¯2+x¯k2+C1σ2+σk2+C2,where x¯ is the mean of value x, k is the number of iterations in the algorithm, x¯k denotes the mean of value xk, σ2 denotes the variance of x, σk2 denotes the variance of xk, σxxk denotes the covariance of x and xk, C1andC2 are stabilization parameters, and for more details, please see [48].


### 4.1. The Parameters Selection

In the field of sparse model research, how to choose the “best” parameters is an open problem. Different methods using the same parameters may get diametrically opposite conclusions. In the compared TV method and MCTV method, the corresponding parameters follow the “best” parameter values provided by the authors [30, 31, 37].For AtanTV method, we take the reconstruction results of the Shepp-Logan phantom (256×256) data under radial mask with 3% sampling ratio as an example to illustrate how to select parameters. Firstly, we consider the setting of parameter λ, and the parameter β adopts a similar method. In order to display the influence of parameter λ independently, we fix other parameters and let
(24)λ∈10,1,0.5,0.1,0.05,0.01,0.005,0.001,0.0005,0.0001,respectively. Figure 5 shows the reconstruction components of Shepp-Logan obtained by the AtanTV method with λ=0.1,0.01,0.001. The plot of PSRN versus the parameter λ values is shown in Figure 6. Through the observation of Figure 6, it is not difficult to find that when λ=0.01, the performance of AtanTV method is the “best.” Therefore, we set the parameter λ=0.01, and similarly we set β=0.01.Figure 5
The recovery results whenλ=0.1,0.01,0.001 are from left to right.Figure 6
PSNR value vs. the parameterλ value.As the parametera of the Arctangent penalty function between 0 and 1/λ, we first let
(25)a∈0,10,20,30,40,50,60,70,80,90,100.The plots of PSRN (RE) versus the parametera values are shown in Figure 7. We can observe that when a=10, the value of PSNR is the largest, and the value of RE is the smallest. Next, we let a=0,1,2,3,4,5,6,7,8,9,10, respectively. Figure 8 presents that the AtanTV method achieves excellent performance when a=5.Figure 7
PSNR (RE) value vs. the parametera∈0,100.Figure 8
PSNR (RE) value vs. the parametera∈0,10.Figures7–8 show that appropriate parameter selection has a certain impact on MRI reconstruction performance.


### 4.2. Convergence Properties

These experiments were tested using the Shepp-Logan phantom256×256 data; the undersampling mask is the radial mask with sampling rate of 0.03, as shown in the first column of Figure 4. Figure 9 shows the RE value curve versus the iteration number. It is easy to see that the RE value generated by the three methods (TV, MCTV, and AtanTV) gradually decrease with the increase of the number of iterations. When iterationnumber=10, the RE value generated by the three methods are approximately equal (RE%≈40). When iteration number=200, the RE value generated by AtanTV is much lower than TV and MCTV. Figure 10 shows the relationship between PSNR and the number of iterations. It can be seen from the curve that when the number of iterations increases, the PSNR values of the three methods increase. When the number of iterations is fixed, the PSNR values are obtained by the three methods such that PSNRAtanTV>PSNRMCTV>PSNRTV. From Figures 9–10, we can see that the curves of RE and PSNR show excellent convergence properties as the number of iterations increases.Figure 9
RE value vs. iteration number.Figure 10
PSNR value vs. iteration number.


### 4.3. Effect of Sampling Ratios

In this subsection, we compare brain data reconstruction results at different sampling masks with different ratios. The results are shown in Figures11–13. It is easy to see that the PSNR curve and SSIM curve increase with the increase of sampling rate, while RE curve is opposite. For a fixed sampling rate, we note that from Figures 11–13 the AtanTV model can obtain a higher PSNR and SSIM values and a lower RE value.Figure 11
PSNR vs. sampling ratios on radial mask.Figure 12
RE vs. sampling ratios on random mask.Figure 13
SSIM vs. sampling ratios on Cartesian mask.In addition, from Figures11–13, we can clearly see that at a lower sampling rate, AtanTV shows obvious competitiveness compared to TV and MCTV methods.


### 4.4. Comparison of Reconstruction Results

In this subsection, we use the sampling masks and MR datasets shown in Figure4 to evaluate our proposed AtanTV method. In those experiments, the stopping criteria for all algorithms are ε=10−6 with a maximum iteration number of 200. The reconstruction results are presented in Figures 14–16. For obvious visual contrast, we use 10× residual diagram to display the restoration results. Figure 14 shows the reconstruction result of Shepp-Logan data under radial mask with 3% sampling ratio, where the AtanTV method produces RE≈0.03, which is much lower than TV and MCTV. In Figure 15, we test the brain data under random sampling mask with 0.3 sampling rate and sampling radius r=0.1. The PSNR values obtained by the three different methods are 38, 41, and 48, respectively. Cartesian undersampling is widely used in practice. For brain angiography image, we use a Cartesian sampling mask to test the image at a sampling rate of 34%. The reconstructed images and errors are shown in Figure 16. The reconstruction results of the three methods are similar, and RE≈0.1. The values of PSNR show that our AtanTV is slightly better than the other two methods. Further, all experimental results including PSNR, RE, SSIM, and CPU time are listed in Table 1. Comparing the results of TV, MCTV, and AtanTV in Table 1, we find that the AtanTV nonconvex model obtains a higher PSNR and SSIM values, which indicates that AtanTV has a better performance.Figure 14
Shepp-Logan reconstruction results with radial mask sampling rate of 3%.Figure 15
Brain reconstruction results with random mask sampling rate of 30%.Figure 16
Brain angiography reconstruction results with Cartesian mask sampling rate of 34%.Table 1
Reconstruction numerical results.
ImageTemplateModelPSNR(dB)RE%SSIMCPU time(s)Shepp-LoganRadial sampling 3%TV26.966318.210.65324.967998MCTV30.610911.970.844528.840455AtanTV43.64362.670.873412.688320Random sampling 30%TV54.51540.760.98034.725769MCTV56.67650.600.985831.025552AtanTV73.22900.090.999712.993860Cartesian sampling 34%TV55.19030.710.98455.318684MCTV56.56830.600.986228.146686AtanTV59.08860.450.991612.763237BrainRadial sampling 3%TV22.586726.040.44846.052898MCTV22.778525.470.572635.939883AtanTV23.576923.240.752813.372252Random sampling 30%TV38.00284.410.86554.965910MCTV40.97573.140.918231.285251AtanTV48.41701.330.993313.048973Cartesian sampling 34%TV31.51869.310.87584.893582MCTV32.88997.950.910228.528819AtanTV33.00047.850.923012.735388Brain angiographyRadial sampling 3%TV22.496037.830.68014.869466MCTV23.164035.030.707429.287268AtanTV23.389534.140.745812.566450Random sampling 30%TV38.63375.900.96644.869629MCTV40.14294.960.972429.679897AtanTV43.11993.520.982612.616800Cartesian sampling 34%TV33.845410.240.94605.528671MCTV35.48458.480.959037.097387AtanTV36.88567.220.964212.737054In addition, we found from Table1 that under the radial mask with sampling rate of 0.03, the PSNR and RE values obtained by three different methods are similar, but the SSIM results are quite different. The SSIM value obtained by AtanTV method is 0.75, which is nearly 30% and 20% higher than TV and MCTV, respectively.


## 4.1. The Parameters Selection

In the field of sparse model research, how to choose the “best” parameters is an open problem. Different methods using the same parameters may get diametrically opposite conclusions. In the compared TV method and MCTV method, the corresponding parameters follow the “best” parameter values provided by the authors [30, 31, 37].For AtanTV method, we take the reconstruction results of the Shepp-Logan phantom (256×256) data under radial mask with 3% sampling ratio as an example to illustrate how to select parameters. Firstly, we consider the setting of parameter λ, and the parameter β adopts a similar method. In order to display the influence of parameter λ independently, we fix other parameters and let
(24)λ∈10,1,0.5,0.1,0.05,0.01,0.005,0.001,0.0005,0.0001,respectively. Figure 5 shows the reconstruction components of Shepp-Logan obtained by the AtanTV method with λ=0.1,0.01,0.001. The plot of PSRN versus the parameter λ values is shown in Figure 6. Through the observation of Figure 6, it is not difficult to find that when λ=0.01, the performance of AtanTV method is the “best.” Therefore, we set the parameter λ=0.01, and similarly we set β=0.01.Figure 5
The recovery results whenλ=0.1,0.01,0.001 are from left to right.Figure 6
PSNR value vs. the parameterλ value.As the parametera of the Arctangent penalty function between 0 and 1/λ, we first let
(25)a∈0,10,20,30,40,50,60,70,80,90,100.The plots of PSRN (RE) versus the parametera values are shown in Figure 7. We can observe that when a=10, the value of PSNR is the largest, and the value of RE is the smallest. Next, we let a=0,1,2,3,4,5,6,7,8,9,10, respectively. Figure 8 presents that the AtanTV method achieves excellent performance when a=5.Figure 7
PSNR (RE) value vs. the parametera∈0,100.Figure 8
PSNR (RE) value vs. the parametera∈0,10.Figures7–8 show that appropriate parameter selection has a certain impact on MRI reconstruction performance.


## 4.2. Convergence Properties

These experiments were tested using the Shepp-Logan phantom256×256 data; the undersampling mask is the radial mask with sampling rate of 0.03, as shown in the first column of Figure 4. Figure 9 shows the RE value curve versus the iteration number. It is easy to see that the RE value generated by the three methods (TV, MCTV, and AtanTV) gradually decrease with the increase of the number of iterations. When iterationnumber=10, the RE value generated by the three methods are approximately equal (RE%≈40). When iteration number=200, the RE value generated by AtanTV is much lower than TV and MCTV. Figure 10 shows the relationship between PSNR and the number of iterations. It can be seen from the curve that when the number of iterations increases, the PSNR values of the three methods increase. When the number of iterations is fixed, the PSNR values are obtained by the three methods such that PSNRAtanTV>PSNRMCTV>PSNRTV. From Figures 9–10, we can see that the curves of RE and PSNR show excellent convergence properties as the number of iterations increases.Figure 9
RE value vs. iteration number.Figure 10
PSNR value vs. iteration number.


## 4.3. Effect of Sampling Ratios

In this subsection, we compare brain data reconstruction results at different sampling masks with different ratios. The results are shown in Figures11–13. It is easy to see that the PSNR curve and SSIM curve increase with the increase of sampling rate, while RE curve is opposite. For a fixed sampling rate, we note that from Figures 11–13 the AtanTV model can obtain a higher PSNR and SSIM values and a lower RE value.Figure 11
PSNR vs. sampling ratios on radial mask.Figure 12
RE vs. sampling ratios on random mask.Figure 13
SSIM vs. sampling ratios on Cartesian mask.In addition, from Figures11–13, we can clearly see that at a lower sampling rate, AtanTV shows obvious competitiveness compared to TV and MCTV methods.


## 4.4. Comparison of Reconstruction Results

In this subsection, we use the sampling masks and MR datasets shown in Figure4 to evaluate our proposed AtanTV method. In those experiments, the stopping criteria for all algorithms are ε=10−6 with a maximum iteration number of 200. The reconstruction results are presented in Figures 14–16. For obvious visual contrast, we use 10× residual diagram to display the restoration results. Figure 14 shows the reconstruction result of Shepp-Logan data under radial mask with 3% sampling ratio, where the AtanTV method produces RE≈0.03, which is much lower than TV and MCTV. In Figure 15, we test the brain data under random sampling mask with 0.3 sampling rate and sampling radius r=0.1. The PSNR values obtained by the three different methods are 38, 41, and 48, respectively. Cartesian undersampling is widely used in practice. For brain angiography image, we use a Cartesian sampling mask to test the image at a sampling rate of 34%. The reconstructed images and errors are shown in Figure 16. The reconstruction results of the three methods are similar, and RE≈0.1. The values of PSNR show that our AtanTV is slightly better than the other two methods. Further, all experimental results including PSNR, RE, SSIM, and CPU time are listed in Table 1. Comparing the results of TV, MCTV, and AtanTV in Table 1, we find that the AtanTV nonconvex model obtains a higher PSNR and SSIM values, which indicates that AtanTV has a better performance.Figure 14
Shepp-Logan reconstruction results with radial mask sampling rate of 3%.Figure 15
Brain reconstruction results with random mask sampling rate of 30%.Figure 16
Brain angiography reconstruction results with Cartesian mask sampling rate of 34%.Table 1
Reconstruction numerical results.
ImageTemplateModelPSNR(dB)RE%SSIMCPU time(s)Shepp-LoganRadial sampling 3%TV26.966318.210.65324.967998MCTV30.610911.970.844528.840455AtanTV43.64362.670.873412.688320Random sampling 30%TV54.51540.760.98034.725769MCTV56.67650.600.985831.025552AtanTV73.22900.090.999712.993860Cartesian sampling 34%TV55.19030.710.98455.318684MCTV56.56830.600.986228.146686AtanTV59.08860.450.991612.763237BrainRadial sampling 3%TV22.586726.040.44846.052898MCTV22.778525.470.572635.939883AtanTV23.576923.240.752813.372252Random sampling 30%TV38.00284.410.86554.965910MCTV40.97573.140.918231.285251AtanTV48.41701.330.993313.048973Cartesian sampling 34%TV31.51869.310.87584.893582MCTV32.88997.950.910228.528819AtanTV33.00047.850.923012.735388Brain angiographyRadial sampling 3%TV22.496037.830.68014.869466MCTV23.164035.030.707429.287268AtanTV23.389534.140.745812.566450Random sampling 30%TV38.63375.900.96644.869629MCTV40.14294.960.972429.679897AtanTV43.11993.520.982612.616800Cartesian sampling 34%TV33.845410.240.94605.528671MCTV35.48458.480.959037.097387AtanTV36.88567.220.964212.737054In addition, we found from Table1 that under the radial mask with sampling rate of 0.03, the PSNR and RE values obtained by three different methods are similar, but the SSIM results are quite different. The SSIM value obtained by AtanTV method is 0.75, which is nearly 30% and 20% higher than TV and MCTV, respectively.


## 5. Conclusion

In this article, we introduced a nonconvex model for under-sampled magnetic resonance imaging reconstruction. In this nonconvex model, an arctangent penalty function is introduced to act as a regularization term to improve the fitting performance of the model (called AtanTV). Compared with the classic TV model, the new proposed AtanTV model can not only prevent the systematic underestimation feature but also maintain the total object function which is convex. Therefore, we use the fast and effective ADMM method to compute the AtanTV model. Experimental results demonstrate that the AtanTV regularization can improve the TV regularization significantly and is comparable with the recently proposed MCTV method [31]. In future work, we aim to develop this proposed method for dynamic MRI and low-rank sparse model, etc.



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*Source: 1758996-2022-04-07.xml*