Hybrid High-Order and Fractional-Order Total Variation with Nonlocal Regularization for Compressive Sensing Image Reconstruction

Abstract

Total variation often yields staircase artifacts in the smooth region of the image reconstruction. This paper proposes a hybrid high-order and fractional-order total variation with nonlocal regularization algorithm. The nonlocal means regularization is introduced to describe image structural prior information. By selecting appropriate weights in the fractional-order and high-order total variation coefficients, the proposed algorithm makes the fractional-order and the high-order total variation complement each other on image reconstruction. It can solve the problem of non-smooth in smooth areas when fractional-order total variation can enhance image edges and textures. In addition, it also addresses high-order total variation alleviates the staircase artifact produced by traditional total variation, still smooth the details of the image and the effect is not ideal. Meanwhile, the proposed algorithm suppresses painting-like effects caused by nonlocal means regularization. The Lagrange multiplier method and the alternating direction multipliers method are used to solve the regularization problem. By comparing with several state-of-the-art reconstruction algorithms, the proposed algorithm is more efficient. It does not only yield higher peak-signal-to-noise ratio (PSNR) and structural similarity (SSIM) but also retain abundant details and textures efficiently. When the measurement rate is 0.1, the gains of PSNR and SSIM are up to 1.896 dB and 0.048 dB respectively compared with total variation with nonlocal regularization (TV-NLR).

1. Introduction

Compressive sensing (CS) [1,2] has been successfully applied in signal acquiring, processing, and compression. By exploiting the redundancy that existed in a signal, CS conducts sampling, and compression at the same time. CS theory demonstrates that a signal can be reconstructed with high probability when it exhibits sparse representation in some transformation domain, which has been widely used in many fields, such as single-pixel imaging [3], remote sensing imaging [4], and medical imaging [5].

Based on the CS theory, if an original signal $x\in R^N$ is sparse or after sparse transformation $\Psi$. The measured value $y=\Phi x,y\in R^M(M\ll N)$ can be obtained through the measurement matrix $\Phi$. CS theory shows that when the sparse transformation matrix $\Psi$ and the measurement matrix $\Phi$ satisfy the restricted isometry property (RIP) [6], the original signal x can be reconstructed by solving the following optimization problem

$$\underset{x}{\min } {\Vert {\Psi }^{T}x\Vert }_0 s.t. y=\Phi x,$$

(1)

where signal $\alpha ={\Psi }^{T}x$, $\alpha$ is the sparse coefficient after sparse transformation. ${\Vert \cdot \Vert }_0$ represents the l₀ norm. Since the solution of Equation (1) is an NP-hard problem, it can be solved by the approximate form l₁ norm

$$\underset{x}{\min } {\Vert {\Psi }^{T}x\Vert }_1 s.t. y=\Phi x.$$

(2)

The above-constrained problem can be constructed by the Lagrangian multiplier method to an unconstrained problem

$$\tilde{x}=\arg \underset{x}{\min }\{{\Vert y-\Phi x\Vert }_2^2 +\lambda {\Vert {\Psi }^{T}x\Vert }_1\},$$

(3)

where ${\Vert y-\Phi x\Vert }_2^2$ is a cost function, $\lambda$ is the Lagrangian parameter. Since prior characteristics exist in natural images generated by natural light imaging, the optimization problem of image compression sensing can be expressed as

$$\tilde{u}=\arg \underset{u}{\min }\{ {\Vert y-\Phi u\Vert }_2^2 +\lambda R(u)\},$$

(4)

where $u$ is a 2D image. $R(u)$ represents a regularization term representing prior information of an image.

The current CS recovery algorithms explore the prior knowledge that a natural image is sparse in some domains, such as discrete cosine transform (DCT) [7], wavelets [8], and gradient-domain utilized by total variation (TV) model [9,10]. Image sparse representation is a key factor to affect image reconstruction quality. Despite high effectiveness in image CS recovery, the TV-based [11] algorithms cannot recover the fine details and textures, reconstructed images suffer from undesirable staircase artifact. This problem has sparked numerous studies in designing regularizers that could efficiently suppress the staircase artifacts while retaining sharp edges. To overcome this drawback, a weighted total variation [12] is proposed to improve the sparsity of the TV norm. Furthermore, an improved total variation by intraprediction is proposed [13]. Besides, total generalized variation (TGV) [14,15] is widely discussed, which is more precise in describing pixels variations in smooth regions, and thus reduces oil painting artifacts while still being able to preserve sharp edges. Except for these TV and its variants, there also exists some combination of TV (or TGV) and different waves, such as TV + wavelet [16] and TGV + shearlet [17]. However, the methods described above are improved on TV, there is still a staircase artifact effect in the results. Further, nonlocal means (NLM) [18] uses the similarity between the surrounding pixels of the image to perform weighted filtering, which effectively utilizes the non-local self-similarity. NLM is successfully applied in the CS image reconstruction, but it often results in the painting-like effect.

Recently, a class of fractional-order TV regularization models has received considerable interest and it is widely used in image denoising [19,20]. Adaptive weighted high-frequency iterative fractional-order TV is proposed, the high frequency gradient of an image is reweighted in iterations adaptively when using fractional-order TV [21]. Another conventional way to suppress the staircase artifact is to use a high-order TV regularization [22,23], there exists a high-order total variation minimization model that removes undesired artifacts for restoring blurry and noisy images [24]. High-order TV regularization can reconstruct piecewise linear regions, but high-order TV may also smooth out the image details, and it may reduce the ability of edges-preserving [23]. In order to make full use of image prior information as much as possible to construct a regularized model. A generalized hybrid non-convex variational regularization model [25] and unidirectional hybrid total variation with nonconvex low-rank regularization [26] have been proposed.

Motivated by the aforementioned studies, both the fractional-order TV and high-order TV can solve the problem of staircase artifacts, but fractional-order TV will cause non-smooth effects just like noise in smooth areas of the image. The ability of high-order TV to reduce the staircase effect is not very ideal since it may smooth the image details. In this paper, a two-dimensional compressive sensing image reconstruction model based on hybrid high-order and fractional-order TV (HoFrTV) with nonlocal regularization is proposed. The proposed algorithm makes the fractional-order and the high-order TV complement each other on image reconstruction, which can effectively solve the problems caused by high-order and fractional-order TV. The proposed algorithm effectively reduces the problem of staircase artifacts while preserving the edges, and suppresses painting-like effects produced by nonlocal means regularization.

To effectively solve the proposed algorithm model, we introduce auxiliary variables to a construct constrained optimization problem in Section 2.2. We divide the proposed model into four subproblems to solve. These subproblems are fractional-order TV model, high-order TV model, nonlocal means regularization model, and the reconstructed image by iterative update respectively. The augmented Lagrangian method (ALM) and the alternating direction method of multipliers (ADMM) are incorporated to solve these problems. Each parameter of the Lagrangian function is optimized for the experiment results empirically. The experimental results show that the proposed algorithm outperforms the current state-of-the-art algorithms, the edges and details of the reconstructed image are more abundant, and the visual effect of the reconstructed image is better.

The remainder of this paper is organized as follows. Section 2 introduces the related regularization model and the proposed HoFrTV model. In Section 3, parameter selection and experimental results are presented. In Section 4, the conclusions are drawn.

2. The Proposed Algorithm Model

2.1. Regularization Model

2.1.1. Fractional-Order Total Variation Model

Fractional total variation can be regarded as the generalization of total variation composed of fractional orders. There are three widely used definitions of the model, including Riemann-Liouville (R-L), Grünwald-Letnikov (G-L), and Caputo model [27]. Here, we choose the G-L model because it is easier to implement for image reconstruction. The G-L model can be defined as

$${}_a\mathrm{D}_t^{\alpha }f(t)=\underset{h\to 0}{\lim }\frac{1}{h^{\alpha }}{\displaystyle \sum _{k=0}^{\left[\frac{t-a}{h}\right]}(-1}{)}^k\left(\begin{array}{c}\alpha \\ k\end{array}\right)f(t-kh),$$

(5)

where α fractional order, t and a are the upper and lower boundaries of the independent variable respectively, and h is the differential step-size, $\left(\begin{array}{c}\alpha \\ k\end{array}\right)$ is the binomial coefficient and defined as $\left(\begin{array}{c}\alpha \\ k\end{array}\right)=\frac{\Gamma (\alpha +1)}{\Gamma (k+1)\Gamma (\alpha -k+1)}$, where $\Gamma (\cdot )$ is the Gamma function. Without loss of generality, let $u$ denote the image. The fractional-order total variation in the horizontal and vertical directions can be expressed as

$$\left[D^{\alpha }u\right]=((D_h^{\alpha }u),(D_v^{\alpha }u)),$$

(6)

and

$$\{\begin{array}{l}(D_h^{\alpha }u)={\displaystyle \sum _{k=0}^{K-1}{(-1)}^k\left(\begin{array}{c}\alpha \\ k\end{array}\right)}u(i-k,j)\\ (D_v^{\alpha }u)={\displaystyle \sum _{k=0}^{K-1}{(-1)}^k\left(\begin{array}{c}\alpha \\ k\end{array}\right)}u(i,j-k)\end{array},$$

(7)

where i and j denote the pixel in the i-th row and the j-th column of an N × N size image. K ≥ 3 is the number of items involved in the computation of the fractional-order derivative, which is usually set as K = N. Based on this definition, the fractional-order semi-norm is defined as

$${\Vert D^{\alpha }u\Vert }_{FrTV}={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N\sqrt{{\left[(D_h^{\alpha }u)\right]}^2+{\left[(D_v^{\alpha }u)\right]}^2}}}.$$

(8)

2.1.2. High-Order Total Variation Model

On the basis of the first-order total variation operators of the image, we can obtain the high-order total variation, defined as

$$\begin{array}{l}{D}^{2}u=(D_{vv}u,D_{vh}u,D_{hv}u,D_{hh}u)\\ \Vert D^{2}u\Vert =\sqrt{{\left|D_{vv}u\right|}^2+{\left|D_{vh}u\right|}^2+{\left|D_{hv}u\right|}^2+{\left|D_{hh}u\right|}^2}\end{array},$$

(9)

where $(D_{vv}u)$, $(D_{vh}u)$, $(D_{hv}u)$, and $(D_{hh}u)$ are high-order total variation operators in the horizontal and vertical directions, respectively. The high-order total variation can be seemed as performing the differential operation again on the first-order total variation.

2.1.3. Nonlocal Mean Regularization Model

The pixels in the image can be estimated by the weighted average of the surrounding pixels of similar neighborhoods in the similar neighborhood of the search window, which can be expressed as follows

$${\widehat{u}}_i={\displaystyle \sum _{j\in \Omega }{w}_{i,j}{u}_j},$$

(10)

where $\Omega$ is a search window area D_s × D_s. let $u_i$ and $u_j$ denote the central pixel of similar neighborhood window d_s × d_s respectively. The weight of $u_j$ to $u_i$, denoted by $w_{i,j}$, which is determined by the similarity of the pixels in a similar neighborhood window. It can be calculated by the Gaussian l₂ distance between pixels in the neighborhood window. It can be written in the following form

$$w_{i,j}=\frac{1}{c}\exp (-\frac{{\Vert u_j-u_i\Vert }_2^2}{h^2}).$$

(11)

Specifically, assume that $u_j$ lies in the search window of $u_i$. The neighborhood window centered on the central pixel $u_j$ can slide in the search window to calculate the similarity between two neighboring windows. $h$ is the factor controlling the Gaussian kernel and $c$ is the normalization constant. Applying nonlocal regularization in the CS reconstruction process, furthermore, it can be rewritten in a matrix form as

$$R(u)={\Vert u-Wu\Vert }_2^2,$$

(12)

here, $W$ represents a matrix composed of $w_{i,j}$ in Equation (10), let $R(u)$ denote nonlocal regularization term. When calculating the second norm, vectorizing the elements in the norm is employed and then the second norm of the vectorized matrix is calculated.

2.2. The Proposed Algorithm Model

Incorporating Equations (8), (9), and (12) into the CS optimization problem jointly, the proposed hybrid high-order and fractional-order total variation with nonlocal regularization model image CS recovery algorithm are formulated as

$$\arg \underset{u}{\min } \tau {\Vert D^{\alpha }u\Vert }_1+\epsilon {\Vert D^{2}u\Vert }_1+\beta {\Vert u-Wu\Vert }_2^2 s.t. y=\Phi u.$$

(13)

Using variable splitting, we introduce auxiliary variables and change Equation (13) to a constrained optimization problem

$$\begin{array}{l}\arg \underset{u}{\min } \tau {\Vert w\Vert }_1+\epsilon {\Vert v\Vert }_1+\beta {\Vert z-Wz\Vert }_2^2 \\ s.t. D^{\alpha }u=w,D^{2}u=v,u=z,y=\Phi u\end{array}$$

(14)

where $\tau$ and $\epsilon$ control the weights of fractional-order and high-order total variation. Note that the problem of Equation (14) is quite difficult to solve directly due to the non-differentiability and non-linearity of the combined regularization terms. An augmented Lagrangian based approach is developed to solve the problem

$$\begin{array}{l}{L}_A(w,v,z,u)=\tau ({\Vert w\Vert }_1-{\gamma }_1^T(D^{\alpha }u-w)+\frac{{\mu }_1}{2}{\Vert D^{\alpha }u-w\Vert }_2^2)\\ +\epsilon ({\Vert v\Vert }_1-{\gamma }_2^T(D^{2}u-v)+\frac{{\mu }_2}{2}{\Vert D^{2}u-v\Vert }_2^2)\\ +\frac{{\mu }_3}{2}{\Vert \Phi u-y\Vert }_2^2 -{\gamma }_3^T(\Phi u-y)+\beta {\Vert z-Wz\Vert }_2^2\\ -{\gamma }_4^T(u-z)+\frac{{\mu }_4}{2}{\Vert u-z\Vert }_2^2\end{array}$$

(15)

where ${\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4$, and $\beta$ are regularization parameters associated with the quadratic penalty terms. ${\gamma }_1,{\gamma }_2,{\gamma }_3$, and ${\gamma }_4$ are the Lagrange multipliers associated with the constraints of Equation (15). The idea of using ADMM is to find a saddle point of $L_A(w,v,z,u)$ which is the solution to the original problem in Equation (13). It can minimize the augmented Lagrangian function $L_A(w,v,z,u)$ alternately, the problem in Equation (15) is decomposed into the four subproblems. We investigated the subproblems one by one.

2.2.1. w and v Subproblems

Given v, z, and u, the optimization problem associated with w can be expressed as

$$\tilde{w}=\arg \underset{w}{\min }\tau ({\Vert w\Vert }_1-{\gamma }_1^T(D^{\alpha }u-w)+\frac{{\mu }_1}{2}{\Vert D^{\alpha }u-w\Vert }_2^2).$$

(16)

According to Lemma 2 [28], the closed solution form of Equation (16) is

$$\tilde{w}=\max \tau \left\{\left|D^{\alpha }u-\frac{{\gamma }_1}{{\mu }_1}\right|-\frac{1}{{\mu }_1},0\right\}·\mathrm{sgn}(D^{\alpha }u-\frac{{\gamma }_1}{{\mu }_1}).$$

(17)

When solving v subproblem, given w, v and z, similarly, the v subproblem becomes

$$\tilde{v}=\arg \underset{v}{\min }\epsilon ({\Vert v\Vert }_1-{\gamma }_2^T(D^{2}u-v)+\frac{{\mu }_2}{2}{\Vert D^{2}u-v\Vert }_2^2).$$

(18)

Same as w subproblems, the closed solution form of (18) can write as

$$\tilde{v}=\max \epsilon \left\{\left|D^{2}u-\frac{{\gamma }_2}{{\mu }_2}\right|-\frac{1}{{\mu }_2},0\right\}·\mathrm{sgn}(D^{2}u-\frac{{\gamma }_2}{{\mu }_2}).$$

(19)

2.2.2. u Subproblems

Fixed w, v, and z, u subproblems is equivalently expressed as

$$\begin{array}{l}\tilde{u}=\arg \underset{z}{\min } \tau (-{\gamma }_1^T(D^{\alpha }u-\tilde{w})+\frac{{\mu }_1}{2}{\Vert D^{\alpha }u-\tilde{w}\Vert }_2^2)+\epsilon (-{\gamma }_2^T(D^{2}u-\tilde{v})+\frac{{\mu }_2}{2}{\Vert D^{2}u-\tilde{v}\Vert }_2^2)\\ +\frac{{\mu }_3}{2}{\Vert \Phi u-y\Vert }_2^2 -{\gamma }_3^T(\Phi u-y)-{\gamma }_4^T(u-z)+\frac{{\mu }_4}{2}{\Vert u-z\Vert }_2^2\end{array}.$$

(20)

We can see that the problem in Equation (20) is a quadratic function optimization problem. To reduce the calculation, the gradient descent method was used

$$\tilde{u}=u-\eta \cdot d,$$

(21)

$\eta$ is the step size. $u$ can be reconstructed by every iterative update, $d$ indicates its gradient,

$$\begin{array}{l}d=\tau {(D^{\alpha })}^T({\mu }_1{D}^{\alpha }-{\gamma }_1-{\mu }_1\tilde{w})+\epsilon {(D^2)}^T({\mu }_2{D}^2-{\gamma }_2-{\mu }_2\tilde{v})\\ -{\gamma }_4+{\mu }_4(u-z)+{\Phi }^T({\mu }_3(\Phi u-y)-{\gamma }_3)\end{array},$$

(22)

where $\eta =abs(d^{T}d/d^{T}Gd)$ is the optimal step and $G={\mu }_1{(D^{\alpha })}^T{D}^{\alpha }+{\mu }_{4}I+{\mu }_3{\Phi }^T\Phi$.

2.2.3. z Subproblems

Similar to other subproblems, z subproblems becomes

$$\tilde{z}=\arg \underset{z}{\min }\beta {\Vert z-Wz\Vert }_2^2 -{\gamma }_4^T(\tilde{u}-z)+\frac{{\mu }_4}{2}{\Vert \tilde{u}-z\Vert }_2^2.$$

(23)

According to [18], the Equation (23) can be further transformed into

$$\underset{z}{\min } \frac{1}{2}{\Vert z-r\Vert }_2^2+\frac{\beta }{{\mu }_4}{\Vert z-Wz\Vert }_2^2,$$

(24)

where $r=(\tilde{u}-\frac{{\gamma }_4}{{\mu }_4})$, $r$ can be regarded as an approximation of z. Since the weight matrix $W$ represents the nonlocal means operator, the Equation (24) can be rewritten as

$$\underset{z}{\min } \frac{1}{2}{\Vert z-r\Vert }_2^2+\frac{\beta }{{\mu }_4}{\Vert z-Wr\Vert }_2^2.$$

(25)

Setting the gradient of the Equation (25) to zero, we acquire the closed-form solution as follows

$$\tilde{z}=\frac{{\mu }_{4}r+2\beta Wr}{{\mu }_4}.$$

(26)

Finally, the Lagrange multipliers are updated by the following

$$\{\begin{array}{l}{\gamma }_1^{k+1}={\gamma }_1^k-{\mu }_1(D^{\alpha }{u}^{k+1}-w^{k+1})\\ {\gamma }_2^{k+1}={\gamma }_2^k-{\mu }_2(D^2{u}^{k+1}-w^{k+1})\\ {\gamma }_3^{k+1}={\gamma }_3^k-{\mu }_3(\Phi u^{k+1}-y)\\ {\gamma }_4^{k+1}={\gamma }_4^k-{\mu }_4(u^{k+1}-z^{k+1})\end{array}.$$

(27)

Once we obtained an efficient solution for each separated subproblem, the overall algorithm will be more efficient to get better reconstruction. Given all the derivations above, the specific implementation process of the proposed algorithm is described in Algorithm 1.

Algorithm 1 HoFrTv algorithm.

Input: The observed measurement $y$, the measurement matrix Φ and ${\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,\beta$ Initialization: $u_0={\Phi }^{T}y$, ${\gamma }_1={\gamma }_2={\gamma }_3={\gamma }_4=0$, $w_0=v_0=z_0=0$  While Outer iteration unsatisfied do   While Inner iteration unsatisfied do    Solve w subproblem via Equation (17)    Solve v subproblem via Equation (19)    Solve u subproblem via Equation (21)    Compute the weight wij via Equation (11)

Solve z subproblem via Equation (26)

end while

Update multipliers via Equation (27)

end while

Output: the reconstructed image

3. Experimental Results and Discussion

In this section, the experimental results are presented to demonstrate the performance of the proposed algorithm HoFrTv model. In our implementation, we chose the USC-SIPI image database, which is a collection of digitized images. The database was maintained primarily to support research in image processing, image analysis, and machine vision. Ten images were selected for verification, the original images are shown in Figure 1.

Reconstructed image quality was measured using the peak signal-to-noise ratio (PSNR) and the structure similarity (SSIM) [29]. The calculation expression is as follows

$$\begin{array}{l}\mathrm{PSNR}=20{\log }_{10}\frac{\mathrm{R}}{\mathrm{RMSE}}\\ \mathrm{RMSE}=\sqrt{\frac{1}{MN}{\displaystyle {\sum }_{i=1}^M{\displaystyle {\sum }_{j=1}^N{(u_{ij}-{\tilde{u}}_{ij})}^2}}}\end{array}.$$

(28)

The root mean square error is the arithmetic square root of the mean square error, it can measure the deviation between the reconstructed image and the original image. Where ${\tilde{u}}_{ij}$ and $u_{ij}$ denote the pixel values of the reconstructed image and the original image, $\mathrm{R}$ is the maximum value of the image gray level range. SSIM is defined as

$$\mathrm{SSIM}=\frac{(2{\mu }_x{\mu }_y+C_1)(2{\sigma }_{xy}+C_2)}{({\mu }_x^2+{\mu }_y^2+C_1)({\sigma }_x^2+{\sigma }_y^2+C_2)},$$

(29)

where ${\mu }_x$ and ${\mu }_y$ are the mean value of the reconstructed image and the original image. ${\sigma }_x$ and ${\sigma }_y$ are the standard deviations of the reconstructed image and the original image. $C_1$ and $C_2$ are small positive numbers to avoid ${\mu }_x^2+{\mu }_y^2$ and ${\sigma }_x^2+{\sigma }_y^2$ being zero, and

$${\sigma }_{xy}={(\frac{1}{N-1})}^2{\displaystyle \sum _{i=1}^N{\displaystyle \sum _j^N(u_{ij}-{\mu }_x)}}({\tilde{u}}_{ij}-{\mu }_y).$$

(30)

The stopping criterion for all of the algorithms tested was set to

$$\frac{{\Vert {\tilde{u}}^t-{\tilde{u}}^{t+1}\Vert }_2}{{\Vert {\tilde{u}}^{t+1}\Vert }_2}\le {10}^{-3},$$

(31)

where ${\tilde{u}}^{t+1}$ and ${\tilde{u}}^t$ are the restored images at the current iterate and previous iterate respectively.

3.1. Parameter Selection

Since $\tau$ and $\epsilon$ were related to the weights of fractional-order and high-order total variation, the respective ranges of $\tau$ and $\epsilon$ were greater than 0 and less than 1. We constrained the summation of these two weights to be 1 in our proposed algorithm model. In the following content, we analyzed the influence of high-order and fractional-order weights parameters on the reconstructed results.

3.1.1. The Influence of High-Order

This subsection discusses the influence of high-order TV when Equation (14) does not exist in fractional-order, which means that the fractional-order is the traditional total variation. In order to investigate its effect, experiments were implemented with different weight parameters $\epsilon$ to high-order TV, the weights parameters value range was greater than 0 and less than 1. We randomly selected Lena image, the results are shown in

Table 1. Peak signal-to-noise ratio (PSNR) (dB)/structural similarity (SSIM) results for the different $\epsilon$ for Lena image.

$\mathit{\epsilon }$	Measurement Rates
	0.1	0.15	0.2	0.25	0.3
0.1	26.539/0.801	28.398/0.845	29.526/0.875	30.699/0.895	31.733/0.916
0.2	26.721/0.807	28.411/0.846	29.545/0.877	30.799/0.897	31.797/0.916
0.3	26.790/0.809	28.485/0.853	29.696/0.879	30.905/0.900	31.896/0.918
0.4	26.949/0.814	28.537/0.853	29.706/0.880	30.957/0.903	32.023/0.920
0.5	26.984/0.815	28.681/0.854	29.805/0.882	31.059/0.905	32.140/0.922
0.6	27.320/0.824	28.763/0.860	29.947/0.886	31.158/0.907	32.235/0.924
0.7	27.235/0.823	28.891/0.856	30.030/0.888	31.277/0.910	32.401/0.927
0.8	26.670/0.803	28.911/0.861	30.130/0.895	31.312/0.912	32.503/0.929
0.9	26.671/0.813	28.677/0.860	30.059/0.893	30.953/0.901	32.583/0.930

.

From Table 1, with the parameter ε increasing, the higher the parameter ε was, the better the result of reconstruction. This is due to high-order TV that can effectively reduce the stair casing artifacts in the reconstructed image. According to the experimental results, when parameter ε was between 0.6 and 0.9, the reconstruction results could achieve stable results. So, the parameter ε value range could be selected from 0.6 to 0.9. According to the experiment in the database, we could make appropriate selection values according to different images. In our experiments, we set ε = 0.7 and τ = 0.3 to balance the results.

3.1.2. The Influence of Fractional-Order

In this experiment, α is the fractional-order. The effect of this parameter on the image reconstruction performance was tested without high-order TV in Equation (14), ranging from 0.5 to 2. Image Lena was tested in the experiment. The results are shown in

Table 2. PSNR (dB)/SSIM results for the different fractional-order for Lena image.

α	Measurement Rates
	0.1	0.15	0.2	0.25	0.3
0.5	20.427/0.580	22.578/0.651	26.361/0.718	25.553/0.758	26.919/0.801
0.7	24.061/0.720	25.876/0.773	27.534/0.822	28.800/0.854	29.841/0.879
0.9	26.065/0.782	27.729/0.828	28.988/0.860	30.252/0.888	31.362/0.908
1	26.483/0.794	28.187/0.839	29.379/0.869	30.567/0.894	31.595/0.912
1.1	26.823/0.803	28.305/0.842	29.631/0.874	30.727/0.897	31.775/0.914
1.3	27.021/0.810	28.506/0.847	29.833/0.879	30.848/0.900	31.865/0.917
1.5	26.996/0.810	28.604/0.852	30.092/0.887	31.024/0.903	32.071/0.922
1.7	26.956/0.809	28.591/0.853	30.125/0.889	31.093/0.906	32.074/0.924
1.9	26.858/0.806	28.561/0.854	30.119/0.889	31.191/0.908	32.252/0.925
2.0	26.700/0.801	28.526/0.852	30.108/0.889	31.250/0.908	32.282/0.926

.

From Table 2, we could see that when α < 1, the reconstructed results were poor. The reason is fractional-order TV loses more details and textures. When α = 1, fractional-order TV converts to traditional TV. When α > 1, the larger the parameter α is, the better the textures and image details. We can see that the PSNR at α = 1 is lower than the PSNR at α > 1. We can select the fractional-order α between α = 1.3 and α = 2. However, when a value of α is too close to 2, the frequency of textures would be enhanced excessively that becomes a kind of noise. Finally, to achieve a good trade-off, in our experiments, we set α = 1.7 to balance the results.

3.1.3. The Influence of Non-Local Mean Regularization Kernel Window and Search Window

Non-local means regularization indicates the estimated value of the current pixel obtained by the weighted average of the pixels in the image that have a similar neighborhood structure. There is no doubt that the radius of the neighborhood kernel window and the radius of the neighborhood search window were vital to the experiment. If their values are too small, the self-similarity of the image cannot be fully utilized, and the characteristics of the non-local means cannot be used fully. If their values are too large, the image search area will become larger and the time will be longer, which leads to the fact that the algorithm efficiency will be lower. There are different kernel windows to search window ratio (k:s) result values for different measurement rates in

Table 3. PSNR (dB)/SSIM and time (s) results for the different k:s ratios.

Rates	PSNR (dB)/SSIM and Time (s)
	k:s = 1:3		k:s = 3:7		k:s = 5:11		k:s = 7:15
0.1	26.448/0.793	198	27.460/0.828	289	27.463/0.828	901	27.510/0.832	3715
0.15	28.553/0.854	116	29.108/0.870	440	29.214/0.870	831	29.161/0.872	3996
0.2	30.097/0.888	93	30.379/0.896	366	30.525/0.896	1327	30.837/0.902	2439
0.25	31.183/0.908	120	31.612/0.916	311	32.158/0.916	1039	32.034/0.921	3639
0.3	32.208/0.925	99	32.821/0.933	208	32.958/0.933	1181	32.869/0.933	3610

. Figure 2 shows the PSNR and time curves with measurement rates = 0.1 and 0.15 for different k:s. We can see from Table 3 and Figure 2 that the performance was the best when k:s was 3:7 considering from time and PSNR. Therefore, the kernel window and search window were set to be 3 and 7 in the following experiments.

3.2. Parameter Verification

3.2.1. Verify Fractional Order Existence Performance

According to Section 3.1, we set fractional-order α = 1.7 in this experiment, in order to verify the validity of fractional-order existence. On the basis of Section 3.1.1, the fractional-order α = 1.7 was added. The test image selection was the same as the previous tested image Lena. The results are shown in

Table 4. PSNR (dB)/SSIM results for the different $\epsilon$ for the Lena image.

ε	Measurement Rates
	0.1	0.15	0.2	0.25	0.3
0.1	27.016/0.811	28.731/0.856	30.155/0.890	31.209/0.910	32.204/0.925
0.2	27.097/0.814	28.795/0.858	30.190/0.891	31.286/0.911	32.241/0.925
0.3	27.186/0.817	28.852/0.859	30.238/0.892	31.348/0.912	32.292/0.926
0.4	27.283/0.820	28.939/0.862	30.291/0.893	31.422/0.913	32.365/0.927
0.5	27.407/0.824	29.056/0.864	30.358/0.895	31.469/0.914	32.450/0.928
0.6	27.497/0.827	29.159/0.867	30.396/0.896	31.581/0.916	32.525/0.929
0.7	27.600/0.831	29.222/0.869	30.400/0.897	31.610/0.917	32.636/0.931
0.8	27.405/0.828	29.139/0.867	30.347/0.894	31.731/0.919	32.714/0.932
0.9	26.883/0.810	28.967/0.864	30.156/0.892	31.499/0.913	32.713/0.932

.

Comparing Table 1 with Table 4, we could see that the image reconstructed results in Table 4 were overall higher. The performance of fractional-order existence was better than that without fractional-order.

3.2.2. Verify High-Order Existence Performance

Similarly, in this experiment. In Section 3.1.2, the weight of high-order TV was 0.7. In order to verify the validity of high-order existence. On the basis of Section 3.1.2, the high-order was added. The test image was Lena. The results are shown in

Table 5. PSNR (dB)/SSIM results for the different fractional-order for the Lena image.

α	Measurement Rates
	0.1	0.15	0.2	0.25	0.3
0.5	21.541/0.626	26.831/0.810	28.642/0.856	29.952/0.885	31.306/0.909
0.7	25.537/0.762	28.106/0.841	29.492/0.872	30.613/0.895	31.920/0.917
0.9	26.482/0.791	28.523/0.854	30.015/0.885	31.189/0.908	32.437/0.926
1	26.745/0.800	28.611/0.853	30.113/0.886	31.337/0.909	32.524/0.927
1.1	27.086/0.814	28.725/0.857	30.195/0.887	31.367/0.91	32.553/0.928
1.3	27.495/0.826	28.996/0.866	30.328/0.892	31.450/0.912	32.556/0.928
1.5	27.552/0.829	28.946/0.865	30.319/0.894	31.445/0.912	32.611/0.929
1.7	27.493/0.829	28.977/0.866	30.347/0.894	31.489/0.916	32.714/0.932
1.9	27.190/0.822	29.053/0.868	30.357/0.895	31.617/0.918	32.769/0.932
2.0	26.975/0.820	28.947/0.867	30.500/0.899	31.787/0.920	32.852/0.933

.

From comparing Table 2 with Table 5, we could see that the performance of high-order existence was better than that without high-order by comparing data one by one.

In order to verify the effect of the proposed algorithm, we gave the visual comparison of different TV modes in algorithms to test images Lena and Peppers. From Figure 3 and Figure 4, traditional TV with nonlocal means regularization in Figure 3a and Figure 4a usually produced undesirable staircase artifact and painting-like effects. Even though fractional-order TV in Figure 3b and Figure 4b could enhance image edges and textures, it could cause non-smooth of smooth areas. Although, the high-order TV in Figure 3c and Figure 4c was capable of alleviating the problem caused by traditional TV, still smoothed out the details of the image, which led to the fact that the effect was not ideal. In Figure 3d and Figure 4d, this was obvious that our proposed algorithm made the fractional-order and the high-order TV complement each other on image reconstruction. From the figures, we could see that our proposed algorithm preserved image edges and textures more effectively, and could alleviate the staircase artifact and painting-like effects produced by nonlocal means regularization.

3.3. Comparison to Other Reconstruction Algorithms

In this experiment, we compared the proposed method with several state-of-the-art algorithms: TVNLR [18], BCS-TV [21], TVAL3 [28], and two non-TV based algorithms: BCS-SPL [30] and MH-BCS [31]. In order to reduce the computational complexity and memory requirements. Images were divided into non-overlapping blocks of size 32 × 32 for all algorithms.

Table 6. The PSNR (dB)/SSIM results of various algorithms with different measurement rates.

Image	Algorithms	Measurement Rates
		0.1	0.15	0.2	0.25	0.3
barbara	TV-NLR	24.926/0.723	26.458/0.781	27.236/0.807	27.970/0.832	29.113/0.859
	MH-BCS	25.605/0.740	27.162/0.798	28.103/0.815	29.126/0.853	30.015/0.853
	BCS-TV	24.215/0.681	25.560/0.738	26.585/0.778	27.421/0.807	28.167/0.830
	BCS-SPL	23.619/0.640	25.067/0.697	26.209/0.739	27.105/0.770	27.807/0.793
	TVAL3	23.563/0.664	25.814/0.751	26.812/0.788	27.480/0.810	28.335/0.834
	HoFrTV	25.706/0.758	27.253/0.807	28.304/0.834	29.317/0.861	30.211/0.879
boat	TV-NLR	24.136/0.674	25.553/0.746	26.475/0.782	27.509/0.819	28.350/0.847
	MH-BCS	24.322/0.682	25.879/0.745	27.170/0.796	28.198/0.831	29.011/0.851
	BCS-TV	23.190/0.623	24.691/0.702	26.013/0.758	27.156/0.800	28.152/0.834
	BCS-SPL	22.974/0.596	24.079/0.647	25.090/0.69	26.088/0.733	26.929/0.766
	TVAL3	23.393/0.631	25.035/0.716	26.421/0.773	27.403/0.810	28.445/0.84
	HoFrTV	24.465/0.699	26.258/0.774	27.523/0.816	28.615/0.851	29.953/0.881
cameraman	TV-NLR	24.561/0.794	26.245/0.838	27.356/0.864	28.904/0.890	29.968/0.909
	MH-BCS	24.366/0.751	26.672/0.813	27.939/0.853	29.229/0.875	30.892/0.912
	BCS-TV	23.613/0.761	25.196/0.807	26.652/0.844	27.864/0.871	28.910/0.892
	BCS-SPL	22.785/0.691	24.284/0.740	25.626/0.783	26.686/0.813	27.744/0.838
	TVAL3	23.169/0.727	25.442/0.817	26.741/0.850	28.018/0.877	29.284/0.900
	HoFrTV	25.102/0.813	26.992/0.858	28.791/0.887	30.244/0.907	31.600/0.926
house	TV-NLR	29.519/0.830	31.809/0.862	33.018/0.879	34.153/0.894	35.306/0.909
	MH-BCS	30.006/0.828	32.348/0.866	33.235/0.878	34.825/0.899	35.609/0.911
	BCS-TV	27.763/0.79	29.784/0.829	31.111/0.852	32.282/0.870	33.413/0.888
	BCS-SPL	26.627/0.742	28.104/0.770	29.815/0.814	31.059/0.838	31.464/0.840
	TVAL3	26.047/0.750	29.885/0.837	31.385/0.860	32.653/0.878	33.584/0.892
	HoFrTV	30.122/0.838	32.446/0.870	33.881/0.887	35.143/0.903	35.953/0.915
lena	TV-NLR	26.272/0.789	28.030/0.840	29.334/0.869	30.373/0.893	31.249/0.909
	MH-BCS	26.773/0.797	28.520/0.853	29.757/0.872	30.815/0.897	31.840/0.918
	BCS-TV	25.486/0.752	27.043/0.805	28.478/0.846	29.569/0.872	30.623/0.894
	BCS-SPL	24.56/0.689	26.142/0.742	27.423/0.784	28.473/0.816	29.443/0.841
	TVAL3	24.510/0.724	27.469/0.818	28.633/0.850	29.644/0.876	30.570/0.895
	HoFrTV	27.493/0.829	29.071/0.869	30.403/0.896	31.745/0.916	32.655/0.931
mandrill	TV-NLR	22.014/0.469	22.483/0.527	23.495/0.602	24.018/0.656	23.835/0.671
	MH-BCS	22.003/0.439	23.087/0.545	24.036/0.602	24.472/0.677	25.172/0.713
	BCS-TV	21.895/0.453	22.695/0.525	23.380/0.587	23.956/0.638	24.499/0.682
	BCS-SPL	22.069/0.449	22.701/0.504	23.264/0.559	23.695/0.599	24.186/0.639
	TVAL3	22.351/0.471	22.875/0.537	23.437/0.592	23.954/0.643	24.590/0.686
	HoFrTV	22.202/0.486	23.189/0.560	23.942/0.621	24.613/0.680	25.231/0.720
peppers	TV-NLR	26.601/0.829	29.119/0.879	30.762/0.906	32.295/0.925	33.845/0.940
	MH-BCS	26.929/0.805	29.035/0.854	30.614/0.884	31.739/0.902	32.938/0.918
	BCS-TV	25.802/0.790	28.270/0.850	30.001/0.882	31.432/0.905	32.799/0.924
	BCS-SPL	24.241/0.695	25.974/0.747	27.280/0.783	28.544/0.812	29.604/0.836
	TVAL3	24.515/0.767	27.949/0.853	29.709/0.885	31.198/0.908	32.467/0.924
	HoFrTV	28.497/0.868	30.991/0.906	32.755/0.929	34.309/0.944	35.732/0.954
ruler	TV-NLR	14.799/0.298	15.496/0.441	16.445/0.546	16.827/0.591	19.348/0.783
	MH-BCS	14.895/0.238	19.312/0.638	20.312/0.638	21.965/0.792	22.895/0.856
	BCS-TV	15.172/0.309	15.811/0.451	16.707/0.539	17.457/0.605	18.059/0.656
	BCS-SPL	15.870/0.401	16.550/0.497	17.493/0.595	18.365/0.661	19.141/0.712
	TVAL3	15.300/0.274	15.392/0.363	16.269/0.483	17.158/0.585	18.019/0.661
	HoFrTV	15.117/0.346	16.188/0.512	17.589/0.618	18.783/0.710	20.373/0.814
testpat	TV-NLR	16.397/0.723	19.372/0.831	19.440/0.792	24.283/0.929	26.333/0.959
	MH-BCS	16. 819/0.779	19.135/0.828	20.142/0.856	22.379/0.879	24.798/0.956
	BCS-TV	15.981/0.709	18.798/0.820	21.241/0.888	23.224/0.932	24.934/0.960
	BCS-SPL	14.734/0.497	16.637/0.571	18.178/0.626	19.342/0.666	20.452/0.696
	TVAL3	14.834/0.613	17.719/0.752	19.338/0.799	22.402/0.900	24.212/0.936
	HoFrTV	17.799/0.803	22.762/0.910	26.047/0.944	28.512/0.965	30.191/0.970
Resolutionchart	TV-NLR	20.726/0.880	25.863/0.950	27.620/0.966	32.921/0.982	35.516/0.988
	MH-BCS	18.511/0.707	20.473/0.783	22.234/0.819	24.008/0.862	25.640/0.886
	BCS-TV	9.173/0.578	9.649/0.6718	10.276/0.721	9.714/0.741	10.012/0.76
	BCS-SPL	16.213/0.550	18.068/0.607	19.215/0.627	20.384/0.654	21.410/0.676
	TVAL3	16.411/0.675	21.741/0.888	24.951/0.942	27.899/0.966	30.475/0.980
	HoFrTV	20.311/0.861	24.204/0.923	26.925/0.956	30.100/0.971	33.498/0.983

displays their PSNR and SSIM values with the growth of the measurement rates for each image. We added additional test images and reconstruction results in the Appendix A. Figure 5 and Figure 6 display PSNR and SSIM values respectively with the growth of the measurement rates for two images. From Table 6 and Figure 5 and Figure 6, it can be seen that the HoFrTV algorithm performed best for the images at different measurement rates. The visual quality of the reconstructed images was used to further verify the effectiveness of the proposed algorithm. To compare the results visually, Figure 7 and Figure 8 optionally display some reconstructed images and local enlargements obtained using the different algorithms with a 0.2 measurement rate. We can clearly see that the visual quality of the images recovered by HoFrTV was better than that of others. HoFrTV could efficiently reconstruct fine details and textures while preserving sharp edges and avoid producing painting-like effects.

3.4. Computational Complexity

The experiments were performed under the MATLABR2018a environment with Intel Corei5-4200 CPU of 3.4 GHz and 4.0 GB RAM.

Table 7. Average reconstruction time(s).

Algorithms		BCS-SPL	TVAL3	MH-BCS	TV-NLR	HoFrTV	BCS-TV
Time (s)	Rate = 0.1	4.8	3.5	15.6	150.6	322.3	392.3
	Rate = 0.15	3.1	3.1	18.3	150.2	327.5	465.4
	Rate = 0.2	2.5	2.4	16.6	148.9	316.4	530.3
	Rate = 0.25	2.4	2.2	18.4	149.4	326.5	627.4
	Rate = 0.3	2.5	2.0	14.2	151.3	324.4	775.6

and Figure 9 give the time of reconstructing image at different measurement rates. From this result, we could see that BCS-SPL, TVAL3, and MH-BCS were faster than the others. Comparing with TVNLR, HoFrTV, and BCS-TV, HoFrTV was faster than BCS-TV and slower than the TVNLR algorithm. It is because at each iteration needs to calculate the high-order and fractional-order total variation in the reconstruction process. However, the HoFrTV algorithm had higher reconstruction quality.

4. Conclusions

In this paper, a hybrid high-order and fractional-order total variation with nonlocal means regularization model was proposed for compressive sensing image reconstruction. ALM and ADMM methods were used to solve this model. Our proposed algorithm makes the fractional-order and the high-order total variation complement each other on image reconstruction. It can solve the problem of non-smooth in smooth areas when fractional-order total variation can enhance image edges and textures. In addition, it also addresses high-order total variation alleviates the staircase artifact produced by traditional total variation, still smoothes the details of the image and the effect is not ideal. Meanwhile, the proposed algorithm suppresses painting-like effects produced by nonlocal means regularization. Experimental results show that, comparing with several state-of-the-art algorithms, the reconstructed images obtained by the proposed approach not only outperformed many existing methods in terms of PSNR and SSIM but also had better visual quality.