In-line phase-contrast computed tomography (IL-PCCT), a density difference within the sample causes a change in the X-ray phase shift, which has passed through the sample [ 13] . During the tomography scans, the projection images are collected from various views evenly distributed over $\pi$-view in BL13W1 of the Shanghai synchrotron radiation facility (SSRF).

A significant proportion of phase-retrieval is performed before IL-PCCT reconstruction. This study uses phase-attenuation duality Paganin (PAD-PA) method [ 14] to solve the problem of phase retrieval. The PAD-PA method can be described by the following formula:

${\phi }_{\theta }\left(x,y\right)=\frac{1}{2}\gamma \ln \left[{ℱ}^{-1}\left(\frac{ℱ\left(I_{\theta }^D\left(x,y\right)\right)}{\pi \gamma \lambda D\left({\tau }^2+{\upsilon }^2\right)+1}\right)\right]$ (2)

where ${\phi }_{\theta }\left(x,y\right)$ denotes the phase-shift function at the projection angle $\theta$, $I_{\theta }^D\left(x,y\right)$ represents the intensity function measured by the detector at a distance $D$, $\lambda$ denotes the X-ray wavelength, $\gamma$ is a constant, and $\left(\tau ,\upsilon \right)$ are the coordinates of $\left(x,y\right)$in the Fourier domain, and $ℱ$ and ${ℱ}^{-1}$ are the 2D Fourier transform and its inverse operators.

PAD-PA is a phase retrieval technique that relies on single-distance diffraction data. This approach exploits the inherent relationship between phase and attenuation in the diffraction pattern to recover the phase information of the sample, making it especially effective for X-ray diffraction imaging.

A CT imaging system can be modeled as the following discrete linear system after phase retrieval:

$A\delta =\phi ,$ (3)

where $A\in R^{M\times N}$ denotes the CT system matrix, $\phi \in R^M$ denotes the projection data from detector after necessary processing, and $\delta \in R^N$ denotes the phase CT image of the object to be reconstructed. $M$ is the number of detector bins and $N$ is the number of the pixels of the CT image. The task of CT imaging aims to reconstruct $\delta$ from the projection data $\phi$ and the system matrix $A$.

Apply the filter to the projection data ${\phi }_{\theta }\left(x,y\right)$ to obtain ${\phi }_{\theta }^{filtered}\left(x,y\right),$the filtered projection data will be back-projected into the image space, with the goal of mapping the intensity distribution of the projections back to the corresponding image locations. The back-projection process is typically implemented through the following integral formula [ 5] :

$\delta \left(x,y\right)=\underset{0}{\overset{\pi }{{\displaystyle \int }}}{\phi }_{\theta }^{filtered}\left(x,y\right)|d\theta$ (4)

SART is an iterative method for image reconstruction that updates all pixels simultaneously in each iteration. It is especially useful for situations with noisy or incomplete data and provides a more efficient approach.

The SART algorithm formula can be expressed as [ 5] :

${\delta }_j^{\left(K+1\right)}={\delta }_j^{\left(K\right)}+\frac{{\lambda }_K}{{{\displaystyle \sum }}_{i=1}^M{a}_{ij}}\underset{i=1}{\overset{M}{{\displaystyle \sum }}}\left[\frac{{\phi }_i-{{\displaystyle \sum }}_{n=1}^N{a}_{in}{\delta }_n^{\left(K\right)}}{{{\displaystyle \sum }}_{n=1}^N{a}_{in}}\right]a_{ij}$ (5)

where the relaxation parameter ${\lambda }_k$controls the step size of each update balancing convergence speed and stability.

The CT reconstruction problem (3) can be solved by the CS-based reconstruction method to minimize the image isotropic TV regularized by the projections. This process is equivalent to solving the following optimization program:

$\delta =\underset{\delta }{argmin}{\left(A\delta -\phi \right)}_2^2+{\lambda }_1^{*}{\delta }_{TV}^{iso}$ (6)

The model described by Equation (4) was investigated using the alternating minimization procedure [8] :

1: a SART step to solve (3);

2: descending the TV-gradient to minimize the image TV. The gradient descent is controlled via the step size parameter ${\lambda }_1^{*}$, If the step size of the gradient descent is too strong the image becomes uniform and inconsistent with the projection data. On the other hand, if the step size of the gradient descent is too small, the algorithm reduces to standard SART with a positivity constraint included. The SART-step loop is executed $K_{max}$ times. $N_{grad}$ denotes total number of gradient descents.

The CT reconstruction problem given in Equation (1) can be solved to minimize the image anisotropic TV regularized. This process is equivalent to solving the following optimization program:

$\delta =\underset{x}{argmin}{\left(A\delta -\phi \right)}_2^2+{\lambda }_2^{*}{\delta }_{TV}^{ani}$ (7)

Equation (5) is decoupled data fidelity minimization and image regularization problems based on proximal forward–backward splitting (PFBS), the SART algorithm is used to solve the data fidelity. ADMM is used to solve the image regularization, The ADMM-step loop is executed $N_2$ times, ${\lambda }_2^{*}$ is the regularization parameter [ 13] .

Three metrics, the MSE, the PSNR, and structural similarity index (SSIM), are used to evaluate the quality of reconstructed images.

The PSNR is a traditional measure of image quality, and a larger PSNR value represents better quality. It is defined as follows:

$MSE\left(x,y\right)=\frac{1}{M\times M}\underset{i=1}{\overset{M}{{\displaystyle \sum }}}\underset{j=1}{\overset{N}{{\displaystyle \sum }}}{\left(x_{i,j}-y_{i,j}\right)}^2$ (8)

$PSNR\left(x,y\right)=10{\log }_{10}\left(\frac{Pea{k}^2}{MSE}\right)$ (9)

where MSE is the mean square error function, x denotes the reference image, y denotes the reconstructed image, and $M\times M$ is the size of $x$ and $y$; $x_{i,j}$ and $y_{i,j}$ represent the pixel intensities of $x$ and $y$ in some pixel $\left(i,j\right)$, respectively, and Peak represents the largest pixel intensity in the normalized image.

We use the structural similarity (SSIM) index [15] as a quantitative measure for the reconstructed image quality. The computed local similarity index is defined on windows $x$ and $y$:

$ssim\left(x,y\right)=\frac{\left(2{\mu }_x{\mu }_y+c_1\right)\left(2{\sigma }_{xy}+c_2\right)}{\left({\mu }_x^2+{\mu }_y^2+c_1\right)\left({\sigma }_x^2+{\sigma }_y^2+c_2\right)}$ (10)

where ${\mu }_x$ and ${\mu }_y$ are the averages of $x$ and $y$; ${\sigma }_x^2$, ${\sigma }_y^2$ are the variances; ${\sigma }_{xy}$ is the covariance of $x$, $y$; and $c_1$, $c_2$ are two variables to stabilize the division with weak denominator. The overall SSIM is the mean of local similarity indices:

$SSIM\left(X,Y\right)=\frac{1}{N}\underset{i=1}{\overset{N}{{\displaystyle \sum }}}ssim\left(x_i,y_i\right)$ (11)

where $X$ is a reference image; $Y$ is a reconstructed image; $x_i,y_i$ are the corresponding windows indexed by $i$; and $N$ is the number of windows. Here, the size of the windows is $8\times 8$.

For conventional absorption X-ray CT, a common constraint is that linear attenuation coefficient has to be positive, as a negative value is physically not possible. However, for the analytic reconstruction algorithm, algorithm, such as the filtered back projection (FBP) algorithm, because the ramp filter zeros the dc term in the frequency domain, each back projection image will have zero average value. This means that the pixels in each back projection image will have negative and positive values. When all the back projections are added to form the final image, some negative locations may become positive and the average value may not be zero, but typically, the final image will still have negative pixels [16] .

In order to research the impact of the negative and positive values for result image of FBP algorithm, no constraints phase information image (negative and positive values), positive phase information image, and negative phase information image are reconstructed by FBP algorithm. The result of those images are shown in Figure 1 .

In this section, we will analyze the influence of non-negative constraints on reconstruction results for two groups of different samples. The first sample is liver sample, the sample structure is relatively simple, mainly includes two parts, the liver region and vascular space from Figure 2(b) and its local amplification area in Figure 2(e) , you can see that for homogeneous sample, nonnegative constraints will not be lost in Figure 2(a) and its local amplification area in Figure 2(d) samples of liver and vascular structure information. As can be seen from Figure 2(c) and its enlarged view, the negative refractive index reconstruction results correspond to the vascular region. Therefore, the structure information of the reconstructed object will not be lost when the simple structure sample is subjected to non-negative constraint.

To further analyze the influence of non-negative constraints on the reconstructed images of samples with complex structures, we also selected the reconstructed image results of bone trabecular samples for analysis. Figure 3(a) did not carry out the reconstruction results of non-negative constraints, and the reconstructed image contained complete structural information of the sample. The results of Figure 3(b) and local amplification Figure 3(e) show that the important structural information of the sample will be lost when the reconstructed image with non-negative constraint is applied to the complex structure sample. It can be seen from Figure 3(c) and its local amplification Figure 3(f) , that the negative refractive index also contains important structural information of the sample with complex structure.

For conventional absorption CT, a simple constraint is that every linear attenuation coefficient has to be positive, as a negative value is physically not possible. This condition can be implemented directly in the back projection step of the iterative method by a simple truncation. The application of this constraint results in a reduction of noise in the reconstructed image, especially in the background. This is illustrated in Figure 4(a) and Figure 4(c) (enlarged image in Figure 4(a) ), where one can clearly see how several noise in the background reconstructed image reconstructed images by SART algorithm. However, several noise can no longer be observed when applying the nonnegative constraint, this is illustrated in Figure 4(b) and Figure 4(d) (enlarged image in Figure 4(b) ).

It is common to enforce non-negative values to accommodate the physical non-negativity of x-ray attenuation. However, non-negativity constraint imposes limitations on the choice of iterative optimization algorithm for IL-PCCT. Compared with the analytic reconstruction algorithm, the main advantages of iterative methods are that regularization techniques are often used in iterative reconstruction algorithms. The choice of regularization term depends on the specific object examined and, on the desire, to preserve or emphasize particular features.

First, we view the application of SART., SART-TV, and SART-ATV algorithm to research the impact of the non-negativity constraint for IL-PCCT. Under the few-view condition, the images with a size of 1821 × 1821 pixels were reconstructed by these algorithms. The parameters were set as follows:

1) SART: $\mu =0.05$and $K_{\text{max}}$; 2) SART-TV: ${\lambda }_1^{*}=0.02$, $N_{grad}=10$; 3) SART-ATV: ${\lambda }_2^{*}=0.00002$, $N_2=20$. All parameters were obtained by optimal selection.

To better quantitively value the reconstruction results by different algorithm, the PSNR, SSIM, MSE and TIME are listed in Table 1 . We see that the reconstructed image obtained by SART-ATV algorithm is of the best quality.

Note: The reconstruction time was measured based on a reconstruction slice.