Filtered back projection (FBP) [2] is a spatial processing technique based on Fourier transform. Firstly, the projection data were filtered to remove noise and improve image quality. Then the backprojection operation was carried out, that is, the value of each pixel was assigned to the detector under the corresponding projection Angle. Finally, the back projection results were superimposed to obtain the reconstructed CT images. It is characterized by convolutional processing of the projections under each acquisition projection Angle before back projection, thus improving the shape artifacts caused by the point spread function, and the reconstructed image quality is better. Despite its overall acceptable performance, CT studies that are reconstructed with FBP can be affected by high image noise, artifacts (e.g. streak artifacts), or poor low-contrast detectability in specific clinical scenarios.

GIF is a local linear filter with good edge preservation and low time complexity. In GIF, the filtered output $f^{GIF}$ is a linear transformation of the guidance image $I$ into a square window ${\omega }_n$ with pixel n as the center and radius r [14] :

$f_i^{GIF}=a_n{I}_i+b_n\forall i\in {\omega }_n$ (1)

where i is pixel index. $\left(a_n,b_n\right)$ are linear coefficients assumed to be constant in ${\omega }_n$. In this work, the size of ${\omega }_n$ is 3 × 3. The $\left(a_n,b_n\right)$ are determined by minimizing the following cost function in ${\omega }_n$:

$E\left(a_n,b_n\right)={\displaystyle \underset{i\in {\omega }_n}{\sum }\left({\left(a_n{I}_i+b_n-f_i\right)}^2+\epsilon a_n^2\right)}$ (2)

where $\epsilon$ is a regularization parameter that penalizes large $a_n$values. $f_i$ is the filtering input of image $f$ at pixel $i$.

Equation (2) represents the linear ridge regression model, and its solution can be expressed by the linear regression:

$a_n=\frac{\frac{1}{\left|\omega \right|}{\displaystyle \underset{i\in {\omega }_n}{\sum }{f}_i}{I}_i-{\bar{f}}_n{\mu }_n}{{\sigma }_n^2+\epsilon }$ (3)

$b_n={\bar{f}}_n-a_n{\mu }_n$ (4)

where ${\mu }_n$ and ${\sigma }_n{}^2$ are the mean value and variance of the guidance image I in ${\omega }_n$, respectively. $f_n$is the mean value of input image $f$ in ${\omega }_n$. $\left|\omega \right|$is the number of pixels in ${\omega }_n$. If the guidance image I is set to the filtering input image $f$. We can obtain:

$a_n=\frac{{\sigma }_n^2}{{\sigma }_n^2+\epsilon }$ (5)

$b_n=\left(1-a_n\right){\mu }_n$ (6)

Because a given pixel $i$ is involved in several windows, the values of $J_i$ are not identical when computed in different windows. A simple strategy is to average all values of $f_i^{GIF}$. After computing $\left(a_n,b_n\right)$ for all windows, the output of the filter can be computed as:

$f_i^{GIF}=\frac{1}{\left|\omega \right|}{\displaystyle \underset{n\in {\omega }_n}{\sum }\left(a_n{I}_i+b_n\right)}={\bar{a}}_n{I}_i+{\bar{b}}_n$ (7)

where ${\bar{a}}_n=1/\left|\omega \right|{\displaystyle {\sum }_{n\in {\omega }_i}{a}_n}$and ${\bar{b}}_n=1/\left|\omega \right|{\displaystyle {\sum }_{n\in {\omega }_i}{b}_n}$ are the average coefficients calculated from all windows overlapping $i$.

A CT imaging system can be modeled as:

$Af=p$ (8)

where $A=\left(a_{ij}\right)$ denotes an $M\times N$ measurement matrix, $a_{ij}$is the contribution of the $j^{th}$ pixel to the $i^{th}$ X-ray, $p$ is the projection data collected by the detector, and $f\in R^N$ is the linear attenuation coefficients of the measured object. The task of CT imaging is to reconstruct $f$ from $p$.

The convergence of the following simultaneous iterative scheme is considered [16] [17] :

$f^{\left(K+1\right)}=f^{\left(K\right)}+{\lambda }_K{V}^{-1}{A}^{*}W\left(p-A{f}^{\left(K\right)}\right)$ (9)

where $K=0,1\dots ,{\lambda }_K>0$ is the relaxation coefficient, $V$ and $W$are two positive definite diagonal matrices of order $N$ and $M$.

The SART algorithm is a special case of Equation (9), and its formula can be expressed as [5] :

$f_j^{\left(K+1\right)}=f_j^{\left(K\right)}+\frac{{\lambda }_K}{{\displaystyle \underset{i=1}{\overset{M}{\sum }}{a}_{ij}}}{\displaystyle \underset{i=1}{\overset{M}{\sum }}\left[\frac{p_i-{\displaystyle \underset{n=1}{\overset{N}{\sum }}{a}_{in}}{f}_n^{\left(K\right)}}{{\displaystyle \underset{n=1}{\overset{N}{\sum }}{a}_{in}}}\right]}{a}_{ij}$ (10)

with $V$ and $W$ defined:

$V=\mathrm{diag}\left(V^1,V^2,\dots ,V^N\right)$ (11)

$W=\mathrm{diag}\left(W^1,W^2,\dots ,W^M\right)$ (12)

and with

$V^j={\displaystyle \underset{i=1}{\overset{M}{\sum }}{a}_{ij}},\text{for j}=1,\dots ,N$ (13)

$\frac{1}{W^i}={\displaystyle \underset{n=1}{\overset{N}{\sum }}{a}_{in}},\text{for}i=1,\dots ,M$ (14)

The SART relaxation parameter ${\lambda }_K$ was chosen by considering the convergence theorem [18] : Theorem 1: Assuming that $W^i>0$ for all $i=1,2,\dots M$, and if, for all $K\ge 0$,

$0\le \epsilon \le {\lambda }_K\le \left(2-\epsilon \right)/\max \left\{W^i|i=1,2,\dots ,M\right\}$ (15)

where $\epsilon$ is an arbitrarily small constant, then any sequence ${\left\{f^{\left(K\right)}\right\}}_{K=0}^{\infty }$ generated by SART converges to the weighted least-squares solution $f^{*}=\arg \min \left\{{\Vert Af-p\Vert }_2|f\in R^n\right\}$.

In order to generate a better SART iterative initial value, we calculate a Mask M and get the guided image $D$ by the following formula:

$D=\left(1-M\right)\cdot F^{GIF}+M\cdot f^{\left(K+1\right)}{}^{GIF}$ (16)

where $F$ denotes the FBP reconstructed image, $F^{GIF}$ is computed using the guided image filtering on $F$. For convenience, $S$ is represented as SART reconstruction image, and we use $S^{GIF}$ to represent $f^{\left(K+1\right)}{}^{GIF}$. In such a way, M adjusts the relative contribution of information from different sources during the generation of initial values in the expectation of obtaining better initial values and thus improving the quality of reconstructed images in the subsequent iterative reconstruction process. We add a threshold to compute M by looking for pixels with a small difference between $F^{GIF}$ and $S^{GIF}$:

$M^{Shad}=\{\begin{array}{c}\begin{array}{cc}1& whe{n}_{}{F}_{{}^{Lin}}^{GIF}-S_{{}^{Lin}}^{GIF}\le {\tau }_{shad}\end{array}\\ \begin{array}{cc}{0}_{{}_{}}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}{}_{}otherwise.& \end{array}& \end{array}& \end{array}\end{array}\end{array}$ (17)

where $F_{{}^{Lin}}^{GIF}$ and $S_{{}^{Lin}}^{GIF}$ are linearized versions of the corresponding $F^{GIF}$ and $S^{GIF}$. The SART algorithm is used to iterate over the initial values to obtain the final reconstructed image.

While the GIF can reduce noise, it cannot add detail information that may be present in the FBP reconstructed image. To transfer the detail information we begin by computing a detail layer from the FBP reconstructed image as the following ratio:

$F^{Detail}=\frac{F+\epsilon }{F^{GIF}+\epsilon }$ (18)

The ratio captures the local detail variation in $F$ and is commonly called a quotient image [19] or ratio image [20] in computer vision. Figure 2 shows that the advantage of using the GIF to rather than a classic low-pass Gaussian filter is that the haloing was reduced.

Although the FBP reconstructed image contains detail information, it also contains noise that may lead to transfer spurious detail. Therefore, $\epsilon$ was added to both the numerator and denominator of the Equation (19) to reject transfer the noise and also avoid division by zero. In this work, $\epsilon =9\times {10}^{-5}$ and $\epsilon =9\times {10}^{-7}$ were used in simulation experiment. To transfer the detail information, the final image $D$ is computed as:

$D=\left(1-M\right)\cdot F^{GIF}\cdot F^{Detail}+M\cdot f^{\left(K+1\right)}{}^{GIF}$ (19)

In the process of our proposed method, the results obtained by FBP reconstruction and SART reconstruction of projection data are denoted as $F$ and $S$. Then $F$ and $S$ are filtered by the guided image, respectively, to obtain the corresponding results $F^{GIF}$ and $S^{GIF}$. In order to better retain and transmit the detail information to improve the delicacy and accuracy of the reconstructed image, the detail information $F^{Detail}$ is calculated from $F$ through Equation (18). In order to provide a better initial value for the following SART reconstruction, this detail layer is transferred to $F^{GIF}$ and combined with $S^{GIF}$ to obtain the final image $D$ as shown in Equation (19). In the process of obtaining the initial value, Mask calculated by Equation (17) is used as a weight, controlling the contribution of the information extracted from two different sources $F^{GIF}$ and $S^{GIF}$ in the initial value. Then $D$ is as the guidance image to filter the image $S^{GIF}$, and the obtained image $L^{GIF}\left(S^{GIF}\right)$:

$L_i^{GIF}\left(f^{\left(K+1\right)}{}^{GIF}\right)=a_n{D}_i+b_n\begin{array}{cc}& \forall i\in {\omega }_n\end{array}$ (20)

where $f^{(K+1)}{}^{GIF}$ is $S^{GIF}$. And $L_{}^{GIF}\left(f^{\left(K+1\right)}{}^{GIF}\right)$ denotes the image $f^{\left(K+1\right)}{}^{GIF}$ after GIF. $i$ is pixel index, $\left(a_n,b_n\right)$ are linear coefficients assumed to be constant in square window ${\omega }_n$ which centered at pixel $n$.

$L_i^{GIF}\left(f^{\left(K+1\right)}{}^{GIF}\right)$ is used as the initial value of SRAT iteration to participate in the following SART reconstruction process for iteration until the final image is obtained.

To explain the SART-DT algorithm more clearly, the corresponding pseudocode is presented, as illustrated in Algorithm 1.