A feature refinement approach for statistical interior CT reconstruction

Without loss of generality, under the assumption of a mono-energetic beam, the x-ray CT measurement can be approximately expressed as a discrete linear system:

$$\begin{eqnarray}&&y=H\mu \end{eqnarray} \tag{ 1 }$$

where $y$ represents the obtained sinogram data (projections after system calibration and logarithm transformation), i.e. $y={{\left(\,{{y}_{1}},{{y}_{2}},...,{{y}_{M}}\right)}^{T}}$, $\mu$ is the vector of attenuation coefficients to be estimated, i.e. $\mu ={{\left({{\mu}_{1}},{{\mu}_{2}},...,{{\mu}_{N}}\right)}^{T}}$, where 'T' denotes the matrix transpose. The operator $H$ represents the system or projection matrix with the size of $M\times N$. The element of $H$ denotes the length of intersection of projection ray $i$ with pixel $j$. In our implementation, the associated element was pre-calculated by a fast ray-tracing technique stored as a file. The goal for CT image reconstruction is to estimate the attenuation coefficients $\mu$ from the measurement $y$ with $H$.

In this study, the proposed CS based statistical interior tomography approach contains three steps. First, an interior projection extrapolation based FBP reconstruction is conducted. Second, a TV based interior tomography is performed under the PWLS criteria with the FBP reconstruction as the initial guess. Third, a feature refinement step is added to recover the losing structure information and preserve details. The above three steps will be introduced in the following sub-sections, respectively.

Statistical iterative reconstruction (SIR) for x-ray CT, utilizing the statistical distribution resulting from the x-ray interaction process, has been extensively explored for radiation dose reduction in CT field (De Man et al 2001, Elbakri and Fessler 2002, Tang et al 2009, Ma et al 2012b, Huang et al 2013, Lauzier and Chen 2013, Niu et al 2014a). Among them, PWLS is one of state-of-the-art approaches. Mathematically, the PWLS criterion for CT image reconstruction can be rewritten as follows (Sukovic and Clinthorne 2000):

$$\begin{eqnarray}&&{{\mu}^{\ast}}=\underset{\mu \geqslant 0}{{\arg \,\min}}\left\{{{\left(y-H\mu \right)}^{{}}}{{^{\prime}}^{{}}}{{\Sigma}^{-1}}\left(y-H\mu \right)+\beta R\left(\mu \right)\right\}\end{eqnarray} \tag{ 2 }$$

where $H$ represents the system or projection matrix, $y$ represents the obtained sinogram data. $R\left(\mu \right)$ is the penalty term. $\beta$ is a hyper-parameter to balance the fidelity term and penalty term. $\Sigma$ is a diagonal matrix with the $i$th element of $\sigma _{i}^{2}$ which is the variance of sinogram data $y$. In the implementation, the variance of $\sigma _{i}^{2}$ was determined by the following mean-variance relationship (Wang et al 2008a, Ma et al 2012a):

$$\begin{eqnarray}&&\sigma _{i}^{2}=\frac{1}{{{I}_{0}}}\exp \left({{\overline{p}}_{i}}\right)\left(1+\frac{1}{{{I}_{0}}}\exp \left({{\overline{p}}_{i}}\right)\left(\sigma _{\text{e}}^{2}-1.25\right)\right)\end{eqnarray} \tag{ 3 }$$

where ${{I}_{0}}$ denotes the incident x-ray intensity, ${{\overline{p}}_{i}}$ is the mean of the sinogram data at bin $i$ and $\sigma _{\text{e}}^{2}$ is the background electronic noise variance.

According to the CS-based interior tomography theory, the interior ROI can be exactly reconstructed via minimizing an objective function which is associated with an appropriate sparsifying transform (Han et al 2009, Yu and Wang 2009, Yang et al 2010, 2012a, 2012b). Among all the existing sparsifying transforms, total variance (TV) is one of most commonly used:

$$\begin{eqnarray}&&\text{TV}\left(\mu \right)=\underset{s,t}{\sum}\sqrt{{{\left({{\mu}_{s,t}}-{{\mu}_{s-1,t}}\right)}^{2}}+{{\left({{\mu}_{s,t}}-{{\mu}_{s,t-1}}\right)}^{2}}+\delta}\end{eqnarray} \tag{ 4 }$$

here $s$ and $t$ are the indices of the location of the attenuation coefficients of the desired-image. $\delta$ is a small constant used for keeping differentiable with respect to the image intensity.

In this paper, the TV regularized PWLS statistical formulation is referred to as 'PWLS-TV', which can be solved using the steepest descent algorithm. Two key ingredients for improving the performance of PWLS-TV for interior tomography: projection extrapolation and interior feature refinement are integrated into the PWLS-TV framework.

In the original PWLS method, the initialization is usually with a smooth FBP image, whose low spatial-frequency components are nearly correct (Wang et al 2006). However, the FBP reconstruction from truncated projection has serious artifacts due to truncation. In this regard, the pre-processing on the truncated projection is needed to obtain a good initial estimation. In this paper, the projection data extrapolation step is performed to mitigate the truncation artifacts and to moderately extend the field of view. Specifically, this step approximates the support of the object by an ellipsoid, which is large enough to contain all object pixels, so that they can be possibly assigned to their correct locations (Wiegert et al 2005, Xu and Tsui 2011, Lauzier et al 2012). The typical procedure is as following: the projection data of an ellipsoid is simulated, and the extrapolation is performed between the boundary of the truncated projection and the projection of the simulated ellipsoid. The FBP algorithm is then used to reconstruct the extrapolated projection. This procedure ensures continuity at the boundary between the measured and extrapolated regions. In this paper, the FBP reconstruction with projection extrapolation is referred to as 'FBP-EP', and when we refer to 'PWLS-TV' in this paper, which always includes FBP-EP initialization.

In clinical practice, the image object is complicated. Small-scale and low-contrast structures are often very important for diagnosis. However, fine structures, such as small vessels and tissues are likely to be lost in the TV minimization reconstruction, especially when dose is very low. This is mainly due to the piecewise smooth assumption (Sidky et al 2006, Sidky and Pan 2008, Yu and Wang 2009, Bian et al 2013, 2014, Han et al 2015). It means the residual image between two successive iterations in PWLS-TV minimization may contain useful structure information, which has been discarded in the conventional PWLS-TV method. Inspired by this observation, an interior feature refinement (IFR) step is proposed to improve the performance of the TV minimization on detail preservation. This step aims to extract the structure information from the residual image and add back to the TV minimized image, which can be described in following equation:

$$\begin{eqnarray}&&{{\mu}_{\text{IFR}}}=\mu +{{f}_{t}}\otimes \nu \end{eqnarray} \tag{ 5 }$$

where ${{\mu}_{\text{IFR}}}$ is the feature refined image, $\mu$ is the TV minimized image before feature refinement, $\nu$ is the residual image between two successive iterations in PWLS-TV minimization. ${{f}_{t}}$ denotes a feature descriptor, ${{f}_{t}}\otimes \nu$ is the detected feature image and the symbol $\otimes$ represents the point-wise multiplication. It is clear that the performance of the IFR step heavily depends on the construction of the feature descriptor. This descriptor can be seen as a filter or mask for extracting the meaningful structure information and discarding the noise and noise-like artifacts. motivated by the idea from the computer vision community (Liu et al 2011), a feature descriptor ${{f}_{t}}$ is designed as:

$$\begin{eqnarray}&&{{f}_{t}}=1-\left|\frac{2{{\sigma}_{qp}}+{{C}_{1}}}{\sigma _{p}^{2}+\sigma _{q}^{2}+{{C}_{1}}}\right|\end{eqnarray} \tag{ 6 }$$

where local statistics ${{\sigma}_{p}}$, ${{\sigma}_{q}}$ and ${{\sigma}_{pq}}$ at pixel $i$ are defined as, ${{\sigma}_{p}}(i)={{\left(\frac{1}{N-1}{{{\sum}^{}}_{i\in {{p}_{i}}}}{{\left(\mu (i)-P(i)\right)}^{2}}\right)}^{1/2}}$, ${{\sigma}_{q}}(i)={{\left(\frac{1}{N-1}{{{\sum}^{}}_{i\in {{q}_{i}}}}{{\left({{\mu}_{d}}(i)-Q(i)\right)}^{2}}\right)}^{1/2}}$ and ${{\sigma}_{qp}}(i)=\frac{1}{N-1}{{{\sum}^{}}_{i\in \left(\,{{p}_{i}},{{q}_{i}}\right)}}\left(\mu (i)-P(i)\right)\left({{\mu}_{d}}(i)-Q(i)\right)$, where $P(i)=\frac{1}{N}{{{\sum}^{}}_{i\in {{p}_{i}}}}\mu (i)$, $Q(i)=\frac{1}{N}{{{\sum}^{}}_{i\in {{q}_{i}}}}{{\mu}_{d}}(i)$, ${{p}_{i}}$ and ${{q}_{i}}$ denote two local image patches with size of $\sqrt{N}\times \sqrt{N}$ centered at pixel $i$ in corresponding interior ROI, extracted from $\mu$ and a degraded images ${{\mu}_{d}}$ obtained by a linear Gaussian filter from $\mu$, respectively. The constants ${{C}_{1}}$ and ${{C}_{2}}={{C}_{1}}/2$ are introduced for numerical stability. The nature of the proposed model structure descriptor is a structure-related map, and it involves a contrast variation component and a structure correlation component. The former calculates the reduction of contrast variation due to the degraded operation, and the later quantifies the structural correction between the original and degraded images (please refer to (Liu et al 2011) for details of the descriptor). The value of each element in the feature descriptor image ${{f}_{t}}$ is in the interval [0, 1], and the larger its value is, the more likely it belongs to part of the structure. In this paper, the proposed CS based statistical interior tomography with feature refinement approach is referred to as 'PWLS-TV-IFR'.

Presented in figure 1 is a flowchart representation of our approach for CT interior image reconstruction. It is comprised of the following three main steps:

• (1)  
  Applying Projection Extrapolation: Given the truncated projection $y$, projection extrapolation is performed by simulating an ellipsoid support. Then the FBP algorithm is used to obtain the initial guess ${{\mu}^{0}}$.
• (2)  
  Minimizing the Cost-Function of PWLS-TV: Given the current image estimation ${{\mu}^{n}}$, the steepest descent scheme is utilized to yield a new image estimation, i.e. $\mu _{P}^{n}$, which can be expressed as follows:

  $$\begin{eqnarray}&&\mu _{P}^{n}={{\mu}^{n}}-{{\eta}^{n}}\times \left({{H}^{T}}\left({{\Sigma}^{-1}}\left(y-H{{\mu}^{n}}\right)\right)\right)-\beta \times \nabla \text{TV}\left({{\mu}^{n}}\right)\end{eqnarray} \tag{ 7 }$$

  where $\nabla \text{TV}\left({{\mu}^{n}}\right)$ represents the gradient of $\text{TV}\left({{\mu}^{n}}\right)$, ${{\eta}^{n}}$ represents the gradient step-size which can be calculated adaptively by the following estimator:

  $$\begin{eqnarray}&&{{\eta}^{n}}=\frac{{{G}^{T}}G}{{{(HG)}^{T}}\left({{\Sigma}^{-1}}(HG)\right)}\,\,\text{with}\,\,\,G\triangleq {{H}^{T}}\left({{\Sigma}^{-1}}\left(H{{\mu}^{n}}-y\right)\right)\end{eqnarray} \tag{ 8 }$$
• (3)  
  Performing Interior Feature Refinement: After each iteration of PWLS-TV, the reconstruction result $\mu _{P}^{n}$ is replaced by $\mu _{\text{IFR}}^{n}$ with a greater number of losing structure are extracted and recovered, which progressively improves the image quality. The feature refinement step is $\mu _{\text{IFR}}^{n}=\mu _{P}^{n}+f_{t}^{n}\otimes {{\nu}^{n}}$, where $\mu _{\text{IFR}}^{n}$ is the feature refined image, ${{\nu}^{n}}={{\mu}^{n}}-\mu _{P}^{n}$ is the residual image between two successive iterations in the process of the PWLS-TV minimization. $f_{t}^{n}$ is the feature descriptor extracted from $\mu _{P}^{n}$, and $f_{t}^{n}\otimes {{\nu}^{n}}$ is the detected structure image. Finally, update ${{\mu}^{n+1}}=\mu _{\text{IFR}}^{n}$.

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Output ${{\mu}^{n+1}}$ until the stopping criterion is satisfied. A modified gradient descent algorithm (O'Sullivan and Benac 2007) was adopted to solve the reconstruction problem. In the implementation, to find a stable convergent solution, the root mean square error (RMSE) metric was used to measure the reconstruction quality.

In summary, the pseudo-code for the presented PWLS-TV-IFR approach can be described as follows:

• 1:  
  Initialization: $\mu _{\text{IFR}}^{0}$  =  FBP-EP $\left\{y\right\}$;
• 2:  
  While a stopping criterion is not met;
• 3:  
  ${{\mu}^{n}}=\mu _{\text{IFR}}^{n-1},$
• 4:  
  calculate the gradient step-size ${{\eta}^{n}}$ using (8),
• 5:  
  obtain a new image estimation $\mu _{P}^{n}$ using (7),
• 6:  
  calculate the residual image ${{\nu}^{n}}={{\mu}^{n}}-\mu _{P}^{n}$,
• 7:  
  calculate the feature descriptor $f_{t}^{n}$ from $\mu _{P}^{n}$ using (6),
• 8:  
  extract the structure information from the residual image and refine the image estimation $\mu _{\text{IFR}}^{n}=\mu _{P}^{n}+f_{t}^{n}\otimes {{\nu}^{n}}$, n  =  n  +  1
• 9:  
  End

In line 1, an initial estimate of the to-be-reconstructed image is set to be the preliminary image reconstructed by the FBP-EP method. Each loop (lines 3–8) is performed by two components: minimizing the cost-function of PWLS-TV (lines 4–5), and feature refinement step (lines 6–8).

Since the feature descriptor which is designed to distinguish structure from noise and artifacts plays an important role in our proposed framework, the selection on several scalar parameters is critical. For example, the image patch size has to be carefully selected because the large one leads to better detail-detection ability, while the small one can benefit the computational load. Practically, the patch size of 3  ×  3 is a good choice for balancing the structure-detection ability and computational efficiency. Additionally, the parameter ${{C}_{1}}$ is included to avoid instability when $\sigma _{p}^{2}+\sigma _{q}^{2}$ is very close to zero. Therefore, it should be assigned a small constant (1  ×  10⁻⁵ in our work).

In this experiment, a digital XCAT phantom (Segars et al 2010) was used for experimental data simulations (as shown in figure 2). We chose a geometry that is representative for a mono-energetic fan-beam CT scanner setup with a circular orbit to acquire 1160 views over 2π. The distance from the rotation center to the curved detector is 570 mm and the distance from the x-ray source to the detector is 1140 mm. Each projection datum along an x-ray through the sectional image was calculated based on the known densities and intersection areas of the x-ray with the geometric shapes of the objects in the sectional image. An equi-spatial virtual detector was assumed, which is centered at the system origin and made perpendicular to the line from the system origin to the x-ray source. The number of channels per view is 168 with a total length of 235 mm. Each phantom is comprised of 512  ×  512 square pixels with the intensity value from 0.004 to 0.019. The size of each pixel is 0.5 mm  ×  0.5 mm. A ROI is defined by the local scanning beam, and only the projection data through the ROI were obtained.

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For the noisy truncated projection data, similar to the studies presented in (Ma et al 2012b, Huang et al 2013, Niu et al 2014a), after calculating the noise-free truncated line integral $y$ as a direct projection operation based on the model (1), the noisy measurement ${{b}_{i}}$ at each bin $i$ was generated according to the statistical model of pre-logarithm projection data:

$$\begin{eqnarray}&&{{b}_{i}}=\text{Poisson}\left({{I}_{0}}\exp \left(-{{y}_{i}}\right)\right)+\text{Normal}\left(0,\sigma _{\text{e}}^{2}\right)\end{eqnarray} \tag{ 9 }$$

where ${{I}_{0}}$ denotes the incident x-ray intensity and $\sigma _{\text{e}}^{2}$ is the background electronic noise variance, which was set to 10 in the experiment. The noisy measurements ${{y}_{i}}$ were calculated by the logarithm transform of ${{b}_{i}}$. In the present study, the normal- and low-dose scanning protocols were simulated by setting the x-ray exposure level to 5.0  ×  10⁵ and 1.0  ×  10⁵, respectively. In the case of fewer projection data, they were generated by under-sampling the 1160 projections of the normal-dose simulation to only 580 and 360 projections evenly over 360°.

In vivo imaging experiments were performed using a Micro-CT Skyscan1076 scanner. In this experiment, a CT scan of an anesthetized mouse was performed and only central slice projection was extracted, which is typical fan-beam geometry. The distance from the object to the curved detector is 165 mm and the distance from the object to the x-ray source is 121 mm. In our experiments, two scans were performed. One used a normal-dose (70 kVp, 225 mAs) protocol where 450 projections were uniformly collected over a 360º range, and the other used a low-dose (70 kVp, 27 mAs) protocol where 360 projections were uniformly collected over a 360° range. For each projection, 4000 detector elements were equi-angularly distributed with the total length of 50.12 mm. We first reconstructed the entire cross-section of the lungs in a 500  ×  500 matrix covering a 12.5 mm  ×  12.5 mm region from each full-scan dataset. Then a circular region around the heart was chosen as an ROI. Finally, only the projection data through the ROI was kept to simulate an interior scan. To evaluate the performance of the proposed algorithm with fewer projections, the views were down sampled from 450 to 225 and 150 for the normal-dose protocol and from 360 to 180 for the low-dose protocol. Figure 3 is the normal-dose image reconstructed by using the FBP method with ramp filtering from the 450-views projection data and the corresponding interior ROI.

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• (1)  
  Evaluation by reconstruction accuracy. To quantitatively evaluate the reconstruction accuracy, the root mean squared error (RMSE) metrics were calculated over a region of interest (ROI). The RMSE is defined as:

  $$\begin{eqnarray}&&\text{RMSE}=\sqrt{\underset{m=1}{\overset{Q}{\sum}}{{\left(\mu (m)-{{\mu}_{x\text{true}}}(m)\right)}^{2}}}\end{eqnarray} \tag{ 10 }$$

  where $\mu$ denotes the to-be-reconstructed image, ${{\mu}_{x\text{true}}}$ denotes the normal-dose image or ground truth image. $m$ indexes the pixels in the ROI and $Q$ is the total number of pixels in the ROI.
• (2)  
  Evaluation by noise reduction. The following two metrics were utilized to evaluate the noise reduction for the comparisons among FBP-EP, ART-TV, PWLS-TV and PWLS-TV-IFR: (1) peak signal-to-noise ratio (PSNR); and (2) mean per cent absolute error (MPAE):

  $$\begin{eqnarray}&&\text{PSNR}=10{{\log}_{10}}\left(\frac{\text{MA}{{\text{X}}^{2}}\left({{\mu}_{x\text{true}}}\right)}{\sum\nolimits_{m=1}^{Q}{{\left(\mu (m)-{{\mu}_{x\text{true}}}(m)\right)}^{2}}/(Q-1)}\right)\end{eqnarray} \tag{ 11 }$$

  $$\begin{eqnarray}&&\text{MPAE}=\frac{100}{Q}\underset{m=1}{\overset{Q}{\sum}}\left|\frac{\mu (m)}{{{\mu}_{x\text{true}}}(m)}-1\right|\end{eqnarray} \tag{ 12 }$$

  where $\mu$ denotes the to-be-reconstructed image, ${{\mu}_{x\text{true}}}$ denotes the ground truth image, and $\text{MAX}\left({{\mu}_{x\text{true}}}\right)$ represents the associated maximum intensity value. $Q$ is the number of pixels in the ROI.
• (3)  
  Noise-resolution tradeoff. The noise-resolution tradeoff curves from PWLS-TV and PWLS-TV-IFR were generated from the simulated projection data using the XCAT phantom. The image resolution was analyzed by the edge spread function (ESF) along the vertical profile. Based on the strategy described in La Riviere and Billmire (2005), by assuming the broadening kernel is a Gaussian function with a standard deviation ${{\delta}_{b}}$, an error function (erf) can be used to represent the ESF function parameterized by ${{\delta}_{b}}$. Consequently, the parameter ${{\delta}_{b}}$ can be calculated by fitting the vertical profiles to the error function, and the associated full-width at half-maximum (FWHM) of the Gaussian broadening kernel is denoted as 2.35${{\delta}_{b}}$, which is used to indicate the resolution of the reconstructed image. In this study, the noise level of the reconstructed image was characterized by the standard deviation of a uniform region of size 40  ×  40 in the background region. By varying the penalty parameter $\beta$, we obtained the noise-resolution tradeoff curves from the images reconstructed using PWLS-TV and PWLS-TV-IFR.

4.1.1. Reconstructions from simulated truncated normal-mAs datasets.

Figure 4 shows the images and zoomed ROIs reconstructed by FBP-EP, ART-TV, PWLS-TV and PWLS-TV-IFR using the simulated truncated normal dose projection data (${{I}_{0}}=5\times {{10}^{5}}$, 1160 views), respectively. It can be observed that the truncation artifacts are mostly corrected on the FBP-EP image. The use of ART-TV and PWLS-TV result in less artifacts but worse resolution and degraded details. The reconstruction using the proposed PWLS-TV-IFR method exhibits noticeable gains than those using ART-TV and PWLS-TV in preserving the small-scale and low-contrast structures, as indicated by the arrows in the zoomed four ROIs. Figure 5 depicts the horizontal profiles of the reconstructions in figure 4. The result reconstructed by ART-TV has an obvious bias while PWLS-TV achieves better performance. Moreover, it is noticeable that the profiles from PWLS-TV-IFR agree much better with the profiles from the reference. In other words, the presented PWLS-TV-IFR method can achieve higher low-contrast detestability as compared with other methods.

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Table 1 lists the RMSE, PSNR, and MPAE measures of the images reconstructed by FBP-EP, ART-TV, PWLS-TV, and PWLS-TV-IFR, as shown in figure 4. It can be seen that the results from ART-TV, PWLS-TV, and PWLS-TV-IFR exhibit an average of more than 20% gains over that from the FBP-EP method in terms of the RMSE and MPAE, and of more than 8% gains in terms of the PSNR. The PWLS-TV-IFR method is found to perform better than both ART-TV and PWLS-TV with less noise and better reconstruction accuracy.

Table 1. Image quality metrics.

Method	RMSE	PSNR	MPAE
FBP-EP	0.028	24.34	0.33
ART-TV	0.010	26.44	0.26
PWLS-TV	0.009	26.82	0.20
PWLS-TV-IFR	0.008	27.53	0.17

The appealing performance of the PWLS-TV-IFR method is mainly due to the contribution of the IFR step. Figure 6 shows the current image estimations ${{\mu}^{n}}$, the images $\mu _{P}^{n}$ after PWLS-TV minimization, the residual images ${{\nu}^{n}}$, the feature descriptors $f_{t}^{n}$ and the detected feature images $f_{t}^{n}\otimes {{\nu}^{n}}$ (which is also the difference image before and after refinement) in the first, tenth and last iteration. It can be observed that in the first iteration, the image ${{\mu}^{0}}$ exhibits many truncated artifacts and noise, most of which are filtered out in the residual image ${{\nu}^{0}}$. The feature descriptor $f_{t}^{n}$ estimated from $\mu_{P}^{n}$ could recognize the structure component from the artifacts and noise. As the iteration continues, the feature descriptor $f_{t}^{n}$ gradually contains more and more detailed structures. Finally, more structural information is recovered and a better reconstruction is obtained. Meanwhile, the evolution of the figures along iterations can also provide an interpretation of the convergence property of our proposed framework.

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To demonstrate the convergence property of our proposed method, figure 7 shows the values of the RMSE and PSNR against the number of iterations. It can be observed that, both PWLS-TV and PWLS-TV-IFR achieve convergence at the 80th iteration, since the convergence of the algorithm is guaranteed by the convexity of the objective function. Obviously, both PWLS-TV and PWLS-TV-IFR exhibit better image quality in the initial stage of the iteration. This could be due to the fact that PWLS-TV and PWLS-TV-IFR use the FBP-EP image as the initial guess, while the ART-TV method uses zero images. Meanwhile, PWLS-TV-IFR outperforms PWLS-TV in just first decades of iterations. It can be contributed to the FBP initialization which provides rich fine-feature information in initial iteration steps for IFR to work quickly, considering the nature of IFR for extracting structure information from the residual image between two successive iterations. The PWLS-TV-IFR method achieves a lower RMSE and higher PSNR after several iterations, implying that our algorithm converges to a stable and better solution.

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The parameter $\beta$ is a hyper-parameter to balance the fidelity term and penalty term, which is determined by optimizing the trade-off between noise and image sharpness through a trial-error process. Figure 8 shows the noise-resolution curves obtained via both PWLS-TV and PWLS-TV-IFR. One vertical profile as indicated by the line in the image located at the bottom-left corner was selected. Additionally, one uniform region near the corresponding profile as indicated by the square in the background was selected for calculating the standard deviation of the reconstructed image. When doing the noise-resolution experiment, the iteration number N was set as 100. By varying the penalty parameter $\beta$ from 5.0  ×  10⁻³ to 5.0  ×  10⁻², we obtained the corresponding noise-resolution tradeoff curves. The noise-resolution curves from PWLS-TV and PWLS-TV-IFR have similar trends, but the PWLS-TV-IFR method shows a slightly higher resolution level in most noise range. This is probability due to the consideration of the feature refinement by the IFR step.

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4.1.2. Reconstruction from simulated truncated low-mAs datasets.

This experiment is expected to evaluate the performance of the proposed PWLS-TV-IFR approach with simulated low-mAs and/or sparse-view projections (${{I}_{0}}=1\times {{10}^{5}}$, 1160, 580 and 360 views). Figures 9–11 show the reconstructed images by different reconstruction methods. It can be observed that the FBP-EP images have mitigated truncation artifacts, but are noisy and affected by undersampling. When the number of views was decreased from 1160 to 360, more streak artifacts appear in the FBP-EP images. The images reconstructed by the other three algorithms degenerated indistinctively, especially the PWLS-TV-IFR ones. In any case with different views, the FBP-EP method exhibits more noise and artifacts. ART-TV and PWLS-TV mitigate both cupping and streaks. However, the small-scale and low-contrast structures are particularly degraded. Comparing with the other three approaches, PWLS-TV-IFR always achieve the best performance in preserving small-scale and low-contrast structures.

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Table 2 lists the RMSE, PSNR, and MPAE measures of the images reconstructed by FBP-EP, ART-TV, PWLS-TV, and PWLS-TV-IFR with different level of views, as shown in figures 9–11, respectively. The quantitative accuracy of each algorithm increases as expected from the qualitative evaluation. For FBP-EP and ART-TV, however, on the low dose dataset, view down-sampling makes the results distinctly worse, especially in the case of 360 views. However, in any case with different views, the ROIs reconstructed by the PWLS-TV-IFR method are always the best one compared with the other methods. Although PWLS-TV does not work as well as PWLS-TV-IFR, it outperforms the ART-TV.

Table 2. Image quality metrics.

Methods	1160 views			580 views			360 views
	RMSE	PSNR	MPAE	RMSE	PSNR	MPAE	RMSE	PSNR	MPAE
FBP-EP	0.035	22.21	0.38	0.046	20.42	0.43	0.061	17.46	0.52
ART-TV	0.017	25.14	0.28	0.022	23.24	0.32	0.031	20.48	0.39
PWLS-TV	0.011	25.99	0.23	0.015	24.87	0.25	0.019	24.41	0.30
PWLS-TV-IFR	0.009	26.34	0.20	0.013	25.47	0.23	0.018	24.84	0.28

4.2.1. Reconstruction from truncated normal-mAs datasets.

Figures 12–14 show the images and zoomed interior ROIs reconstructed by FBP-EP, PWLS-TV and PWLS-TV-IFR using the truncated normal-mAs projection data with different views (225 mAs, 450, 225 and 150 views), respectively. It is observed that the images reconstructed by FBP-EP contain severe noise which leads to a low visibility of details. The results reconstructed by the other two algorithms show a lower level of noise. For the FBP-EP method, the down-sampling from 450 to 225 and 150 views results in some streak artifacts. The PWLS-TV results contain less artifacts but with degraded resolution and missing details. The reconstructions using the presented PWLS-TV-IFR method exhibit noticeable gains compared with those from FBP-EP and PWLS-TV in suppressing noise and preserving the small-scale and low-contrast structures, as indicated by the arrows in the zoomed ROIs.

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4.2.2. Reconstruction from truncated low-mAs datasets.

Figures 15 and 16 show the images and zoomed interior ROIs reconstructed by FBP-EP, PWLS-TV and PWLS-TV-IFR using the truncated low-dose projection data with different views (27 mAs, 360 and 180 views), respectively. Comparing with the normal-dose reconstructions, the images reconstructed by FBP-EP display a higher noise level, and contain some streak artifacts, especially in the 180-views case. On the other hand, the reconstructions using the PWLS-TV method exhibit less artifacts and noise but with worse resolution and detail loss. The less views that are used, the more the details become degraded. As shown in the zoomed ROIs, the reconstructions using the presented PWLS-TV-IFR method result in better performance than the PWLS-TV method in achieving higher small-scale and low-contrast detestability.

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