I Introduction
Xray computed tomography (CT) involves radiation exposure that may potentially increase the risk of genetic, cancerous, and other diseases in a patient [1]. To reduce these risks, lowdose CT (LDCT) imaging has become an attractive solution. However, LDCT with standard image reconstruction commonly results in high noise in the reconstructed images, which compromises the diagnostic performance. It is desirable to develop more advanced image processing methods to suppress the noise and improve the image quality in LDCT.
In general, there are two categories of image processing methods for improving LDCT. The first category is tomographic image reconstruction from projection data, including analytical reconstruction in combination with sinogram denoising (e.g., [2, 3, 4]), modelbased iterative reconstruction (e.g., [5, 6, 7, 8, 9]), and deep learning based reconstruction (e.g., [14, 11, 12, 10, 13]). These methods have the advantage of exploiting the raw projection data more extensively but have the disadvantage that the access to raw CT projection data remains a resource barrier to many research groups. In contrast, postreconstruction image denoising (e.g., [15, 16, 17]), the other category, directly deals with the reconstructed images, and are more widely accessible to the research community. A product, once developed, can also be more easily integrated into an existing clinical CT workflow.
It has been demonstrated that deeplearning (DL) image denoising has a strong potential to improve LDCT [22, 21, 23, 19, 18, 24, 20]. The image quality by deeplearning image denoising can be equivalent to or even better than the stateoftheart iterative CT reconstruction [24, 25]. A deeplearning model directly learns the endtoend relationship between a noisy image and its clean reference image using, for example, deep neural networks (e.g., [21, 19, 18, 20]). Most existing network architectures improve the capacity of the neural network by adding more network layers. However, increased number of layers does not always improve the learning performance in practice[18]. A recent work of Shan et al. [24] demonstrated an alternative solution that uses a modularized adaptive processing neural network (MAPNN). Instead of adding more new layers, MAPNN repeatedly adds the same network module with shared weights (hence like “clones”) to increase the network depth and has demonstrated improved image quality for LDCT denoising.
All the aforementioned deep learning methods for LDCT have a sequentialtype layout as illustrated in Figure 1(a). From this perspective, the MAPNN model is a sequentialclone network which uses multiple clones of the basic network module in a sequence of depth. Other earlier network models for LDCT denoising such as the residual encoderdecoder convolutional neural network (REDCNN)
[18] can be considered as a special case of the sequentialclone networks, in which only one clone of the basic network module (i.e., REDCNN itself) is used. Increasing the number of clones from 1 deepens the network and has the potential to improve the training.Nevertheless, such sequentialclone architectures have two major limitations. First, the ability of forward propagating the raw image information is limited. The original noisy input image is used only once, i.e., in the first clone. As analyzed later in this paper, such a usage is very different from conventional modelbased image denoising algorithms and can be less effective to propagate the useful information of the noisy input image into later clones. Second, the network structure is also inefficient for back propagating the gradient of the loss function to earlier clones. The loss layer only utilizes the output image of the last clone and is far away from the earlier clones, which makes it difficult to backpropagate the gradient information of the loss to the earlier clones without causing a vanishing gradient. As a result of these two limitations, the overall learning performance of the sequentialclone model for deeplearning image denoising can have been compromised.
In this paper, we propose a parallelclone network method to overcome the limitations. The new model exploits a parallel use of the noisy input image for the clones and incorporates the output image of all clones into the training loss function, also in parallel. The use of parallel input ensures an efficient forward propagation of useful information of the noisy input image into all the clones. The use of parallel output leads to an efficient backpropagation of the gradient of the loss function to the earlier clones. In addition, the proposed method also explores highlevel feature transfer for the communications between the clones. The proposed parallelclone model is expected to bring substantial improvements over the existing sequentialclone framework for LDCT image denoising.
This paper is organized as follows. Section II introduces the backgroud materials that led to the development of the proposed method. Section III describes the detail of the proposed parallelclone network model. Results of the training and testing on the Mayo CT Dataset are given in Section IV. Finally conclusions are drawn in Section V.
Ii Background
Iia Modelbased Image Denoising
The forward model for traditional image denoising methods can be expressed as
(1) 
where and
denote the noisy CT image and the corresponding clean image to be estimated, respectively.
represents a degradation matrix and is equal to the identity matrix in this image denoising work.
represents the additive noise.The commonly used leastsquare image denoising problem is formulated as
(2) 
where the model consists of two components. The first is a data fidelity term and the second is a regularization item for exploiting the prior information with the regularization parameter. Iterative algorithms are commonly used to solve the optimization problem.
IiB EndtoEnd Deep Learning
A learned model using neural networks predicts a denoised image from the noisy image using
(3) 
where denotes the endtoend mapping from to with the neural network weights to be trained from available data sets. A highquality reference image available from the training dataset may be equivalently expressed as
(4) 
where accounts for the difference between the prediction and the truth . The mean squared error (MSE) between them is then defined by
(5) 
In LDCT image denoising, corresponds to the normaldose image and corresponds to the lowdose image . The training problem for deeplearning image denoising is then formulated as the following optimization if the MSE is used as the loss function:
(6) 
where denotes the th image pair of low dose and normal dose in the training dataset. is the total number of training pairs. Once the model parameter set is trained, the final image estimate predicted from a noisy lowdose image is obtained using .
IiC SequentialClone Neural Network
To increase the depth of a neural network model, more layers can be added but with an increasing number of unknown model parameters. An alternative is to repeatedly use a network module. This concept has been explored in general deeplearning models such as the ResNet [26] and unrolled deep learning for image reconstruction (e.g., [11, 27]). A recent development of this concept for LDCT denoising is the modularized adaptive processing neural network (MAPNN) [24] illustrated in Figure 1(a). Mathematically the MAPNN is expressed as
(7) 
where denotes the repeated use of the modularized network for times:
(8) 
which is equivalent to a sequence of “clones”:
(9) 
Each clone is an individual denoiser but shares the same model structure and same parameters with other clones. The denoised image of a clone is the input of the next clone. If MSE is used, then the training loss for the sequentialclone network has the form of
(10) 
If only one clone is used, i.e., , the MAPNN is then the same as traditional deep neural network models.
Iii Proposed ParallelClone Network
Below we first describe the critical components that can be applied to the sequentialclone model individually. We then assemble them to form the proposed parallelclone model as shown in Figure 1(b).
Iiia Use of Coupled Input
Residual mapping is popular in deep learning after the ResNet work [26]. The sequential model in Eq. (9) can be equivalently rewritten as the following residual mapping format,
(11) 
where denotes the residual mapping between the two adjacent clones and is mathematically equivalent to [24] . Note that the noisy input image appears only once in the sequence, i.e., in the first clone () but not in any subsequent clones.
In comparison, conventional modelbased image denoising commonly employs an optimization algorithm with the following iterative update:
(12) 
where denotes the image estimate at iteration . represents the residual update determined by a specific algorithm. For example, the gradient descent algorithm for the leastsquare image denoising has the form of
(13) 
where is the step size and is the gradient of the regularization term. Another example is the alternating direction method of multipliers (ADMM) [28], by which the iterative update can be described as
(14) 
where and represent the updating matrices.
One common feature of these modelbased iterative denoising algorithms is that the previous iterate and the noisy input image are coupled together to update the image estimate at next iteration . This coupled input is originated from the data fidelity term in the objective function Eq. (2). From the Bayesian perspective, the data fidelity carries useful information of the statistical distribution of measurements. We hypothesize the coupled use of and is more beneficial than using alone.
Considering each clone in the sequential model as an unrolled “iteration”, we can modify the sequentialclone model by including the noisy input image into each clone,
(15) 
Compared with Eq. (11), here we use a different notation to denote the residual mapping now taking two inputs (i.e., and ) without significantly changing the structure of the modularized neural network . The inputs and can be coupled using a concatenation operation ,
(16) 
which are then passed into the subsequent convolutional layers in the model . We expect this modification inspired from modelbased image denoising can improve the residual mapping of each clone.
IiiB Auxiliary Output Loss
Compared to conventional modelbased image denoising, deep learning has the advantage of endtoend training. In the sequentialclone model, this is reflected in the training loss by comparing the output image of the last clone to the reference image, i.e., the normaldose image in LDCT. Taking the MSE as an example, the training loss function for the sequentialclone network model is equivalent to
(17) 
which only takes into account the output of the last clone .
A general challenge for deep learning is the vanishing gradient problem
[29], which holds true for the sequentialclone model. In a gradientbased algorithm, efficient back propagation of the gradient to the earlier clones in the sequence is challenging because the earlier clones are further away from the final layer of loss function than the later clones.We note that in the sequentialclone network model, not only the last clone but also each of the earlier clones produces an auxiliary output image, which has not been utilized by the training process. In fact, auxiliary output has been utilized in previous works for the task of image recognition, e.g., GoogLeNet [30]. Hence we propose to incorporate all the auxiliary output images into the loss function for the sequentialclone model. The MSE training loss is then of the form
(18) 
in which the auxiliary output image of each clone contributes to the training loss in a parallel way. It becomes more straightforward to backpropagate the gradient information of the loss function from the output layer to the earlier clones. We expect the use of parallel auxiliary output loss can reduce the impact of the gradient vanishing problem.
IiiC BruteForce Residual Mapping
The residual mapping function used in the sequential models mainly accounts for the difference between two adjacent clones,
(19) 
which leads to the following form for the last clone,
(20) 
Substituting the above expression of into the conventional loss function in Eq. (17), we have
(21) 
which indicates that the totalresidual image is approximated by a sum of the residual images from all the clones. We call in this case incremental residual mapping.
If the parallel auxiliary output loss in Eq. (18) is used, then the training loss becomes to
(22) 
in which still represents an incremental residual mapping, though the accumulation is different in different clones.
Instead of using the mapping to represent the difference between two adjacent clones, i.e., , we employ a different residual mapping model in this work,
(23) 
where the residual mapping is changed to reflect the difference between the output image of clone and the noisy input image . Substituting the new expression into the parallel auxiliary output loss, we obtain
(24) 
which indicates becomes to directly predict the totalresidual image in each clone . To be differentiated from the incremental residual mapping model, we call the new model as bruteforce residual mapping in this paper.
In this work, we combine the bruteforce residual mapping with the coupled input model to explore the benefit of parallel input as a part of the parallelclone network model.
IiiD ClonetoClone Feature Transfer
In the sequentialclone model in figure 1(a), adjacent clones (e.g., and ) are connected using the intermediate denoised image. Other than the output image , additional highlevel features also exist from the clone and can be transferred to clone . We hypothesize the transfer of intermediate highlevel feature can be more useful than just transferring the output image. Thus we use an a more general expression for the model of clone :
(25) 
where , the transferred information from clone , can be the output image or a highlevel feature set.
In order to jointly use and if the latter represents highlevel features, the clone model first extracts a feature set from using
(26) 
where denotes a lowlevel feature extraction operation and is composed of a convolutional layer
and a rectified linear unit (ReLU) layer
in this work. The extracted feature set matches the dimension of but is more focused on the lowlevel information of the input image, such as edge and corner [31]. and can be then concatenated,(27) 
to form the input for the subsequent convolutional layers.
IiiE Combined ParallelClone Network Model
Combing all the aforementioned components together, we obtain a general expression for the proposed parallelclone network model,
(28) 
The objective function for the corresponding optimization problem is then defined by
(29) 
A graphical illustration of the parallelclone model is provided in Fig. 1(b).
This parallelclone network model has three unique features: (1) parallel input, (2) paralleloutput loss, and (3) clonetoclone feature transfer. The parallel input feeds the noisy input image
to all the clones in parallel to enable the bruteforce residual mapping and the use of coupled input, both improving the forward propagation of image information to the loss layer. The paralleloutput loss incorporates all the auxiliary outputs into the training loss, which can improve the backpropagation of the gradient for the earlier clones. The clonetoclone feature transfer connects adjacent clones to allow deeper learning.
The parallelclone model is equivalent to the sequentialclone model if the input image is only fed to the first clone and only the output of the last clone is used in the training.
IiiF Example Architecture and Implementation
An example of the specific architecture of the proposed parallelclone network is shown in figure 2. The model consists of three modules in each clone: a lowlevel representation extraction module to obtain the lowlevel feature set , a highlevel representation learning module to obtain the highlevel feature , and an image recovery module to get the output image . Different clones have the same model structure with shared weights.
The image recovery module is implemented using a deconvolution layer
, a residual connection, and a
:(30) 
where outputs the residual image from the highlevel feature set .
The highlevel representation learning for clone is implemented by a multilayer convolutional neural net (CNN),
(31) 
where its input is a concatenation of the lowlevel feature and the highlevel feature transferred from the clone , see Eq. (27).
In theory, any deeplearning model can be used as the CNN module for the clone networks. Fig. 3 shows two examples adapted from the REDCNN model [18] and the CPCE model [24], both following an encoderdecoder architecture. Details of the models are referred to the original papers of these models. The encoders consist of a series of followed by to suppress image noise and artifacts from lowlevel to highlevel step by step. In the decoders, a series of are used with residual mapping to recover the structural details. As and are symmetric in the models, the number of is the same as the number of . The deconvolution in combination with symmetric shortcut connections are beneficial for detail preservation [18].
Iv Experiments and Results
Iva Clinical CT Dataset and Implementation
The 2016 NIHAAPMMayo Clinic Low Dose CT Grand Challenge dataset [32] was used for evaluating the proposed parallelclone network and other models. The dataset includes the normaldose abdominal CT scans and synthetic quarterdose CT scans of ten patients. Each scan consists of about 210 to 340 transverse image slices, each with a matrix size of 512512 pixels. Nine out of ten were randomly selected and used for training and the remaining one was used for testing. The process was repeated for three times.
For each training, ten image slices were randomly selected from each of the nine patients to generate image patches of size 5555 with an interval of 4 pixels. The resulting total number of image patches used for training was about 1.6 million. For testing, the trained model was directly applied to the full image slices from the testing patient scan.
We used the Adam optimization method [33]
to train different network models with a minibatch of 128 patches in each iteration. Sixty epochs were run. The initial learning rate was set to
and slowly decreased to . The number of convolutional kernels in each layer was 48 except for the last layer, which has only one layer. The kernel size of all layers was set to 33 with a convolutional stride of 1 and no padding. All the networks were implemented using PyTorch on a PC with an Intel i99920X CPU with 64GB RAM and a NVIDIA GeForce RTX 2080 Ti GPU.
IvB Approach of Comparison
We first conducted an ablation study to demonstrate the improvement from the use of parallel input, parallelout loss, and clonetoclone feature transfer by using the sequentialclone network (SCN) model as the baseline. The REDCNN model was used as the basic CNN module for both the SCN model and parallelclone network (PCN) model. The output image of the last clone was used as the final output of each model unless specified otherwise. We also investigated the effect of hyperparameters in the PCN model, such as the number of layers for the CNN module and the number of clones.
The PCN model was further compared with three popular DLbased denoising methods: the denoising convolutional neural network (DnCNN) [34], REDCNN [18] and MAPNN [24]. DnCNN is one of the representative DL models for general image denoising. REDCNN is a typical example specifically developed for LDCT denoising. The MAPNN reflects the most recent example of a sequentialclone model for LDCT image denoising.
Note that the DnCNN, REDCNN, and proposed PCN models were trained using the MSEbased loss. While the original MAPNN was trained using an advanced loss function that combines the basic MSE loss with perceptual losses, we also trained another MAPNN using the MSE loss only.
IvC Evaluation Metrics
Three common image quality metrics, including the root mean square error (RMSE), peak signal to noise ratio (PSNR), and structural similarity index measure (SSIM)
[14], were used to assess the training convergence and testing image quality.(32) 
(33) 
(34) 
where and are the normaldose image and predicted lowdose image with the total number of pixels in the region for quality evaluation. and
denote the mean value and standard deviation inside a sliding window.
respresents the covariance between and . is the maximum pixel value. and are two SSIM parameters defined as and , respectively.IvD Comparison Between Sequential and Parallel Models
Fig. 4(a) shows the plots of the training convergence of RMSE as a function of epoch number for different clone models based on the clonetoclone image transfer. The RMSE was calculated based on the training image patches and averaged over the the 3 times experiments. For each DL model, the number of clones was four and the number of layers in the basic REDCNN module was ten. Compared with SCN, the use of parallel output (PO) and coupled input (CI) incrementally improved the RMSE. Further combination with the bruteforce residual mapping model, which leads to the full PCN, achieved a significant acceleration of the training convergence as compared to the SCN model.
Fig. 4(b) further shows the comparison based on the clonetoclone feature transfer (FT). Replacing the image transfer with FT can improve the RMSE, though not for earlier epochs. On top of that, the use of paralleloutput loss further improved the convergence. An even larger improvement was obtained with the use of parallel input, which includes both the bruteforce residual mapping model and coupled input. The most significant improvement came from the full PCN model which integrates all the three critical modifications (FT, PO, and PI) in the model. The convergence rate of the PCN was dramatically faster than the SCN. The PCN only took about 5 epochs to reach a similar RMSE value as the SCN at 60 epochs, suggesting a speedup factor of about ten.
Fig. 5 shows the results from the evaluation on the testing data. Note that here the image quality was evaluated on the whole image slices. The relationships between different models are consistent with the results of training convergence shown in Fig. 4.
Fig. 6 shows the denoised results of a specific testing image by the SCN model and PCN models with 60 epochs. All the PCN models were implemented with the clonetoclone feature transfer. For better display, the region of a liver metastasis was magnified in each image. Compared to the normaldose reference image, the SCN reduced the noise but suffered from artifacts. The use of paralleloutput loss or parallel input alone improved the image denoising according to quantitative PSNR. The full PCN model which integrates the parallel input, paralleloutput loss, and clonetoclone feature transfer together achieved the best result in terms of quantitative PSNR and visual quality. These results are further confirmed in Fig. 7 which shows a clonewise comparison of PSNR for the SCN and different PCN models that consist of four clones.
Table I summarizes the results of different quality metrics (PSNR, SSIM, and RMSE) from the testing dataset. The mean and standard deviation (SD) were shown for each metric. This comparison further confirmed the improvement of the full PCN model and its individual components (paralleloutput loss, parallel input, and clonetoclone feature transfer) as compared to the baseline SCN model.
Methods  PSNR  SSIM  RMSE 

SCN  43.58241.1476  0.96240.0054  0.00690.0011 
PCN with PO  44.01791.2873  0.97050.0052  0.00610.0007 
PCN with PI  44.02681.3675  0.97180.0056  0.00600.0008 
Full PCN  45.42351.1394  0.98460.0047  0.00540.0006 
IvE Effect of Clone Settings
Fig. 8(a) shows the effect of the number of CNN layers on the image PSNR of the PCN model applied on the testing dataset. The number of clones was fixed at 4. The result suggests a 10layer REDCNN can work well for the PCN model. Adding more layers did not improve the performance significantly. The result is consistent with [18].
Fig. 8(b) shows the effect of the number of clones on the PCN model performance. The PSNR of the output image of each clone was plotted versus the clone index in each PCN model. The total number of clones was varied from 1 to 5. The number of CNN layers was set to 10 based on the result from Fig. 8(a). The curves show that the PSNR of the earlier clones in a PCN model was improved as the number of clones increased. The PSNR of the last clone in each PCN reached its maximum when the number of clones was 3. The differences in peak PSNR were minor among the PCN models with 3 clones, 4 clones, and 5 clones.
Table II compares the choices of the basic CNN module for quantitative image quality evaluation of the PCN. Two basic models were compared, including CPCE [24] and REDCNN [18]. The number of CNN layers was set to 10 in each comparing model and the number of clones was set to 4. The result suggests the REDCNN was better than CPCE to serve as the basic CNN module for the parallelclone network.
IvF Comparison with Other DL Models
Table III summarizes the results of PSNR, SSIM, and RMSE for denoising the testing dataset using five different models: the proposed PCN, the original DnCNN, REDCNN, and MAPCNN. The PCN was implemented with 3 clones. The result of MAPNN trained using the MSE loss was also included as MAPNN in the study. Among the different models, the proposed PCN achieved the best quality as assessed by all the three metrics. Compared to DnCNN and REDCNN which are equivalent to a PCN with single clone, the use of multiple clones in the proposed PCN led to a significant improvement. The comparison of PCN with the sequentialclone models (MAPNN and MAPNN) indicates the parallel structure of PCN can be superior to the sequential structure. Note that the improvement of MAPNN over MAPNN was mainly from the use of a more advanced loss function in the former. This suggests that a combination of advanced loss functions with PCN would be able to further improve the performance of the PCN model, which will be explored in our future work.
Module  PSNR  SSIM  RMSE 

CPCE  44.63651.1332  0.97820.0050  0.00560.0007 
REDCNN  45.42351.1394  0.98460.0047  0.00540.0006 
Method  PSNR  SSIM  RMSE 

DnCNN [34]  44.13051.2569  0.97390.0057  0.00570.0007 
REDCNN [18]  44.52381.1924  0.97560.0055  0.00550.0006 
MAPNN [24]  45.46121.3908  0.98290.0054  0.00540.0007 
MAPNN  43.36981.1724  0.96120.0055  0.00700.0011 
Proposed (PCN)  45.77751.1057  0.98550.0045  0.00530.0006 
Different DL models have different model complexities. Fig. 9 shows the PSNR achieved by each DL model versus the number of trainable parameters) in the model. In addition to the use of 48 convolutional kernels, we also include the result for the use of 64, 80, and 96 kernels in the PCN. The result indicates the increased number of kernels has a minimal effect on PSNR once it exceeds 48. The baseline SCN was composed of four REDCNN clones, but its performance was worse than the original REDCNN [18], mainly because the former used a smaller kernel size and a much less number of kernels. Compared to the MAPNN, the use of advanced loss functions in the MAPNN improved PSNR but also largely increased the model complexity. In comparison, the PCN achieved better PSNR performance with fewer parameters.
Fig. 10 shows the denoised images obtained by different models. The DnCNN and REDCNN generally oversmoothed the liver background. The MAPNN had a closer image appearance to to the normaldose CT reference image, but some details were lost or with lower contrast, as pointed by the arrows. In comparison, the PCN provided generally higher image quality and better visual quality.
These results together indicate the proposed PCN model can outperform existing DL models for LDCT image denoising.
V Conclusion
In this paper, we have developed a simple yet efficient parallelclone network architecture for LDCT image denoising. The model uses modularized clones with shared weights and exploits the benefits of parallel input, parallelout loss, and clonetoclone feature transfer. It has a similar or less number of unknown parameters as compared to conventional deep learning models but can significantly improve the learning process. Experimental results using the Mayo LDCT dataset demonstrated the improvement of the proposed parallelclone network model over conventional sequential models.
Acknowledgment
The authors thank Dr. Cynthia H. McCollough and the Mayo team for providing the LDCT dataset used in this study.
References
 [1] D. J. Brenner and E. J. Hall, “Computed tomography  an increasing source of radiation exposure,” New England Journal of Medicine, vol. 357, no. 22, pp. 22772284, Nov. 2007.
 [2] P. J. La Rivière, “Penalized‐likelihood sinogram smoothing for low‐dose CT,”Medical Physics, vol. 32, no. 6, pp. 16761683, 2005.
 [3] J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted leastsquares approach to sinogram noise reduction and image reconstruction for lowdose Xray computed tomography,” IEEE Trans. Med. Imag., vol. 25, no. 10, pp. 12721283, Oct. 2006.
 [4] A. Manduca, L. Yu, J. D. Trzasko, N. Khaylova, J. M. Kofler, C. M. McCollough, and J. G. Fletcher, “Projection space denoising with bilateral filtering and CT noise modeling for dose reduction in CT,”Medical Physics, vol. 36, no. 11, pp. 49114919, Oct. 2009.
 [5] JB Thibault, KD Sauer, CA Bouman, J Hsieh, “A threedimensional statistical approach to improved image quality for multislice helical CT,” Medical Physics, vol. 34, no. 11, pp. 45264544, 2007.
 [6] IA Elbakri, JA Fessler, “Statistical Image Reconstruction for Polyenergetic XRay Computed Tomography,” IEEE Transactions on Medical Imaging, vol. 21, no. 2, pp. 8999, 2002.
 [7] E. Y. Sidky and X. Pan, “Image reconstruction in circular conebeam computed tomography by constrained, totalvariation minimization,” Phys. Med. Biol., vol. 53, no. 17, pp. 47774807, 2008.
 [8] Q. Xu, H. Yu, X. Mou, L. Zhang, J. Hsieh, and G. Wang, “Lowdose Xray CT reconstruction via dictionary learning,” IEEE Trans. Med. Imaging, vol. 31, no. 9, pp. 16821697, Sep. 2012.
 [9] S. Ye, S. Ravishankar, Y. Long, and J. A. Fessler, “SPULTRA: Lowdose CT image reconstruction with joint statistical and learned image models,” IEEE Transactions on Medical Imaging, vol. 39, no. 3, pp. 729741, Mar. 2020.
 [10] D. Wu, K. Kim, G. El Fakhri, and Q. Li, “Iterative lowdose CT reconstruction with priors trained by artificial neural network,” IEEE Transactions on Medical Imaging, vol. 36, no. 12, pp. 24792486, Dec. 2017.
 [11] H. Chen, Y. Zhang, Y. Chen, J. Zhang, W. Zhang, H. Sun, Y. Lv, P. Liao, J. Zhou, and G. Wang, “LEARN: Learned experts’ assessmentbased reconstruction network for sparsedata CT,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 13331347, Jun. 2018.
 [12] H. Gupta, K. H. Jin, H. Q. Nguyen, M. T. McCann, and M. Unser, “CNNbased projected gradient descent for consistent CT image reconstruction,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 14401453, Jun. 2018.
 [13] P. Bao, H. Sun, Z. Wang, Y. Zhang, W. Xia, K. Yang, W. Chen, M. Chen, Y. Xi, S. Niu, J. Zhou, and H. Zhang, “Convolutional sparse coding for compressed sensing CT reconstruction,” IEEE Transactions on Medical Imaging, vol. 38, no. 11, pp. 26072619, Nov. 2019.
 [14] Y. Li, K. Li, C. Zhang, J. Montoya, and G. H. Chen, “Learning to reconstruct computed tomography images directly from sinogram data under a variety of data acquisition conditions,” IEEE Transactions on Medical Imaging, vol. 38, no. 10, pp. 24692481, Oct. 2019.
 [15] P. F. Feruglio, C. Vinegoni, J. Gros, A. Sbarbati, and R. Weissleder, “Block matching 3D random noise filtering for absorption optical projection tomography,” Phys. Med. Biol., vol. 55, no. 18, pp. 54015415, 2010.
 [16] M. Aharon, M. Elad, and A. Bruckstein, “KSVD: An algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 43114322, Nov. 2006.
 [17] Z. Li, L. Yu, J. D. Trzasko, D. S. Lake, D. J. Blezek, J. G. Fletcher, C. H. McCollough, and A. Manduca, “Adaptive nonlocal means filtering based on local noise level for CT denoising,” Medical Physics, vol. 41, no. 1, pp. 011908101190816, Dec. 2013.
 [18] H. Chen, Y. Zhang, M. K. Kalra, F. Lin, Y. Chen, P. Liao, J. Zhou, and G. Wang, “LowDose CT with a residual encoderdecoder convolutional neural network,” IEEE Transactions on Medical Imaging, vol. 36, no. 12, pp. 25242535, Dec. 2017.
 [19] J. M. Wolterink, T. Leiner, M. A. Viergever, and I. Isgum, “Generative adversarial networks for noise reduction in lowdose CT,” IEEE Transactions on Medical Imaging, vol. 36, no. 12, pp. 25362545, Dec. 2017.
 [20] Q. Yang, P. Yan, Y. Zhang, H. Yu, Y. Shi, X. Mou, M. K. Kalra, Y. Zhang, L. Sun, and G. Wang, “Lowdose CT image denoising using a generative adversarial network with wasserstein distance and perceptual loss,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 13481357, Jun. 2018.
 [21] E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for lowdose CT via wavelet residual network,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 13581369, Jun. 2018.

[22]
H. Shan, Y. Zhang, Q. Yang, U. Kruger, M. K. Kalra, L. Sun, W. Cong, and G. Wang, “3D convolutional encoderdecoder network for lowdose CT via transfer learning from a 2D trained network,”
IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 15221534, Jun. 2018.  [23] L. Huang, H. Jiang, S. Li, Z. Bai, and J. Zhang, “Two stage residual CNN for texture denoising and structure enhancement on low dose CT image,” Computer Methods and Programs in Biomedicine, vol. 184, p. 105115, Feb. 2020.
 [24] H. Shan, A. Padole, F. Homayounieh, U. Kruger, R. D. Khera, C. Nitiwarangkul, M. K. Kalra, and G. Wang, “Competitive performance of a modularized deep neural network compared to commercial algorithms for lowdose CT image reconstruction,” Nature Machine Intelligence, vol. 1, no. 6, pp. 269276, Jun. 2019.

[25]
Y. Nakamura, T. Higaki, F. Tatsugami, J. Zhou, Z. Yu, N. Akino, Y. Ito, M. Iida, and K. Awai, “Deep learning–based CT image reconstruction: Initial evaluation targeting hypovascular hepatic metastases,”
Radiology: Artificial Intelligence
, vol. 1, no. 6, p. e180011, Oct. 2019. 
[26]
K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,”
2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
, Jun. 2016.  [27] H. K. Aggarwal, M. P. Mani, and M. Jacob, “MoDL: Modelbased deep learning architecture for inverse problems,” IEEE Transactions on Medical Imaging, vol. 38, no. 2, pp. 394 405, Feb. 2019.
 [28] S. H. Chan, X. Wang, and O. A. Elgendy, “PlugandPlay ADMM for iImage restoration: fixedpoint convergence and applications,” IEEE Transactions on Computational Imaging, vol. 3, no. 1, pp. 8498, Mar. 2017.
 [29] Y. Bengio, P. Simard, and P. Frasconi, “Learning longterm dependencies with gradient descent is difficult,” IEEE Transactions on Neural Networks, vol. 5, no. 2, pp. 157166, Mar. 1994.
 [30] C. Szegedy, Wei Liu, Yangqing Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Jun. 2015.
 [31] M. D. Zeiler and R. Fergus, “Visualizing and Understanding Convolutional Networks,” Lecture Notes in Computer Science, pp. 818833, 2014.
 [32] C. H. McCollough, A. C. Bartley, R. E. Carter, B. Chen, T. A. Drees, P. Edwards, D. R. Holmes, A. E. Huang, F. Khan, S. Leng, K. L. McMillan, G. J. Michalak, K. M. Nunez, L. Yu, and J. G. Fletcher, “Lowdose CT for the detection and classification of metastatic liver lesions: Results of the 2016 Low Dose CT Grand Challenge,” Medical Physics, vol. 44, no. 10, pp. e339e352, Oct. 2017.
 [33] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimiza tion,” in Proc. Int. Conf. Learn. Represent., 2015. [Online]. Available: https://arxiv.org/abs/1412.6980.
 [34] K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian denoiser: residual learning of deep CNN for image denoising,” IEEE Transactions on Image Processing, vol. 26, no. 7, pp. 31423155, Jul. 2017.
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