Low-dose CT reconstruction network based on the unfoldment of second-order TGV

Abstract

Background

Total generalized variation (TGV) based CT iterative reconstruction algorithm has the ability to effectively suppress the staircase effects caused by the piecewise constant assumption of total variation regularization. By unrolling the model-based iterative reconstruction to networks, the deep unrolling approach can further improve image quality within a finite number of iterations by data-driven training. However, most deep unrolling approaches focus on unrolling the data fidelity term into deep neural networks, which limit the performance of the deep unrolling approach.

Objective

To address this issue, we unrolled both the data fidelity term and the TGV term to construct a novel low-dose CT reconstruction network, called TGV based deep unrolling approach (TGV-DU).

Methods

The Chambolle-Pock algorithm was employed to solve the TGV based CT iterative reconstruction problem to obtain a single-loop CT iterative reconstruction algorithm, which is easy to be unrolled to neural networks. In the proposed algorithm, the parameterized mapping that updates primal variables and dual variables across successive iterations was implemented by convolutional neural networks and was dynamically learned from big data.

Results

To validate the effectiveness of our proposed algorithm, we perform the experiment on the “Low-Does CT Image and Projection Data” dataset. The results show that the proposed TGV-DU outperforms other state-of-the-art methods quantitatively and qualitatively.

Conclusions

Experiments show that our proposed algorithm can effectively alleviate the piecewise smoothness while preserve more structural details.

Keywords

computed tomography low-dose total generalized variation deep learning unrolling approach

Introduction

Computed tomography (CT) is a non-invasive diagnostic technique widely used in medical diagnostics. However, its use of ionizing radiation has raised significant concerns due to the associated carcinogenic risk, particularly from radiation exposure.¹ In view of these considerations, X-ray exposure doses should be minimized during CT scanning to ensure safety. This may lead to poor image quality during reconstruction, loss of tissue structural details, and the presence of excessive noise and artifacts.² Ultimately, these factors may trigger misdiagnosis. Thus, reducing the CT radiation dose while ensuring the acquisition of high-quality images that meet clinical diagnostic requirements has always been an important research focus and challenge in the field of CT. Numerous CT reconstruction algorithms are proposed to enhance image quality for low-dose computed tomography (LDCT).³

Traditional CT reconstruction algorithms are categorized into analytical and model-based iterative reconstruction (IR) methods. Filtered back-projection (FBP) is a classical analytical reconstruction algorithm, but as the decreasing of dose and angle, the reconstructed image shows a lot of noise and artifacts. Model-based IR has the potential to reduce the radiation dose while maintaining the quality of the image.⁴ In model-based IR algorithms, the objective function consists of a data fidelity term and a regularization term. The data fidelity term is utilized to quantify the consistency between the reconstructed image and the actual measured projection data, to ensure that the reconstructed image accurately reflects the real projection data. The regularization term is employed to incorporate a priori knowledge and impose constraints on the smoothness and structural information of the reconstructed image, thereby yielding better reconstructed image quality. Total variation (TV) regularization, a widely adopted technique in CT reconstruction, effectively preserves edge structures by imposing gradient sparsity constraints while suppressing noise. Due to the piecewise constant assumption there are usually staircase effects in the reconstructed image. To improve this drawback, the second-order total generalized variation (TGV) was adopted as the regularization term of the objective function. However, model-based IR methods typically necessitate laborious hyperparameter turning and high computational demands due to their iterative optimization frameworks.⁵

With the rise of artificial intelligence (AI), deep learning techniques are widely applied in the fields of computer vision, image segmentation and medical image reconstruction. Over the past few years, many researchers have been dedicated to the development of deep learning-based LDCT methods. These methods can be roughly categorized into four classes. The first class of methods recovers the corrupted projection data by filtering the projection data in the projection domain through neural networks.^6,7,8 The second class of methods employs various post-processing networks to directly enhance the reconstructed images for removing noise and artifacts.^9,10,11,12 The third class utilizes two neural networks on dual domain to learn the sinogram and CT image information respectively.^13,14,15,16 The fourth class of methods unrolls the specific model-based IR methods into networks which effectively enhances the performance of CT reconstruction networks. Chen et al. unrolled the steepest gradient descent method to construct a sparse-data CT reconstruction network.¹⁷ In this algorithm, the regularization term of the objective function was directly replaced by the learning experts. To further improve the feature extraction capability of this unrolling network, Xia et al. adopted a graph convolution to extract the features of the low-dimensional manifold for LDCT.¹⁸ Adler and Öktem proposed a learned primal-dual algorithm for tomographic reconstruction, generalizing the Chambolle-Pock (CP) algorithm by replacing the proximal operators with parametrized operators where the parameters are learned from training data.¹⁹ Zhang et al. unrolled ISTA into deep network and developed an effective strategy to solve the proximal mapping using nonlinear transforms.²⁰ Xiang et al. employed fast iterative shrinkage/thresholding algorithm (FISTA) to solve the inverse problem and unrolled the obtained iterative steps into a deep neural network. In this algorithm, it alternately minimized the differentiable part using the gradient information and the non-differentiable part using a proximal operator represented by a learned network.²¹ Wang et al. unrolled the alternating direction method of multipliers (ADMM) to construct the architecture of the unrolling network.²² In this network, the regularization term and the regularization parameters were adaptively obtained from the training data with the architecture of U-Net. He et al. similarly unrolled the ADMM-based iterative reconstruction into a CT reconstruction network.²³ This method took both the projection and image domains into account and utilized residual neural networks to directly replace the regularization term in the projection and image domains respectively. Pan et al. employed the block-coordinate descent method to unroll the objective function based in residual measurements.⁴⁰ This unrolling network integrated neural network prior as a regularizer which was undertaken by the convolution-assisted transformer subnetworks. Komolafe et al. utilized the gradient descent method to solve the Field of Expert based objective function.⁴¹ The regularization term is implemented through a cascaded convolutional and deconvolutional network similar to a classic CNN structure. All of these methods only acknowledged the significance of the data consistency in the unrolling networks, but ignored to explore the prior information which plays a crucial role in the iterative algorithm. Without unfolding the regularization term in the objective equation and directly replacing it with a neural network, deep unrolling networks suffer from a limited performance.

To address the aforementioned limitations of deep unrolling approaches, we constructed the CT reconstruction network by unrolling both the data fidelity term and regularization term. Zhang et al. proposed a deep unrolling algorithm by enabling the direct unrolling of TV prior terms to convolutional neural networks (CNNs).²⁴ To further eliminate the undesired staircase effect caused by TV based models, we utilize the second-order TGV as the regularization term to formulate the objective function for the CT reconstruction problem. The CP algorithm was employed to solve the TGV based CT reconstruction problem to obtain a single-loop CT iterative reconstruction algorithm, which is easy to be unrolled to neural networks. Leveraging the remarkable learning capacity of CNNs, this framework achieves high-fidelity reconstructions by adaptively optimizing parameter spaces during the process of solving inverse problem. The main contributions of our work are as follows.

In view of the staircase effect caused by TV regularization, we adopted the second-order TGV as regularization term of the objective function by incorporating higher-order derivatives. Due to the strong convergence, we employed the CP algorithm to solve above objective function to obtain a single-loop algorithm which is easy to be unrolled to networks.

Different from the other data-driven deep learning-based algorithm, our proposed backbone network architecture, combining model-driven and data-driven approaches, is built by unrolling the iterative algorithm into deep neural networks which retains interpretability of physical models and optimizes the process of reconstruction by outstanding non-linear feature extraction capability of CNNs.

Distinguished from the other deep unrolling approach, TGV-DU took both the data fidelity term and prior information into account when unrolled a model-based IR algorithm into networks rather than replaced the prior term with a CNN directly.

Method

TGV-based iterative algorithm

TV-based reconstruction model has been widely adopted for its edge-preserving properties. However, the assumption of segmentation constants for TV minimization usually leads to significant staircase effect in the reconstructed images.²⁵ In order to eliminate the staircase effect, high-order partial differential equation have been applied to image reconstruction.²⁶ Bredies et al. proposed the concept of the TGV which provides a balance between the first and second derivative of a function. The second-order TGV is defined as²⁷:

T G V_{α}^{2} (u) = min_{v} α_{1} \int_{Ω} | \nabla u - v | d x + α_{0} \int_{Ω} | ε (v) | d x

(1)

where

Ω

is the domain denoting the minimum value to be taken,

\nabla u

denotes the gradient of variable u and

ε (v) = \frac{1}{2} (\nabla v + \nabla v^{T})

denotes the symmetrized derivative.

α_{0}

and

α_{1}

provide a way of balancing between the first and second derivative of a function. Through the definition of the second-order TGV, it can be seen that

\nabla^{2} u

is much smaller than

\nabla u

in the smooth region to achieve the minimization at

\nabla u = v

. Whereas in the neighborhood of edges,

\nabla^{2} u

is larger than

\nabla u

and can be better minimized at

v = 0

. Through the balance of these two terms via the weights

α_{0}

and

α_{1}

T G V_{α}^{2}

has the remarkable ability to characterize the gradient information around the neighborhood of edges and alleviate the staircase effect in smooth regions. Therefore, adopting TGV as a regularization term in the objective function will make the reconstructed images more natural and reliable.

The objective function of the iterative model based on second-order TGV can be written as²⁸:

min_{u, v} \frac{1}{2 λ} ‖ A u - g ‖_{2}^{2} + α_{1} ‖ \nabla u - v ‖_{1} + α_{0} ‖ ε (v) ‖_{1}

(2)

where A is the system matrix, g is the sinogram data, u is the image to be reconstructed,

\nabla u

denotes the gradient of image u. The first-order divergence operator

d i v_{1}

is defined as

d i v_{1} = - \nabla *

, where

\nabla *

is the adjoint operator of

\nabla

. Similarly, the generalized second-order divergence operator

d i v_{2}

is defined as

d i v_{2} = - ε *

, where

ε *

is the adjoint operator of

ε

. The concrete operations of

ε

and

d i v_{2}

are shown in (3) and (4) respectively.²⁶

λ, α_{0}, α_{1}

are hyperparameters.

ε (v) = \frac{1}{2} (\nabla v + \nabla v^{T}) = (\begin{matrix} \partial_{1}^{+} v_{1} \\ \frac{1}{2} (\partial_{2}^{+} v_{1} + \partial_{1}^{+} v_{2}) \end{matrix} \begin{matrix} \frac{1}{2} (\partial_{2}^{+} v_{1} + \partial_{1}^{+} v_{2}) \\ \partial_{2}^{+} v_{2} \end{matrix})

(3)

d i v_{2} (q) = (\begin{matrix} \partial_{1}^{-} q_{11} + \partial_{2}^{-} q_{12} \\ \partial_{1}^{-} q_{21} + \partial_{2}^{-} q_{22} \end{matrix})

(4)

where the forward and backward difference operators are simplified defined as

(u \in R^{m \times n})

(\partial_{1}^{+} u)_{i, j} = {\begin{matrix} u_{i + 1, j} - u_{i, j}, 1 \leq i < m \\ 0, i = m \end{matrix}

(5)

(\partial_{2}^{+} u)_{i, j} = {\begin{matrix} u_{i, j + 1} - u_{i, j}, 1 \leq j < n \\ 0, j = n \end{matrix}

(6)

(\partial_{1}^{-} u)_{i, j} = {\begin{matrix} u_{i, j} - u_{i - 1, j}, 1 \leq i < m \\ 0, i = m \end{matrix}

(7)

(\partial_{2}^{-} u)_{i, j} = {\begin{matrix} u_{i, j} - u_{i, j - 1}, 1 \leq j < n \\ 0, j = n \end{matrix}

(8)

The above minimization problem can be seen as an optimization problem solved by CP algorithm due to its convergence guarantee. The CP algorithm is used to solve a general form of the primal minimization problems in the following.

min_{x} {F (K x) + G (x)}

(9)

In the TGV-based minimization problem, $u, v$ are primal variables, $p, q, r$ are introduced dual variables which are defined as follows.

\begin{aligned} x = (\begin{matrix} u \\ v \end{matrix}), y = (\begin{matrix} p \\ q \\ r \end{matrix}), (\begin{matrix} p \\ q \\ r \end{matrix}) = (\begin{matrix} \nabla \\ 0 \\ A \end{matrix} \begin{matrix} - 1 \\ ε \\ 0 \end{matrix}) (\begin{matrix} u \\ v \end{matrix}), \\ K = (\begin{matrix} \nabla \\ 0 \\ A \end{matrix} \begin{matrix} - 1 \\ ε \\ 0 \end{matrix}) \end{aligned}

(10)

The next step is to derive the dual problem of the primal problem by computing the convex conjugate function of F and G, defined in (11). Based on the TGV prior, the convex function F and G and their convex conjugation which play a vital role to update the values of variables in the CP algorithm are defined as follows.

H * (z) = max_{z^{'}} {{⟨ z, z^{'} ⟩}_{Z} - H (z^{'})}

(11)

\begin{aligned} F_{1} (x) = ‖ x ‖_{1}, F_{2} (x) = ‖ x ‖_{1}, F_{1} (x) = \frac{1}{2 λ} ‖ x ‖_{2}^{2}, \\ G_{1} (x) = 0, G_{2} (x) = 0, \\ F_{1} * (p) = δ_{P} (p), F_{2} * (q) = δ_{Q} (q), F_{3} * (r) = \frac{1}{2} ‖ r ‖_{2}^{2} \end{aligned}

(12)

Subsequently, the above minimization problem can be a special instance of CP algorithm²⁹ and the primal and dual problems are connected in the following saddle point optimization：

min_{u, v} max_{p, q, r} {⟨ r, A u - g ⟩ + ⟨ p, \nabla u - v ⟩ + ⟨ q, ε (v) ⟩ - δ_{P} (p) - δ_{Q} (q) - \frac{λ}{2} ‖ r ‖_{2}^{2}}

(13)

where the exponential functions are defined respectively as

δ_{P} (p) = {\begin{matrix} 0, ‖ p ‖_{\infty} \leq a_{1} \\ \infty, else \end{matrix}, δ_{Q} (q) = {\begin{matrix} 0, ‖ q ‖_{\infty} \leq a_{2} \\ \infty, else \end{matrix}

(14)

The key to solving (13) lies in determining the proximal mapping of the above variables to update the variables. The definition of the proximal mapping is as follows:

p r o x_{σ} [H] (z) = \arg min_{z^{'}} {H (z^{'}) + \frac{‖ z - z^{'} ‖_{2}^{2}}{2 σ}}

(15)

In TGV-based problem, we computed the proximal mapping of the primal and dual variables to obtain the iterative reconstruction algorithm.

{\begin{matrix} u^{n + 1} = \arg min_{u} {⟨ p^{n}, \nabla u - v^{n} ⟩ + ⟨ r^{n}, A u - g ⟩ + \frac{1}{2 τ} ‖ u - u^{n} ‖_{2}^{2}} \\ v^{n + 1} = \arg min_{v} {⟨ p^{n}, \nabla u^{n} - v ⟩ + ⟨ q^{n}, ε (v) ⟩ + \frac{1}{2 τ} ‖ v - v^{n} ‖_{2}^{2}} \\ {\bar{u}}^{n + 1} = u^{n + 1} + θ (u^{n + 1} - u^{n}) \\ {\bar{v}}^{n + 1} = v^{n + 1} + θ (v^{n + 1} - v^{n}) \\ p^{n + 1} = \arg max_{p} {⟨ p, \nabla {\bar{u}}^{n + 1} - {\bar{v}}^{n + 1} ⟩ - δ_{P} (p) - \frac{1}{2 σ} ‖ p - p^{n} ‖_{2}^{2}} \\ q^{n + 1} = \arg max_{q} {⟨ q, ε ({\bar{v}}^{n + 1}) ⟩ - δ_{Q} (q) - \frac{1}{2 σ} ‖ q - q^{n} ‖_{2}^{2}} \\ r^{n + 1} = \arg max_{r} {⟨ r, A {\bar{u}}^{n + 1} - g ⟩ - \frac{λ}{2} ‖ r ‖_{2}^{2} - \frac{1}{2 σ} ‖ r - r^{n} ‖_{2}^{2}} \end{matrix}

(16)

We further calculated the above formula and listed the specific steps in Algorithm 1.

It can be seen that there are numerous parameters in the above iterative algorithm which necessitate much labors and high computations to select a best value. Inspired by the outstanding learning ability of networks, this single-loop iterative algorithm is easy to be unrolled into a network where train these parameters from big data automatically.

LDCT reconstruction network based on second-order TGV unrolling

We constructed the architecture of TGV-DU based on the iterative algorithm in Algorithm 1. The overall structure of our proposed TGV-DU network is illustrated in Figure 1. In this network, each block of the network corresponds to one iteration of the Algorithm 1 which is illustrated in Figure 1(a). Each block is composed of multiple subnetworks which keep the mapping relation between variables but generalize derived operations to have learnable parameters. The subnetworks are demonstrated in Figure 1(b). Inspired by the powerful representation³⁰ and the universal approximation property³¹ of CNN, we adopted a combination of three linear convolution operators with bias terms separated by batch normalization (BN) layers and nonlinear activation layers which construct the architecture of subnetworks with purpose to learn parameters. The pixel size and number of channels of each convolution are shown in Figure 1(c). The update steps between Line 4 and Line 10 are generalized to $C N N_{u}, C N N_{v}, C N N_{\bar{u}}, C N N_{\bar{v}}, C N N_{p}, C N N_{q}, C N N_{r}$ respectively which are utilized to learn the weighting parameters between different variables. With fixed parameters, we generalize the forward projection operation A and back-projection operation $A^{T}$ to $C N N_{A}$ and the $C N N_{A^{T}}$ which utilize the ray-driven and pixel-driven methods respectively.³² The operators $\nabla, d i v_{1}, ε, d i v_{2}$ are implemented by four fixed-parameters convolution layers $C N N_{\nabla},$ $C N N_{d i v_{1}}, C N N_{ε}, C N N_{d i v_{2}}$ respectively. The fixed convolution templates are shown in Figure 2. The parameters of $C N N_{A}, C N N_{A^{T}}, C N N_{\nabla}, C N N_{d i v_{1}}, C N N_{ε}, C N N_{d i v_{2}}$ are fixed which do not need to update in the process of iteration.

Figure 1.

Illustration of our proposed TGV-DU. (a) the overall framework of N iterations (b) the detailed structure of the n-th iteration block (c) the structure of CNN in (b).

Figure 2.

Fixed convolutional kernel, (a), (b): the convolution kernel of operation $\nabla$ and $ε$ (c), (d): the convolutional kernel of operation $d i v_{1}$ and $d i v_{2}$ .

$C N N_{u}$ : This subnetwork aims to compute the $u^{n}$ according to Line 4 of Algorithm 1. Given the input $u^{n - 1}, d i v_{1} p^{n}$ and $A^{T} r^{n}$ , the output of this layer is defined as :

u^{n + 1} = u^{n} + τ^{n} (d i v_{1} p^{n} - A^{T} r^{n})

(17)

where

d i v_{1} p^{n}

denotes the output of the layer

C N N_{d i v_{1}}

with the input

p^{n}

and

A^{T} r^{n}

represents the output of the layer

C N N_{A^{T}}

with the input

r^{n} . τ^{n}

.is the learnable parameter trained by this subnetwork which is not constrained to be the same to enhance the capability of network compared with traditional iterative algorithm. In the first iteration,

u^{n}, p^{n}

and

r^{n}

are initialized to zeros.

$C N N_{v}$ : Similarly, the operation in Line 5 of Algorithm 1 can be seen as a weighted sum of the inputs $v^{n}, p^{n}$ and $d i v_{2} q^{n}$ implemented by this subnetwork. $d i v_{2} q^{n}$ is the output of the layer $C N N_{q}$ which will be introduced later. In the first iteration, $v^{n}$ and $q^{n}$ are initialized to zeros.

$C N N_{\bar{u}}, C N N_{\bar{v}}$ : The operation in Line 6 and Line 7 are replaced by $C N N_{\bar{u}}$ and $C N N_{\bar{v}}$ respectively where the learnable parameter is $θ^{n}$ .

$C N N_{p}$ : The operation in Line 8 of Algorithm 1 can be simplified as:

p^{n + 1} = \frac{p^{n} + σ^{n} (\nabla {\bar{u}}^{n + 1} - {\bar{v}}^{n + 1})}{η^{n}}

(18)

where

η = max (1_{I}, | p^{n} + σ (\nabla {\bar{u}}^{n + 1} - {\bar{v}}^{n + 1}) | / α_{1})

. Given three inputs

p^{n}, {\bar{v}}^{n + 1}

and the gradient of

{\bar{u}}^{n + 1}

, the output of this layer is

p^{n + 1}

. The parameters

σ^{n}

and

η^{n}

are trained from big data.

$C N N_{q}$ : By performing the operation of $d i v_{2}$ on both sides of the equation, the operation in Line 9 of Algorithm 1 is defined as:

d i v_{2} q^{n + 1} = \frac{d i v_{2} q^{n} + σ^{n} d i v_{2} ε ({\bar{v}}^{n + 1})}{ζ^{n}}

(19)

where

ζ = max (1_{I}, | q^{n} + σ ε ({\bar{v}}^{n + 1}) | / α_{0})

. The operator

- d i v_{2} ε

denotes the calculation of the generalized second-divergence defined in (4) for an image. In the first iteration,

- d i v_{2} ε

is initialized to zeros, therefore

d i v_{2} q^{n + 1}

incorporates the second-divergence information of an image. Given the inputs

d i v_{2} q^{n}

and

d i v_{2} ε ({\bar{v}}^{n + 1})

, this layer has the ability to learn the weighted parameters in (19).

$C N N_{r}$ : The update step of $r^{n}$ is simplified as:

r^{n + 1} = \frac{(r^{n} + σ (A {\bar{u}}^{n + 1} - g))}{γ}

(20)

where

γ = 1 + σ λ

, As mentioned above, it can be considered as a weighted sum of the

r^{n}

, the forward projection of image

{\bar{u}}^{n + 1}

and sinogram data g.

C N N_{r}

is employed to learn weighting parameters. In the first iteration,

r^{n}

is initialized to zero.

In conclusion, seven kinds of CNNs were proposed to implement the proximal operators of the vectors $p^{n}, q^{n}, r^{n}, u^{n}, q^{n}, r^{n}, u^{n}$ and $v^{n}$ , unrolling both the data fidelity and TGV regularization terms in the objective function (2) to CNNs. The parameterized mapping that updates primal variables and dual variables across successive iterations was implemented by convolutional neural networks and was dynamically learned from big data. Instead of enforcing the fixed form of updates such as $u^{n} + τ^{n} (d i v_{1} p^{n} - A^{T} r^{n})$ explicitly derived from iterative algorithm, we allow the network to learn the relationship between the inputs by itself. The unrolling network also adopted a residual connection that makes the network a residual network. Considering of the definition of proximal operators, the updated values of variables are close to the previous values. And there are plenty of machine learning literatures show that networks of this structure are easier to train.³³

Our proposed TGV-DU is an unrolling approach with 10 iterations which incorporates both the data consistency and prior information of an image by combining model-driven and data-driven approaches. Each iteration block is composed of seven subnetworks which has the same structure. The structure of CNNs is employed to enhance the feature extraction capabilities of the network. The sinogram data g is inputted into the network. There are three outputs ${\bar{u}}^{n + 1}, p^{n} p^{n}$ and $d i v_{2} q^{n + 1}$ for the TGV-DU. According to the definition of $p^{n}$ , the result of the iteration should be close to 0. Hence, we employed it as a part of the loss function which ensure the consistency in the gradient domain. As shown in (19), the physical interpretation of $d i v_{2} q^{n + 1}$ is the generalized second-divergence of an image. To better preserve structural and textural information, the loss function incorporates the divergence information to improve the quality of the image. Therefore, the Euclidean loss function of the proposed network can be written as:

L o s s = ‖ {\bar{u}}^{n + 1} - \hat{u} ‖_{2}^{2} + β_{0} ‖ p^{n} - 0 ‖_{2}^{2} + β_{1} ‖ d i v_{2} q^{n + 1} - d i v_{2} \hat{q} ‖_{2}^{2}

(21)

where 0 is a tensor of the same dimensional size as

p^{n}

and

d i v_{2} \hat{q}

is the generalized second-order divergence of the input of the labeled image. In this paper, both

β_{0}

and

β_{1}

were set to 0.5 in this paper.

Datasets

Mayo dataset

In this paper, we mainly utilized the public dataset “Low-Does CT Image and Projection Data”³⁴ to verify the effectiveness of our proposed TGV-DU network. The chest CT scan images of 14 patients were selected as the training set and the chest CT scan images of 1 patient were selected as the test set. Since the CT scan images of this dataset were obtained by spiral scanning, we performed the projection of the obtained CT images with fan-beam physical geometry to obtain the CT projection data under fan-beam geometry. In this case, the distance from the ray source to the center of rotation is set to 595 mm, the distance from the ray source to the center of the detector is 1068 mm. The detector is a line-array detector having 768 detectors with a spacing of 1 mm and the ray source is scanned around the patient doing 360 equally spaced scans at an angle with a spacing of 1°, obtaining the size of the sinogram data of 768*360. The size of the reconstructed image of 512*512 with the size of each pixel size is set to 0.5859*0.5859. Since it is difficult to obtain both normal-dose and low-dose scanning images and sinogram data, we simulated the low-dose sinogram data from chosen CT images by fan-beam scanning. Adding Poisson noise and electronic noise to the noise-free sinogram data obtained from the above fan-beam scanning can better simulate the clinical environment and generate the low-dose sinogram data under different X-ray incident intensity with the formula¹⁸:

g = \ln \frac{I_{0}}{P o s s i o n (I_{0} \exp (- \tilde{g})) + N o r m a l (0, ε^{2})}

(18)

where

\tilde{g}

is the noise-free sinogram data of the fan-beam geometry obtained from the simulation at the standard dose.

P o s s i o n (\cdot)

is the Poisson function, and

N o r m a l (\cdot)

is the standard normal distribution function with variance

ε

. In this paper,

ε

was set to

\sqrt{10}

.¹⁸

I_{0}

is the number of incident photons of X-rays, which was set to

1.0 \times 10^{6}

in the normal dose under a 360-view scanning condition. We simulated the low-dose datasets of different dose levels, namely 15, 10 and 5%. To add noise of the three levels to the datasets, parameters

I_{0}

was set to

1.5 \times 10^{5}, 1.0 \times 10^{5}

, and

0.5 \times 10^{5}

respectively. To further validate the effectiveness of the proposed loss function, we also performed the experiments with the different loss function.

Piglet dataset

To further provide the robustness and practical viability of the proposed unrolling network, we executed the supplementary experiments on a publicly accessible Piglet Dataset.⁴² This dataset comprises authentic porcine CT scans acquired with a GE Discovery CT750 HD scanner, setting the tube voltage to 100kVp and a slice thickness to 0.625 mm. Normal dose reference images were obtained at a tube current of 300 mAs. Corresponding LDCT images were generated by reducing the tube current to 50%, 25%, 10%, 5% of the normal dose. For model training, image pairs consisting solely of LDCT images obtained at 150mAs (50% dose) and corresponding normal dose CT images at 300mAs were utilized. The training set comprised 525 randomly selected image pairs. The trained model was subsequently tested on test sets encompassing all four doses (50%, 25%, 10%, 5%), with each test set containing 40 image pairs.

Implementation details

We performed the training and testing of the deep network within the PyTorch framework,³⁵ using the Adam optimization³⁶ algorithm for gradient descent optimization of the loss function. The initial learning rate of this network was set to $1.0 \times 10^{- 4}$ and was decreased by 20% after the completion of each epoch of learning. The batchsize for network training was set to 1. The experiments were performed on a workstation computer platform with an Intel^® Core™ i7-9700 K CPU, 32 GB RAM, and an NVIDIA RTX 3090 GPU. In order to verify the effectiveness of the proposed algorithm, the proposed algorithm was compared with four state-of-the-art networks, including FBPConv-Net,³⁷ Learned Primal-Dual (LPD),¹⁹ FISTA-Net,²¹ DRCR,³⁸ PWLS-SRDTGV,⁴³ MAIR-Net.⁴⁴ To further evaluate the performance of our proposed TGV-DU, peak signal-to-noise ratio (PSNR), structural similarity (SSIM)³⁹ and root mean squared error (RMSE) were employed for quantitative analysis of images.

Results

Results on mayo dataset

Comparison of results at 15% dose

Two representative slices from the testing dataset are selected for visual comparisons of different DL-based methods in Figure 3 at 15% dose. It can be observed that the result of FBPConv is affected by streak-like artifacts which degrade the quality of images. In the results of LPD, there are a certain degree of improvement in image quality compared with the FBPConv, but there are still deficiencies in the accurate restoration of the subtle structures which are vital to diagnosis. Although the results of FISTA show certain noise suppression and texture preservation capabilities, there seem to be some additional structures which are nonexistent in the reference images. The DRCR algorithm has improved the effect of the artifact removal and detail restoration. But there are some oversmoothed effects in the results of DRCR. While the iterative reconstruction-based PWLS-SRDTGV method preserves significant structural details, it exhibits less effective denoising performance compared to deep learning approaches. Conversely, MBIR achieves near-optimal noise suppression for low-dose conditions but yields inaccurate tissue details within ROI regions—particularly at locations indicated by red arrows. As can be seen in Figure 3, the results show that our proposed TGV-DU not only alleviates the effect of noise and artifacts but also keeps richer subtle details. To further evaluate the effectiveness of our proposed TGV-DU, Table 1 illustrates the PSNR, SSIM and RMSE metrics of different methods from the first slice. The bold values in the tables represent the best qualitative indicator values. From Table 1, it is not difficult to see that our proposed method achieves the best performance in metrics which is consistent with the visual inspection in Figure 3. Different from other images, the medical images pay great attention to details which may contain medical diagnostic information. To further demonstrate the detail-preserving capability of our proposed TGV-DU, the magnified regions of interest (ROIs) marked by red rectangles in the first row are shown in the second row. As indicated by red arrows, it can be observed that TGV-DU can better reconstruct the subtle details compared with other methods where are slightly blurred. To quantify the detail preservation capability of different methods in ROI, a horizontal profile, located at the 120th row and 20th to 50th columns in the second row of Figure 3, is plotted in Figure 4. As demonstrated by the results, TGV-DU exhibits superior consistency relative to the reference image compared to other approaches.

Figure 3.

Results of different algorithms at 15% dose. The ROIs, marked with red rectangular boxes in the first and third rows of NDCT, are shown as magnified views below each corresponding image (the second and forth row respectively). The display window is [-160,240]HU.

Figure 4.

Intensity distribution of different algorithms at 15% dose.

Table 1.

Comparison of metrics for different algorithms at 15% dose in Figure 3.

	FBPConv	LPD	FISTA	DRCR	PWLS-SRDTGV	MAIR	TGV-DU
PSNR↑	32.9865	33.4040	34.7350	34.7151	32.2721	35.5440	36.0117
SSIM↑	0.7756	0.7838	0.8562	0.8210	0.7906	0.8530	0.8585
RMSE↓	0.0206	0.0284	0.0184	0.0184	0.0242	0.0161	0.0159

Comparison of results at 10% dose

Figure 5 demonstrates two representative images for qualitative comparison in the first and third rows at 10% dose. As the number of incident photons of X-rays decreases, the results of FBPConv become even worse. Although some improvement has been achieved for noise suppression, the details in the results of LPD, FISTA and DRCR are blurred to varying degrees. Compared to deep learning methods, the iterative PWLS-SRDTGV reconstruction preserves structural detail effectively but achieves weaker denoising effect. Conversely, MBIR provides near-optimal noise suppression under low-dose conditions yet compromises the fidelity of tissue details within ROIs. To further verify the detail preservation capability of our proposed TGV-DU, the ROIs labeled by the red rectangles in the first and third rows of Figure 5 are magnified in the second and forth rows of Figure 5. As indicated by red arrows in zoomed regions, it can be observed that the results of FBPConv, LPD and FISTA cannot suppress the noise and artifacts in certain degree. DRCR has achieved some improvement in noise suppression, but it seems to erode the details indicated by red arrows due to the oversmoothed effects. Our proposed TGV-DU seems to achieve a better balance between the artifact removal and detail preservation in visual effect. To further illustrate the effectiveness of the proposed algorithm, Table 2 lists the quantitative results of metrics on the first slice demonstrated in Figure 5. TGV-DU scores the highest PSNR, SSIM and lowest RMSE compared with other methods which confirm the superiority of the proposed approach. To further inspect the detail preservation ability of different methods, a horizontal profile, located at the 126th row and 55th to 100th columns in the second row of Figure 5, is plotted in Figure 6. Among all approaches, TGV-DU is the most closely aligned with the reference curve.

Figure 5.

Results of different algorithms at 10% dose. The ROIs, marked with red rectangular boxes in the first and third rows of NDCT, are shown as magnified views below each corresponding image (the second and forth row respectively). The display window is [-160,240]HU.

Figure 6.

Intensity distribution of different algorithms at 10% dose.

Table 2.

Comparison of metrics for different algorithms at 10% dose in Figure 5.

	FBPConv	LPD	FISTA	DRCR	PWLS-SRDTGV	MAIR	TGV-DU
PSNR↑	32.0180	33.7556	33.6442	32.0456	31.6135	33.6809	34.3025
SSIM↑	0.7500	0.7838	0.7820	0.7890	0.7178	0.8095	0.7942
RMSE↓	0.0252	0.0206	0.0209	0.0251	0.0262	0.0207	0.0193

Comparison of results at 5% dose

As the number of incident photons decreases further, the results of different approaches at 5% dose are demonstrated in Figure 7. Due to the server noise and artifacts, the results of FBPConv, LPD, FISTA and DRCR become even worse. The quality of the image is drastically reduced which is difficult to obtain useful information. When incident photons drop to ultra-low dose level (5% of standard), PWLS iterative reconstruction suffers from severely elevated noise that critically degrades image quality. While MBIR effectively suppresses this noise, it eliminates essential structural details within ROIs, particularly at the regions marked by blue arrows. The results of TGV-DU achieve better performance in terms of noise suppression and detail preservation. To further demonstrate the effectiveness of TGV-DU, the zoomed regions are displayed in the second and forth rows of Figure 7 marked by blue rectangles in the first and third rows of Figure 7. Two selected structures of vessel are indicated by blue arrows in zoomed region. It can be observed that these structures almost disappear in the results of FBPConv, LPD, FISTA and DRCR. But they are well reconstructed by TGV-DU and have the closest structures of the reference images. Table 3 demonstrates the quantitative analysis with respect to three chosen metrics of the first slice. It is obvious that TGV-DU achieves the highest results of both PSNR and SSIM, lowest result of RMSE, which illustrates the effectiveness of our proposed approach. To further verify the detail preservation capability of different approaches, a horizontal profile, located at 29th row and 20th to 60th columns in the second row of Figure 7, is plotted in Figure 8. It is obvious that TGV-DU exhibits the smallest deviation from the reference curve, outperforming the other approaches.

Figure 7.

Results of different algorithms at 5% dose. The ROIs, marked with red rectangular boxes in the first and third rows of NDCT, are shown as magnified views below each corresponding image (the second and forth row respectively). The display window is [-160,240]HU.

Figure 8.

Intensity distribution of different algorithms at 5% dose.

Table 3.

Comparison of metrics for different algorithms at 5% dose in Figure 7.

	FBPConv	LPD	FISTA	DRCR	PWLS-SRDTGGV	MAIR	TGV-DU
PSNR↑	30.9875	31.3785	30.2466	30.1489	28.3394	31.7738	33.1936
SSIM↑	0.7302	0.7662	0.7641	0.7757	0.6924	0.7709	0.7987
RMSE↓	0.0288	0.0331	0.0384	0.0338	0.0385	0.0265	0.0220

Results on piglet dataset

To demonstrate the robustness of the proposed unrolling network, we conducted additional experiments on the piglet dataset. The network was trained exclusively at the 50% dose level, which was then utilized to reconstruct images from test data acquired at 50%, 25%, 10% and 5% dose levels. Comparative reconstruction results by different methods are presented in Figure 9. FBPConv and FISTA exhibit server noise and artifacts which significantly degrade the image quality. While LPD demonstrates effective noise suppression, it suffers from excessive smoothing leading to structural blurring and consequent anatomical distortion. DRCR achieves improved retention of fine structural details compared to LPD, yet perceptible residual noise and artifacts persist in its outputs. The proposed method addresses these limitations by effectively suppressing noise and artifacts while simultaneously preserving critical structural details to a greater extent than the benchmark approaches. This qualitative superiority is consistently demonstrated in representative regions of interest (ROIs), which clearly illustrate the enhanced capability of the proposed algorithm in maintaining diagnostic image integrity. Corroborating the visual assessment, quantitative analysis confirms the superiority of the proposed approach. It achieves significantly higher PSNR values coupled with substantially lower RMSE values compared to other methods. This quantitative advantage is shown in the upper left corner of Figure 9.

Figure 9.

Result of different methods on the piglet dataset. The magnified ROI is displayed in the upper right corner. The metrics of the whole slice is illustrated in the upper left corner.

Effect of different network parameters

In this section, we conducted two experiments to analyze the values of the parameters involved in the training process on the Mayo dataset at 15% dose, including the size of the convolution kernel and the number of iterations. The convolutional kernel sizes were set to 3, 5 and 7. The number of iterations was set to6, 8 and 10. The results are shown in Table 4 and 5. As indicated in Table 4, the performance metrics exhibit a gradual decline with increasing convolutional kernel size. This degradation suggests a plausible explanation: the available training data may be insufficient to effectively optimize the substantially increased number of trainable parameters associated with larger kernels. Conversely, Table 5 demonstrates a consistent improvement in performance metrics as the number of iterations increases. This positive correlation indicates that additional iterations enhance the model's representational capacity and optimization efficacy within the constraints of the architecture. Based on these empirical observations, the network configuration employing a convolutional kernel size of 3 × 3 coupled with 10 iterations was selected as the optimal configuration for the proposed unrolling network.

Table 4.

The result comparisons of different convolutional kernel size.

Kernel size	3*3	5*5	7*7
PSNR	35.3965	34.6274	30.2568
SSIM	0.8500	0.8392	0.7626

Table 5.

The result comparisons of different iterations.

Iterations	N = 6	N = 8	N = 10
PSNR	34.7976	35.3318	35.3965
SSIM	0.8371	0.8497	0.8500

Reconstruction results with different loss functions

To demonstrate the effectiveness of the proposed loss function in this paper, we train the network with different loss functions at 15% dose, as shown in Table 6. One representative slice from testing dataset is chosen for visual comparison in Figure 10 (a)–(e). It can be seen that although the network training with the incomplete loss function achieves good reconstruction results, the proposed loss function shows more remarkable success in edge preservation and detail reconstruction. To further visualize the texture preservation capability, the difference images are shown in Figure 10 (f)–(i). It can be seen that the difference image of (d) and the label image has the least gray level difference region, and the recovery is better at the edges and details, which fully demonstrates that the reconstructed image of the normal case has a very high degree of similarity to the target image, and among all the loss function settings, the case has the best performance in terms of the accuracy and fidelity of image reconstruction. performance is optimal. Table 7 illustrates the values of PSNR, SSIM, and RMSE metrics with different loss function settings, which shows that the adopted loss function has higher PSNR and SSIM and lower RMSE.

Figure 10.

Reconstructed and difference images with different loss functions. The first row's display window is [-160,240]HU. The second row's display window is [-30,30] HU after applying a thresholding of 40HU.

Table 6.

The definition of different loss function.

	${\bar{u}}^{n + 1} - \hat{u}$	$p^{n} - 0$	$d i v_{2} q^{n + 1} - d i v_{2} \hat{q}$
Loss1	✓	✗	✗
Loss2	✓	✓	✗
Loss3	✓	✗	✓
Normal loss	✓	✓	✓

Table 7.

Comparison results of metrics with different loss functions.

	Loss1	Loss2	Loss3	Normal loss
PSNR↑	33.6459	34.3979	33.8183	34.8376
SSIM↑	0.8148	0.8363	0.8126	0.8403
RMSE↓	0.0209	0.0191	0.0205	0.0175

Ablation studies

To validate the impact of different neural network modules on TGV-DU algorithm performance within the iterative unrolling framework, we conducted ablation studies using three distinct reconstruction networks (Net1, Net2, Net3). Table 8 lists the neural module configurations for each network, where ✗ indicates not to replace the variable optimization steps in the iterative reconstruction algorithm with a neural network module and ✓ denotes to replace the optimization steps with a neural network module. Performance was evaluated under 15% dose on the Mayo dataset. It can be observed that the visual quality of the reconstruction results continuously improves as the number of neural network modules increases from Figure 11. The values of PSNR and SSIM metrics of different networks are shown in Table 8. Consistent with visual observations, TGV-DU with full neural module replacement achieved optimal quantitative metrics. These ablation experiments confirm the efficacy of substituting optimization steps in iterative reconstruction algorithms with neural modules.

Table 8.

Ablation studies: quantitative comparison of different networks on the Mayo dataset at 15% dose.

Method	$C N N_{p}$	$C N N_{q}$	$C N N_{r}$	PSNR	SSIM
Network1	✗	✗	✗	32.2011	0.7870
Network2	✓	✗	✗	33.4957	0.8162
Network3	✓	✓	✗	32.0068	0.7846
TGV-DU	✓	✓	✓	35.3965	0.8500

Figure 11.

Ablation studies: visual comparison of different networks on Mayo dataset at 15% dose. The display window is [-160,240]HU.

Discussion

In this work, we proposed a novel LDCT network by unrolling the second-order TGV based IR algorithm, dubbed TGV-DU. The second-order TGV-based IR algorithm has the potential to alleviate the staircase effect caused by TV regularization. The CP algorithm was employed to solve the objective function to obtain a single-loop iterative algorithm. Inspired by the outstanding non-linear feature extraction capability of CNNs, this single-loop iterative algorithm can be unrolled into network easily which effectively improve the quality of imaging. However, most unrolling approaches have potential limitations. The regularization term, as a critical component of objective function, exerts a decisive influence on noise suppression and detail preservation of the reconstructed images in IR algorithm. While, some unrolling approaches only unroll the data fidelity term but replace the regularization term with a neural network directly which heavily limits the performance of the network. Incorporating the prior information into the unrolling network can further improve the performance of CT reconstructions as demonstrated in Figure 3, 5 and 7.

Our previously proposed network PD-Net unrolled both the data fidelity term and TV term into network. To further alleviate the staircase effects caused by TV, we adopted the second-order TGV-based model and implemented the proximal mapping of both the data fidelity term and regularization term into learnable convolutional neural networks. One of the core advantages of TGV-DU is that TGV regularization term provides a way of balancing between the first and second derivative, significantly suppresses noise at low doses, and excels in detail preservation. Table 9 illustrates the quantitative results compared with PD-Net at 15% dose. It can be observed that TGV-DU achieves the better score in terms of PSNR and RMSE, PD-Net scores the higher SSIM metric. The proposed TGV-DU method exhibits opportunities for further refinement and enhancement.

Table 9.

Comparison with PD-Net.

	PSNR	SSIM	RMSE
PD-Net	34.9175	0.8714	0.0180
TGV-DU	36.0117	0.8585	0.0159

Although we achieved better results on low-dose simulation datasets, validation on real LDCT datasets in a clinical setting remains a focus for future research. In particular, clinical data with more projection

amplitude and complexity will pose a higher challenge to the generalization ability of TGV -DU, so further clinical validation experiments are important for the feasibility of this method.

Conclusion

To make full use of the interpretability of physical models and the outstanding non-linear feature extraction capability of CNNs, we combine model-driven and data-driven approaches to constructed our proposed network architecture which unroll the TGV-based iterative algorithm into deep neural networks for LDCT reconstruction. The proposed network effectively suppresses low-dose noise while preserving critical edge details. Its three-layer CNN architecture hierarchically captures distinct feature representations, which are progressively refined and synthesized into the final high-fidelity output. The experimental results show that TGV-DU not only performs well in terms of visual effects, but also surpasses multiple the-state-of-art algorithms in terms of quantitative evaluation metrics, demonstrating a superior reconstruction performance. It is also a feasible strategy to set different regularization terms for various task scenarios in our future research.

Footnotes

Acknowledgements

The authors would like to thank the institution, the Mayo Clinic, for providing the data (Low-Dose CT Image and Projection Data) used in this study. This work was supported in part by the National Natural Science Foundation of China under Grant 62401517, in part by the Natural Science Foundation of Shanxi Province under Grant 202303021211148, 202203021222038, and 201901D211246, in part by the Research Project Supported by Shanxi Scholarship Council of China under Grant 2021-111 and in part by the Innovation Project for Graduate Student of Shanxi Province under Grant 2024KY582.

ORCID iD

Zhiguo Gui

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Tsalafoutas

Hassan Kharita

Al-Naemi

, et al. Radiation dose monitoring in computed tomography: status, options and limitations. Phys Med 2020; 79: 1-15.

Chen

Zhou

, et al. Deep learning-based algorithms for low-dose CT imaging: a review. Eur J Radiol 2024; 172: 1-13.

Liu

Leng

, et al. Radiation dose reduction in computed tomography: techniques and future perspective. Imaging Med 2009; 1: 65-84.

Willemink

de Jong

Leiner

, et al. Iterative reconstruction techniques for computed tomography part 1: technical principles. Eur Radiol 2013; 23: 1623-1631.

Chen

Xia

Yang

, et al. SOUL-Net: a sparse and low-rank unrolling network for spectral CT image reconstruction. IEEE Trans Neural Netw Learn Syst 2024; 35: 18620-18634.

Ghani

Karl

. Deep learning-based sinogram completion for low-dose CT. In: 2018 IEEE 13th Image, Video, and Multidimensional Signal Processing Workshop (IVMSP). 2018; pp.1-5.

Yang

, et al. Low-Dose CT denoising via sinogram inner-structure transformer. IEEE Trans Med Imaging 2023; 42: 910-921.

Zhang

, et al. Learning to reconstruct computed tomography images directly from sinogram data under A variety of data acquisition conditions. IEEE Trans Med Imaging 2019; 38: 2469-2481.

Chen

Zhang

Kalra

, et al. Low-Dose CT with a residual encoder-decoder convolutional neural network. IEEE Trans Med Imaging 2017; 36: 2524-2535.

10.

Fan

Shan

Kalra

, et al. Quadratic autoencoder (Q-AE) for low-dose CT denoising. IEEE Trans Med Imaging 2020; 39: 2035-2050.

11.

Yang

Yan

Zhang

, et al. Low-dose CT image denoising using a generative adversarial network with wasserstein distance and perceptual loss. IEEE Trans Med Imag 2018; 37: 1348-1357.

12.

Liu

Yan

, et al. DD-DCSR: image denoising for low-dose CT via dual-dictionary deep convolutional sparse representation. IEEE Trans Comput Imaging 2024; 10: 899-914.

13.

Jiao

Gui

, et al. A dual-domain CNN-based network for CT reconstruction. IEEE Access 2021; 9: 71091-71103.

14.

Niu

, et al. DRONE: dual-domain residual-based optimization NEtwork for sparse-view CT reconstruction. IEEE Trans Med Imaging 2021; 40: 3002-3014.

15.

Pan

Zhang

, et al. Multi-domain integrative swin transformer network for sparse-view tomographic reconstruction. Patterns 2022; 3: 1-14.

16.

Liu

, et al. Hybrid-Domain neural network processing for sparse-view CT reconstruction. IEEE Trans Radiat Plasma Med Sci 2021; 5: 88-98.

17.

Chen

Zhang

Chen

, et al. LEARN: learned Experts’ assessment-based reconstruction network for sparse-data CT. IEEE Trans Med Imaging 2018; 37: 1333-1347.

18.

Xia

Huang

, et al. MAGIC: manifold and graph integrative convolutional network for low-dose CT reconstruction. IEEE Trans Med Imaging 2021; 40: 3459-3472.

19.

ÖKETM O

ALDERJ

. Learned primal-dual reconstruction. IEEE Trans Med Imaging 2018; 37: 1322-1332.

20.

Zhang

Ghanem

. ISTA-net: Interpretable optimization-inspired deep network for image compressive sensing. In: 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2018: pp.1828-1837.

21.

Xiang

Dong

Yang

. FISTA-Net: learning a fast iterative shrinkage thresholding network for inverse problems in imaging. IEEE Trans Med Imaging 2021; 40: 1329-1339.

22.

Wang

Zeng

Wang

, et al. ADMM-based deep reconstruction for limited-angle CT. Phys Med Biol 2019; 64: 1-13.

23.

Yang

Wang

, et al. Optimizing a parameterized plug-and-play ADMM for iterative low-dose CT reconstruction. IEEE Trans Med Imaging 2019: 38: 371-382.

24.

Zhang

Ren

Liu

, et al. A total variation prior unrolling approach for computed tomography reconstruction. Med Phys 2023; 50: 2816-2834.

25.

Niu

Gao

Bian

, et al. Sparse-view x-ray CT reconstruction via total generalized variation regularization. Phys Med Biol 2014; 59: 2997-3017.

26.

. Parameter selection and solution algorithm for TGV-based image restoration model. SN Appl Sci 2019; 1: 766.

27.

Bredies

Valkonen

. Inverse problems with second-order Total Generalized Variation constraints. arXiv 2005.09725.

28.

Knoll

Bredies

Pock

, et al. Second order total generalized variation (TGV) for MRI. Magn Reson Med 2011; 65: 480-491.

29.

Sidky

Jørgensen

Pan

. Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm. Phys Med Biol 2012; 57: 3065-3091.

30.

Dong

Loy

, et al. Learning a deep convolutional network for image super-resolution. In: Computer Vision – ECCV 2014. Zurich, Switzerland: Springer, 2014, pp.184-199.

31.

Hornik

Stinchcombe

White

. Multilayer feedforward networks are universal approximators. Neural Netw 1989; 2: 359-366.

32.

Zeng

Gullberg

. Unmatched projector/backprojector pairs in an iterative reconstruction algorithm. IEEE Trans Med Imag 2000; 19: 548-555.

33.

Zhang

Ren

, et al. Deep residual learning for image recognition. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 2016: pp.770-778.

34.

Moen

Chen

Holmes

, et al. Low-dose CT image and projection dataset. Med Phys 2021; 48: 902-911.

35.

Ionescu

Vantzos

Sminchisescu

. Matrix backpropagation for deep networks with structured layers. In: 2015 IEEE International Conference on Computer Vision (ICCV). 2015: pp.2965-2973.

36.

Kingma

. Adam: A method for stochastic optimization. arXiv:1412.6980.

37.

McCann

Jin

Unser

. Convolutional neural networks for inverse problems in imaging: a review. IEEE Signal Process Mag 2017; 34: 85-95.

38.

, et al. DRCR Net: dense residual channel re-calibration network with non-local purification for spectral super resolution. In: 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). 2022: pp. 1258-1267.

39.

Wang

Bovik

Sheikh

, et al. Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 2004; 13: 600-612.

40.

Pan

Gao

, et al. Iterative residual optimization network for limited-angle tomographic reconstruction. IEEE Trans Image Process 2024; 33: 910-925.

41.

Komolafe

Wang

Tian

, et al. MDUNet: deep-prior unrolling network with multi-parameter data integration for low-dose computed tomography reconstruction. Mach Vis Appl 2024; 35: 95.

42.

Piglet Dataset [Online]. Available: http://homepage.usask.ca/∼xiy525/publication/sagan/ 2018.

43.

Niu

Zhang

Qiu

, et al. Evaluation of low-dose computed tomography reconstruction using spatial-radon domain total generalized variation regularization. Phys Med Biol 2024; 8: 105005.

44.

Guo

Liu

Zhang

, et al. MAIR-Net: a sparse-view CT reconstruction network based on a combination of mixed attention and iterative optimization learning. J Instrum 2024; 19: 1-22.