Structure-guided computed tomography reconstruction from limited-angle projections

Abstract

Limited-angle computed tomography (CT) imaging is one of the common imaging problems. The reconstructed images often encounter obvious artifacts and structure degradation. In recent years, the recoverability prior of image structure has been widely explored in limited-angle CT reconstruction, and the image quality has been greatly improved. However, the artifacts and structure degradation still exist. In this study, we establish a new reconstruction model based on weighted relative structure (wRS) determined by image gradients, which serves as weights to guide image reconstruction in order to reduce artifacts and preserve structures. Then, we develop an efficient algorithm using a surrogate function to solve this model. Moreover, this method is compared with some of other popular reconstruction methods, such as anisotropic total variation method and image gradient L₀ norm minimization method and so on. Experiments on digital phantoms, real carved cheese and walnut projection are reported to demonstrate its superiority. Several quantitative indices including RMSE, PSNR, and SSIM of the reconstruction images from 90°-data of FORBILD head phantom are 0.0120, 43.52, and 0.9961. The experimental results indicate that the image obtained by our method is the closest to reference image. By comparing reconstruction images or their residual images, images reconstructed from real CT data, the experimental results of the residual images and the respective quantitative data analysis also demonstrate that the images reconstructed using our new method suffer from less artifacts and structure degradation.

Keywords

Inverse problem image reconstruction computed tomography relative structure anisotropic total variation

1 Introduction

Computed tomography (CT) is widely used in medical diagnosis, industrial non-destructive testing and security check. Traditional CT reconstruction needs complete projection data. For example, circular fan-beam CT reconstruction requires collecting projection data in the range of 360 degrees or at least 180 plus the fan angle. For example, in head and neck surgery, their structures are complex and the operation is difficult. Using C-arm CT for limited-angle CT scanning can realize intraoperative imaging and provide rich geometric information required by the operation. In heart CT imaging [1], collecting projection data at limited view angles can shorten scanning time and reduce motion artifacts of the image. In early screening of breast cancer, limited-angle CT scanning can be used to collect projection data, so as to adapt to the special shape of the breast and reduce the radiation dose. Therefore, limited-angle CT imaging can meet some practical application needs in medical diagnosis. In addition, CT imaging technology is widely used in industrial product quality control and safety inspection [2]. For example, on-line detection of industrial products and security check of baggage at airports or stations require rapid non-destructive testing. Linear computed tomography (LCT) [3] can better serve the above situations. However, LCT is often limited by the field angle of the X-ray source and detector array size. In addition, for plate-like objects such as printed circuit boards (PCB) and biological fossils, limited-angle CT imaging can adapt to their special geometric structures [4]. To sum up, the research on limited-angle CT imaging technology can provide solutions for various needs. CT reconstruction using projection data within a limited angular range is called “limited-angle CT reconstruction", which is an ill-posed inverse problem. The images reconstructed by traditional algorithms, such as filtered back projection (FBP) or simultaneous algebraic reconstruction technique (SART), often encounter obvious artifacts and structure degradation.

Missing projection data is one of the key features of limited-angle CT imaging. Therefore, a natural method is to recover the missing data and then reconstruct CT image using analytical algorithm, such as FBP. Recently, with the development of deep neural network, missing data can be recovered by generative adversarial networks benefiting from its ability of image restoration [5, 6]. In addition, other methods based on deep learning technology have also been used to reduce artifacts and reconstruct degraded structures and details. For example, the advantages of deep learning algorithm in projection data domain and image domain are combined to generate a neural network to suppress artifacts [7]. In limited-angle CT reconstruction, the visible boundary information of the image is taken as the prior knowledge of the structure, and the convolution neural network is used to replace regularization term [8]. The reconstruction method based on deep learning technology can effectively suppress artifacts using the nonlinear fitting ability of neural network, which often requires large number of training data. However, for some special objects, such as biological fossils and other plate-like objects, it is difficult to obtain large number of label images. Therefore, this work plans to establish a new reconstruction model based on rich prior knowledge of CT images for limited-angle CT reconstruction.

Iterative reconstruction can integrate rich prior knowledge to improve CT image quality. As far as we know, image sparsity prior is widely used in CT reconstruction [9 –13]. Total variation- (TV-) based method [13, 14] has been used for limited-angle CT reconstruction, and acquires better image quality than FBP and SART. Some methods are proposed to improve TV-based methods [10 , 15–17]. A typical one is image gradient L₀ norm based method [10], which has stronger ability to suppress artifacts and preserve image structures. However, these methods only use image sparsity to reconstruct CT images, and do not integrate the characteristics of limited-angle CT scan or images into reconstruction model. As shown in Fig. 1, the scanning angular range of limited-angle CT is a significantly reduced one, less than 180 degrees. The incident X-rays are only tangent to structures in some directions. It has been shown that image structures tangent to incident X-rays are easier to reconstruct; on the contrary, it is difficult to reconstruct [18]. Therefore, the images acquired by traditional algorithm suffer from artifacts and structure degradation. Based on the theoretical analysis, Chen et al. [19] defined anisotropic total variation (ATV) term for limited-angle CT reconstruction, which first integrates structure recoverability into reconstruction model. Reweighted anisotropic total variation (RwATV) [20] is proposed as an improved version of ATV term, which combines the merits of image sparsity and reweighted technique. Following this idea, Xu et al. [21] use visible edges as prior information to recover degraded structures. They propose a reconstruction method regularized by alternating edge-preserving diffusion and smoothing (AEDS) based on image gradient L₀ norm. Then, the local statistical information of gradient image is used to define anisotropic relative total variation (ARTV) [22] to guide limited-angle CT reconstruction. Directional-TV [23] is proposed for limited-angle CT and DTV algorithm is designed to solve the optimization problem. Recently, anisotropic structure property based method [24] is proposed for limited-angle CT. However, limited-angle CT reconstruction still attracts researchers’ attention to remove artifacts and recover structures [25, 26].

Fig. 1

Schematic diagram of limited angle CT scanning: the scanning angular range [α, π - α] is significantly less than 180 degrees.

In this work, we propose a new reconstruction model based on defined weighted relative structure (wRS) term for limited-angle CT reconstruction. Structure information is the key feature of a CT image, and it is often encoded by image gradient. However, limited-angle CT images often encounter shading artifacts and structure degradation, which means that the image sparsity is inconsistent with the desired image. Therefore, structures with larger gradient magnitude need to be preserved and the artifacts with smaller magnitude should be suppressed. Ideally, if there are wise weights indicating which to suppress and which to preserve, image reconstruction would be easier. However, it is difficult to determine the weights without the ground truth. How to make effective rules of weight construction from input images is a key question. The innovation of this study can be summarized as three points: 1) We define a new regularizer wRS to diminish the gap between the reconstructed image and the desired one. Image gradient values are used as weights/indications to guide image reconstruction. They are updated as the image is updated. 2) We introduce two weights w_x and w_y to encode the image structure reliability along x- and y- directions. 3) We propose the wRS regularized image reconstruction model, and develop an effective algorithm to solve this model. This problem is non-convex, so it is difficult to directly solve. We use a surrogate function for the wRS, and reformulate it into the product of non-linear terms and quadratic terms. Then the problem can be solved alternately so as to obtain desired CT images. Experiments on digital phantoms and real CT data are reported to verify the effectiveness of the proposed method. Commonly used quantitative metrics are used to evaluate different methods, such as TV, ATV, ADM-L₀, and ARTV methods. Reconstruction result indicates that the image obtained by our method suffers from less artifacts and structure degradation, and is the closest to reference image.

Following the Introduction above in Section I, a new reconstruction model regularized by wRS and an algorithm used to solve this model are presented in Section II. Then, experiments on digital phantoms and real CT data are reported in Section III and Section IV. Finally, we discuss and conclude this work in Section V.

2 Methodology

2.1 Discretization of CT imaging model

Without loss of generality, we use circular fan-beam CT projection data for image reconstruction. Then CT imaging model can be approximated by a linear equation, $A f = p$ (1) where $A$ is the system matrix with a dimension of M×N, f =(f₁, f₂, ⋯ , f_N) ^T is the representation of the objective. Projection data p =(p₁, p₂, ⋯ , p_M) ^T is calculated by multiplying □□ and f according to the rule of matrix multiplication. CT reconstruction is to find image f via solving equation (1). In limited-angle CT, the scanning angular range is smaller than 180 degrees, that is to say, M < N. Therefore, solving Equation (1) is to reconstruct a high-dimensional CT image using low-dimensional projection data, which is an ill-posed inverse problem. CT images reconstructed by traditional algorithm, such as FBP and SART, often encounter obvious artifacts and structure degradation. As shown in Fig. 2, CT image reconstructed by SART (20 iterations) from projection data in the range of [45°, 135°] suffers from obvious artifacts and structure degradation. We endeavor to reconstruct artifact-free CT images without obvious structure degradation from limited-angle CT data.

Fig. 2

CT images reconstructed by SART (20 iterations) encounter obvious artifacts and structure degradation. (a) Shepp-Logan phantom; (b) image reconstructed from projections in the range of [45°, 135°]; (c) and (d) are their respective gradient image. The yellow line over image (c) and (d) means that image profiles along this line will be plotted in Fig. 3.

2.2 The proposed reconstruction model

In this subsection, we propose our reconstruction model for limited-angle CT reconstruction using image gradient information and their reliability along two orthogonal directions. Then we develop an effective algorithm to solve this model.

2.2.1 Image reconstruction model regularized by wRS

We know image structures can be displayed via image gradient magnitude with different values. The desired CT image often has clear structures, which means significant differences between structures and other information, such as artifacts. Therefore, it is reasonable to reestablish the natural differences between image structures and artifacts. In other words, structures with larger gradient magnitude need to be highlighted and the artifacts with smaller magnitude should be suppressed. In this way, we look forward to diminish the gap between the reconstructed image and the desired one. Based on the above analysis, we take the gradient magnitude of the reconstructed image as weights to recover image structures and suppress artifacts.

However, as shown in Fig. 3, the gradient magnitude in Fig. 3(a) seriously deviates from the reference one due to existing artifacts and structure degradation. As shown in Fig. 3(b), the gradient information of most structures is consistent with the reference one. Based on this observation, we infer that the gradient magnitudes of some structures are not completely credible due to artifacts or structure degradation. In the scanning mode shown in Fig. 1, the scanning angular range is symmetrical about the x-axis. Currently, the structure in horizontal direction (Fig. 3(a)) is not completely credible, while the structure information in vertical direction (Fig. 3(b)) is credible. Based on the above analysis, we define the weighted relative structure (wRS) as follows, $wRS (f) = \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{| D_{d} f_{i} |}{max (| D_{d} f_{i} |, ɛ)}$ (2) where operator $D$ is used for calculation of image differences and $D_{d} f_{i}$ represents the forward difference along direction d (d ∈ { x, y }) of image pixel f_i. Parameter ɛ is a small positive value, $| D_{d} f_{i} |$ represents the magnitude information of image partial derivative. And $1 / max (| D_{d} f_{i} |, ɛ)$ represents the weight/indication to diminish the gap between the reconstructed image and the desired one. When $| D_{d} f_{i} |$ is large, $1 / max (| D_{d} f_{i} |, ɛ)$ generates a small weight, then the image gradient will be preserved. When $| D_{d} f_{i} |$ is small, $1 / max (| D_{d} f_{i} |, ɛ)$ generates a large weight, then the image gradient will be smoothed. Parameter w_d is the confidence coefficient for image structure in x- or y-direction. Without loss of generality, we take CT scanning shown in Fig. 1 as an example for image reconstruction. Image structures in vertical direction are credible, so w_y is fixed to 1 in this work. Image structures in horizontal direction are not completely credible, then w_x (0 < w_x < 1) is to be determined in different experiments.

Fig. 3

Gradient image profiles along the horizontal direction (a) and the vertical direction (d).

Based on the analysis of the structure of limited-angle CT image, we define wRS to reduce the gap between the reconstructed image and the desired one. In this section, we propose the limited-angle CT reconstruction model regularized by wRS as follows, $f^{*} = \underset{f ⩾ 0}{arg min} | | A f - p | |_{2}^{2} + λ \cdot \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{| D_{d} f_{i} |}{max (| D_{d} f_{i} |, ɛ)}$ (3)

We call this model wRS model for short. The first term in optimization model (3) is the data fidelity term and the second term is the regularization term. In this paper, the confidence coefficient w_y is fixed to 1 and 0 < w_x < 1. The regularization intensity in x-direction is weak, and data fidelity constrain plays a major role. This model, in conjunction with an appropriate regularization parameter λ, has the potential to reconstruct a desired CT image.

2.2.2 Algorithm for wRS model

When weights $1 / max (| D_{d} f_{i} |, ɛ)$ in model (3) are fixed, wRS model degenerates into reweighted anisotropic total variation (RwATV) [20] model. For RwATV method, a convex problem is to be solved in each iteration. However, wRS model is nonconvex and nonlinear. It is difficult to find a decent solution. At this time, a surrogate function is used to reformulate wRS term into a product of non-linear terms and quadratic terms, $\begin{matrix} wRS (f) = \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{| D_{d} f_{i} |}{max (| D_{d} f_{i} |, ɛ)} \\ = \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{| D_{d} f_{i} | \cdot max (| D_{d} f_{i} |, ɛ)}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} \\ = \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{(D_{d} f_{i})^{2} + | D_{d} f_{i} | \cdot max (ɛ - | D_{d} f_{i} |, 0)}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} \end{matrix},$ (4)

When $ɛ < | D_{d} f_{i} |$ , wRS (f) can be formulated as follows, $wRS (f) = \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{(D_{d} f_{i})^{2}}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)},$ (5)

When $ɛ ⩾ | D_{d} f_{i} |$ , we have the following formulation, $\begin{matrix} \frac{(D_{d} f_{i})^{2} + | D_{d} f_{i} | \cdot max (ɛ - | D_{d} f_{i} |, 0)}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} = \frac{(D_{d} f_{i})^{2} + | D_{d} f_{i} | \cdot (ɛ - | D_{d} f_{i} |)}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} \\ = \frac{| D_{d} f_{i} | \cdot ɛ}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} \\ ⩾ \frac{(D_{d} f_{i})^{2}}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)} \end{matrix},$ (6)

The difference between the left and right terms of inequality (6) is $\frac{| D_{d} f_{i} | \cdot (ɛ - | D_{d} f_{i} |)}{max (| D_{d} f_{i} |, ɛ) \cdot ɛ}$ $(ɛ ⩾ | D_{d} f_{i} |)$ . Considering this situation of $ɛ ⩾ | D_{d} f_{i} |$ , we have $ɛ^{2} + 4 | D_{d} f_{i} |^{2} ⩾ 4 | D_{d} f_{i} | \cdot ɛ$ and $ɛ^{2} \cdot max (| D_{d} f_{i} |, ɛ) ⩾ 4 | D_{d} f_{i} | \cdot (ɛ - | D_{d} f_{i} |) \cdot max (| D_{d} f_{i} |, ɛ)$ . Thus, the following inequality holds: $\frac{| D_{d} f_{i} | \cdot (ɛ - | D_{d} f_{i} |)}{max (| D_{d} f_{i} |, ɛ) \cdot ɛ} ⩽ \frac{ɛ}{4 max (| D_{d} f_{i} |, ɛ)}$ . We can see that the largest difference the left and right terms of inequality (6) is 1/4. Based on this result, we employ $\frac{(D_{d} f)^{2}}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)}$ term as the surrogate function of wRS(f). For short, we call it wRS surrogate (wRSS). We plot the shapes of function wRS(f) and wRSS(f), and their difference image in Fig. 4. We can see that their difference mainly occurs in relatively flat areas, and the largest difference is 0.375.

Fig. 4

Difference between wRS and wRSS: (a) the shape of wRS: $\frac{| u |}{max (| u |, 0.1)} + \frac{1}{2} \frac{| v |}{max (| v |, 0.1)}$ ; (b) the shape of wRSS: $\frac{| u |^{2}}{max (| u |, 0.1) \cdot max (| u |, 0.1)} + \frac{1}{2} \frac{| v |^{2}}{max (| v |, 0.1) \cdot max (| v |, 0.1)}$ ; (c) the difference between wRS and wRSS. The largest difference value is $\frac{1}{4} + \frac{1}{2} \cdot \frac{1}{4} = 0.375$ .

Then the wRS model can be transformed as follows, $f^{*} = \underset{f ⩾ 0}{arg min} | | A f - p | |_{2}^{2} + λ \cdot \sum_{i = 1}^{N} \sum_{d \in {x, y}} w_{d} \cdot \frac{(D_{d} f)^{2}}{max (| D_{d} f_{i} |, ɛ) \cdot max (| D_{d} f_{i} |, ɛ)}$ (7)

We can see that wRSS consists of non-linear term $1 / max (| D_{d} f_{i} |, ɛ)$ and one quadratic term $(D_{d} f)^{2}$ . Then, wRS model can be reformulated as follows, $f^{*} = \underset{f ⩾ 0}{arg min} | | A f - p | |_{2}^{2} + λ \cdot (w_{x} \cdot f^{T} D_{x}^{T} Q_{x} Q_{x} D_{x} f + w_{y} \cdot f^{T} D_{y}^{T} Q_{y} Q_{y} D_{y} f)$ (8) where the symbol “T” represents transposition operation, $D_{x}$ and D_y are two Toeplitz matrices corresponding to difference operator $D_{x}$ and $D_{y}$ . Q_x and Q_y are two diagonal matrices and their nonzero entries are $1 / max (| D_{x} f_{i} |, ɛ)$ and $1 / max (| D_{y} f_{i} |, ɛ)$ , respectively.

Now we develop an algorithm to solve wRS model (8). Theoretically, we can calculate the derivative of the objective function, and then find the function that makes it zero so as to obtain the solution. In this way, we have $[A^{T} A + λ \cdot (w_{x} \cdot D_{x}^{T} Q_{x} Q_{x} D_{x} + w_{y} \cdot D_{y}^{T} Q_{y} Q_{y} D_{y})] \cdot f = A^{T} p .$ (9)

Then image f can be obtained via following formulation, $f = [A^{T} A + λ \cdot (w_{x} \cdot D_{x}^{T} Q_{x} Q_{x} D_{x} + w_{y} \cdot D_{y}^{T} Q_{y} Q_{y} D_{y})]^{- 1} \cdot A^{T} p .$ (10)

Let $L = w_{x} \cdot D_{x}^{T} Q_{x} Q_{x} D_{x} + w_{y} \cdot D_{y}^{T} Q_{y} Q_{y} D_{y}$ . Computing the inverse matrix of $[A^{T} A + λ \cdot L]$ is difficult. The system matrix $A$ has the dimension of M × N. For a real CT scan, we assume that the detector has one thousand units and acquires projection data at an interval of 0.5 degrees in a 120°-rotation, so Mis in the order of 10⁵. For image f of 512×512 pixels that is to be reconstructed, N is in the order of 10⁵. Matrix $A$ ^T $A$ has a huge dimension. To avoid calculating this inverse matrix, we prefer to iteratively solve this problem.

First, we introduce a variable u such that f = u, then problem (10) turns into problem (11), $min_{f ⩾ 0, u, b} | | A f - p | |_{2}^{2} + λ \cdot (w_{x} \cdot u^{T} D_{x}^{T} Q_{x} Q_{x} D_{x} u + w_{y} \cdot u^{T} D_{y}^{T} Q_{y} Q_{y} D_{y} u) + η \cdot | | f - u - b | |_{2}^{2} .$ (11) where η is a parameter and b represents the error between u and f. Vector f in the data fidelity term and the regularization term of (10) are separable. We can solve this problem in an alternative way. And problem (11) is decomposed into three sub-problems as follows, ${\begin{matrix} f^{k + 1} = \underset{f ⩾ 0}{arg min} | | A f - p | |_{2}^{2} + η \cdot | | f - (u + b) | |_{2}^{2} \\ u^{k + 1} = \underset{u}{arg min} (λ / η) \cdot (w_{x} \cdot u^{T} D_{x}^{T} Q_{x} Q_{x} D_{x} u + w_{y} \cdot u^{T} D_{y}^{T} Q_{y} Q_{y} D_{y} u) + | | u - (f - b) | |_{2}^{2} \\ b^{k + 1} = b^{k} - (f^{k + 1} - u^{k + 1}) \end{matrix}$ (12)

Sub-problem 1 has two terms. SART is used to minimize the first term and generates an intermediate image f^k+1/-2, and then sub-problem 1 turns into the following problem, $f^{k + 1} = \underset{f ⩾ 0}{arg min} | | f - f^{k + 1 / 2} | |_{2}^{2} + η \cdot | | f - (u + b) | |_{2}^{2}$ (13)

Its closed form solution is as follows, $f^{k + 1} = \frac{f^{k + 1 / 2} + η \cdot (u + b)}{1 + η} .$ (14)

Now we turn to solving sub-problem 2. We calculate its derivative and let it be zero, then we have $[I + (λ / η) \cdot (w_{x} \cdot D_{x}^{T} Q_{x} Q_{x} D_{x} + w_{y} \cdot D_{y}^{T} Q_{y} Q_{y} D_{y})] \cdot u = f - b .$ (15)

Let $L = w_{x} \cdot D_{x}^{T} Q_{x} Q_{x} D_{x} + w_{y} \cdot D_{y}^{T} Q_{y} Q_{y} D_{y}$ , we know [I + (λ/ - η) · L] is a symmetric positive definite Laplacian matrix. Some algorithms [22 , 27–29] can be used to solve this equation iteratively.

Finally, we turn to updating the error term b via the following rule, $b^{k + 1} = b^{k} - (f^{k + 1} - u^{k + 1}) .$ (16)

Details of this algorithm are listed in Algorithm 1.

Algorithm 1: wRS
Input:f, u, b, p, w_x, w_y, λ, ɛ, η
fork from 1 to N, do
Update f^k+1/-2 via SART algorithm.
Update f^k+1 via Equation (14).
Update u^k+1 via solving Equation (15).
Update b^k+1 via Equation (16).
end
output:f^N.

2.3 Comparisons of wRS and related works

In this section, we compare the new method with some previous methods: TV, ATV, ADM-L₀, and ARTV. TV is defined as the L₁ norm of image gradients. TV method delivers equal punishments on all image gradients to enhance image sparsity. The wRS is defined based on the weights constructed by image gradients, and is closer to L₀ norm. Besides, it makes use of the structure reliability prior by introducing weights w_x and w_y. Anisotropic total variation (ATV) term integrates structure reliability into reconstruction model for the first time. It is defined based on TV, and combines the merits of image sparsity and structure reliability prior. However, ATV measure is still heavily dependent on image gradient magnitudes. The wRS is similar to the weighted anisotropic total variation except that the weights are constructed by image gradients, and are dynamically updated. ADM-L₀ uses image gradient L₀ norm minimization to enhance structure sparsity. However, it does not integrate the image structure reliability into reconstruction model. The wRS method benefits from the use of the structure reliability prior, is better in alleviating structure degradation. Anisotropic relative total variation (ARTV) term is defined based on the local statistical information of gradient image and it integrates the structure reliability prior. For ARTV term, the windowed weighting strategy is suitable for image with repetitive textures. However, this method may smooth out strong but relatively dense structures [30]. We can find that wRS term is similar to ARTV term, however, they are different: ARTV is defined on a local window, while wRS is defined on the pixel level. So ARTV is a local method and wRS is a global one. It is worth mentioning that ARTV turns into wRS when facing images without repetitive patterns.

3 Experiments

In this section, we first validate the effectiveness of our proposed model and the developed algorithm via experiments on two digital phantoms (Fig. 5). The two phantoms consist of 256×256 pixels. The scanning parameters are shown in Table 1. Poisson and Gaussian noises are added to noise-free projections. The incident photon number is 10⁵. The mean value and variance of Gaussian noise is 0 and 0.1% ×||p||_∞, respectively. Then, experiments on real CT data of a carved cheese [31] and a walnut are reported to prove the feasibility of the proposed method in practical problems. Some classical limited-angle CT reconstruction methods, such as TV, ATV, ADM-L₀ and ARTV, are used for comparison. Unless otherwise stated, the number of iterations of all methods is 2000. For numerical simulation experiments, we mainly use root mean squared error (RMSE), peak signal to noise ratio (PSNR) and structure similarity index measure (SSIM) [32] to evaluate different methods. For real data experiments, we mainly use close-ups of selected regions and residual images for evaluation.

Fig. 5

(a) FORBILD head phantom, (b) NCAT phantom.

Table 1

Scanning parameters in simulation experiments

Parameter	Values
Distance from source to rotation axis (mm)	400
Distance from source to detector (mm)	800
Detector units	512
Length of detector (mm)	512
Size of reconstructed image	256×256
Angular increment (degree)	1

3.1 Reconstruction of the FORBILD head phantom and NCAT phantom

We first report the reconstructions of the FORBILD head phantom from 120°-data. The reconstruction parameters are listed in Table 2. The scanning angular range is symmetrical about the coordinate axis. In an actual CT scanning, the angular range may not be symmetrical about one coordinate axis, but we can always establish an appropriate coordinate system to make it meet our requirements. Without loss of generality, the used angular range is [30°, 150°] and it is symmetrical about the x-axis in this section. The number of iterations of all methods is 2000 in this section. The images reconstructed by different methods and their residual images are shown in Fig. 6 and Fig. 7, respectively. Images acquired by competing methods encounter obvious artifacts, which can be observed in their residual images. The wRS method removes more artifacts. And its reconstruction is the closest to the reference image (Fig. 6(a)) compared to other methods. Its residual image in Fig. 7(f) confirms this result. In addition, image profiles along the yellow solid line indicated in Fig. 6(a) are plotted in Fig. 8 for all reconstructions. We can observe that the wRS-reconstruction profile is the closest to the truth profile. The wRS method reduces more shading artifacts. When comparing these images with the FORBILD head phantom, we find that image structures with low degree of confidence are better recovered by wRS method. In addition, RMSE, PSNR, and SSIM of the reconstructed images are listed in Table 3, which shows that wRS method performs the best.

Table 2
Reconstruction parameters in experiments on FORBILD head phantom using 120°-data

Methods TV ATV ADM-L₀ ARTV wRS

Parameters λ= 0.1 K_TV = 20 λ= 0.2, K_ATV = 20, w_x = 0.01, w_y = 1 λ= 0.003, κ= 3, η= 0.1 λ= 0.001, σ= 1, ς= 0.001, K_ARTV = 6, w_x = 0.1, w_y = 1 λ= 0.2, K_wRS = 6, η= 0.03, ɛ= 0.005, w_x = 0.015, w_y = 1

Methods	TV	ATV	ADM-L₀	ARTV	wRS
Parameters	λ= 0.1 K_TV = 20	λ= 0.2, K_ATV = 20, w_x = 0.01, w_y = 1	λ= 0.003, κ= 3, η= 0.1	λ= 0.001, σ= 1, ς= 0.001, K_ARTV = 6, w_x = 0.1, w_y = 1	λ= 0.2, K_wRS = 6, η= 0.03, ɛ= 0.005, w_x = 0.015, w_y = 1

Fig. 6

Reconstructions of FORBILD head phantom from 120°-data. From left to right and top to bottom, the images are reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS; where (a) is the reference image. The yellow solid line over image (a) means that image profiles along this line will be plotted in Fig. 8.

Fig. 7

The residual images of those images in Fig. 6.

Fig. 8

Profiles of the images shown in Fig. 6 along the row indicated by the yellow solid line over Fig. 6(a), and the wRS-reconstruction profile is the closest to the truth profiles.

Then, projection data in [45°, 135°], symmetrical about the x-axis, is used for CT reconstruction. The reconstructions and their close-ups are shown in Fig. 9 and Fig. 10, respectively. The wRS method removes more artifacts than other competing ones, and image structures are well preserved. Also, their close-ups confirm this result. In addition, RMSE, PSNR, and SSIM of the reconstructed images are listed in Table 3, which shows that wRS method performs the best compared to other ones.

Fig. 9

Reconstructions of FORBILD head phantom from 90°-data. From left to right and top to bottom, the images are reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS; where (a) is the reference image. The yellow line over image (a) means that image profiles along this line will be plotted in following sections.

Fig. 10

The close-ups of images in Fig. 9.

Table 3

RMSE, PSNR, and SSIM of reconstruction images from 120°- and 90°-data of FORBILD head phantom

	120 degrees					90 degrees
	TV	ATV	ADM-L₀	ARTV	wRS	TV	ATV	ADM-L₀	ARTV	wRS
RMSE	0.1025	0.0567	0.0808	0.0632	0.0039	0.1643	0.0962	0.0708	0.0248	0.0120
PSNR	24.89	30.04	26.95	29.09	53.20	20.79	25.44	28.10	37.23	43.52
SSIM	0.9272	0.9620	0.9459	0.9603	0.9975	0.8412	0.8890	0.9451	0.9935	0.9961

We then report the reconstructions of the NCAT phantom from 120°-data, and the scanning angular [30°, 150°] is symmetrical about the x-axis. The images reconstructed by different methods and their residual images are shown in Fig. 11 and Fig. 12, respectively. Images acquired by the competing methods encounter obvious artifacts and the wRS method removes more artifacts. When comparing their residual images, we find that wRS-reconstruction is the closest to the reference image (Fig. 11(a)). In addition, image profiles along the yellow line indicated in Fig. 11(a) are plotted in Fig. 13 for all reconstructions. We can see the wRS-reconstruction profile is the closest to the truth profiles, which confirms that wRS method reduces more artifacts and image structures with low degree of confidence are better recovered than other methods as compared to the phantom itself. In addition, RMSE, PSNR, and SSIM of the reconstructed images are listed in Table 4, which shows that wRS method performs the best compared to other ones.

Fig. 11

Reconstructions of NCAT phantom from 120°-data. From left to right and top to bottom, the images are reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS; where (a) is the reference image. The yellow line over image (a) means that image profiles along this line will be plotted in following sections.

Fig. 12

The residual images of those images in Fig. 11.

Fig. 13

Profiles of the images shown in Fig. 11 along the row indicated by the yellow solid line over Fig. 11(a), the wRS-reconstruction profile is the closest to the truth profiles.

Table 4

RMSE, PSNR, and SSIM of reconstruction images from 120°-data of NCAT phantom

	TV	ATV	ADM-L₀	ARTV	wRS
RMSE	0.0329	0.0349	0.0128	0.0075	0.0063
PSNR	29.65	29.14	37.88	42.52	43.95
SSIM	0.9617	0.9528	0.9774	0.9968	0.9970

From the reconstruction results, we know wRS method can obviously reduce the shading artifacts and preserve image structures. We analyze the reasons as follows. First, the wRS makes full use of image sparsity; it is closer to L₀ norm than L₁ norm. Second, wRS introduce parameter w_d to encode the degree of confidence for image structure along x- or y-direction, which is conductive to removing shading artifacts.

The wRS model is non-convex, different initialization may generate different images. Here we take the CT image reconstructed by SART (20 iterations), average image [33] and a random image as the initial image of Algorithm 1. For the average image, the pixel value is calculated by averaging the sum of measured projection values. The reconstructed images are shown in Fig. 14. Three typical initializations do not generate obvious difference. We conclude that the wRS model is insensitivity to initial images. We can get the same result by observing their RMSE curve as shown in Fig. 15. Therefore, in this work, image reconstructed by SART (20 iterations) is used as the initial image.

Fig. 14

Impact of initial image: (a), (b) and (c) are respectively initialized by average image, random image and image reconstructed by SART (20 iterations).

Fig. 15

RMSE curves in the experiments with different initial images.

3.2 Reconstruction of real CT projections

In order to see whether the wRS method can suppress shading artifacts in real limited-angle CT reconstruction, we carried out experiments on real CT data of a carved cheese [31] and a walnut. The carved cheese data is open and is available at http://www.fips.fi/dataset.php. The walnut projection is collected by a micro-CT system and supplied by Chongqing University. The geometry of the measurement setup in the carved cheese experiments is given in the reference [31]. The exposure time is 1000 ms. The X-ray tube acceleration voltage is 40 kV and tube current 1 mA. The geometry of the measurement setup in the walnut experiments is listed in Table 5. Their reference images are shown in Fig. 16. The cheese was been carved with two letters “CT” as shown in Fig. 16(a). It can simulate different materials including calcium-containing organic compounds and air. Besides, it has special structures; the shape of the carved letters is suitable for testing limited-angle tomography. Walnut structure is approximately symmetrical, like a skull, as shown in Fig. 16(b). The flesh part contains various non-convex structures; besides, the dense shell, flesh and inner wood membrane form different contrasts. So we choose these two materials for limited-angle CT reconstruction. We can see that the shape of the carved cheese is close to a circle. Therefore, without loss of generality, the scanning angular range in the experiments on the carved cheese is set to [30°, 150°], that is symmetrical about the x-axis. The reconstruction parameters are listed in Table 6. CT images reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS methods from the 120°-data are shown in Fig. 17.

Table 5
Scanning parameters in the walnut experiments

Parameter Values

Distance from source to rotation axis (mm) 280.7

Distance from source to detector (mm) 654.5

Detector units 512

Length of detector (mm) 114.89

Size of reconstructed image 384×384

Angular increment (degree) 0.7031

Parameter	Values
Distance from source to rotation axis (mm)	280.7
Distance from source to detector (mm)	654.5
Detector units	512
Length of detector (mm)	114.89
Size of reconstructed image	384×384
Angular increment (degree)	0.7031

Table 6

Parameters in experiments on the carved cheese

Methods	TV	ATV	ADM-L₀	ARTV	wRS
Parameters	λ= 0.1 K_TV = 20	λ= 0.15, K_ATV = 20, w_x = 0.01, w_y = 1	λ= 0.000015, κ= 4, η= 0.2	λ= 0.000003, σ= 2, ς= 0.001, K_ARTV = 6, w_x = 0.3, w_y = 1	λ= 0.001, K_wRS = 6, η= 0.015, ɛ= 0.0005, w_x = 0.05, w_y = 1

Fig. 16

Reference CT images of a carved cheese (a) and a walnut (b).

Fig. 17

Images of the carved cheese from 120°-data. From left to right and top to bottom, the images are reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS; (a) is the reference image. The display window is [0, 0.05].

In order to observe the differences of these images, we show the close-ups of their respective residual images in Fig. 18. The reconstructed images deliver us a general understanding of the performance of different methods. TV- and ATV-reconstruction images are prone to obvious artifacts and structure blur. ADM-L₀ is a competitive method, as no obvious artifacts and blur appear in the reconstructed image as shown in Fig. 17(d), which directly reflects the effectiveness of image sparsity prior. ARTV-reconstruction image encounters obvious fake or wrong information (Fig. 17(e)). Finally, we see that wRS-reconstruction image resembles that of ADM-L₀. When comparing the close-ups of their residual images, we see that fewer artifacts appear in the residual image of wRS method (Fig. 18(f)). The residual images of TV- and ATV-reconstruction images confirm that TV and ATV method are not good at suppressing artifacts in limited-angle CT. And ARTV performs not well. In summary, the proposed wRS method well suppresses artifacts and preserves image structures.

Fig. 18

The close-ups of the residual images of images in Fig. 17. The display window is [–0.02, 0.02].

In addition to the experiments on the carved cheese data, we carried out experiments on a walnut. In the experiment on walnut, the scanning angular range is [30°, 150°], that is symmetrical about the x-axis. The reconstruction parameters are listed in Table 7. The CT images reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS methods are shown in Fig. 19. And the close-ups of their residual images are listed in Fig. 20. Obvious artifacts and structure blur can be found in TV- and ATV-reconstruction images, and they also remain in their residual images. When observing ADM-L₀-, ARTV- and wRS-reconstruction results, we find that there are no obvious artifacts in their reconstructed images. That is to say, these methods can significantly reduce artifacts. When comparing their residual images, we observe that ADM-L₀- and wRS-reconstruction image structures almost coincide with the reference image. ARTV-reconstruction image encounters fake or wrong information, as shown in Fig 19(e) and Fig. 20(e). When considering artifacts, we observe that there are the least artifacts in the image reconstructed by wRS method and its residual image. This result confirms that wRS method reduces more artifacts, and image structures with low degree of confidence are better recovered than other methods as compared to the reference image.

Table 7

Reconstruction parameters in experiments on the walnut

Methods	TV	ATV	ADM-L₀	ARTV	wRS
Parameters	λ= 0.055 K_TV = 20	λ= 0.1, K_ATV = 20, w_x = 0.02, w_y = 1	λ= 0.00005, κ= 2, η= 0.05	λ= 0.000005, σ= 2, ς= 0.00002, K_ARTV = 5, w_x = 0.2, w_y = 1	λ= 0.00015, K_wRS = 6, η= 0.02, ɛ= 0.0004, w_x = 0.3, w_y = 1

Fig. 19

Images of the walnut from 120°-data. From left to right and top to bottom, the images are reconstructed by TV, ATV, ADM-L₀, ARTV, and wRS; (a) is the reference image. The display window is [0, 0.04].

Fig. 20

The residual images of those images in Fig. 19. The display window is [–0.005, 0.005].

Although there is no ground truth in real CT applications, the image reconstructed by SART (20 iterations) is used as the reference image. Then RMSE, PSNR, and SSIM of the CT images in the real CT experiments are listed in Table 8. The best results are highlighted in bold. ADM-L₀ and wRS methods outperform other method in terms of these quantitative data. This result can also be confirmed by graphic comparisons from Fig. 17 and Fig. 19. When comparing ADM-L₀ and wRS, quantitative data indicates that they acquire images of similar quality, though ADM-L₀ sometimes perform better in terms of some index. Graphic comparisons show that wRS method acquires CT image that is visually closer to the reference image and the fewest artifacts remain in its residual image. ADM-L₀ recovers most image structures; however, some shading artifacts are not well suppressed, as shown in Fig. 18(d) and Fig. 20(d). Although ADM-L₀ can reconstruct high-quality CT images, the wRS method is a competitive method.

Table 8

RMSE, PSNR, and SSIM of CT images in real CT experiments

Target	Measure	TV	ATV	ADM-L₀	ARTV	wRS
	RMSE	0.0029	0.0037	0.0026	0.0030	0.0025
Cheese	PSNR	26.45	24.49	27.34	26.30	27.97
	SSIM	0.5967	0.5841	0.6234	0.6339	0.6323
	RMSE	0.0027	0.0029	0.0023	0.0026	0.0022
Walnut	PSNR	24.34	23.57	25.77	24.51	26.15
	SSIM	0.6354	0.6151	0.6572	0.6186	0.6520

For the residual images in real CT experiments, we list their RMSE, entropy and variance in Table 9 to analyze their differences. The reference image for RMSE calculation is the ideal residual image, that is, all pixel values are zero. The entropy of image has been used to measure motion artifacts in magnetic resonance imaging [34] and to measure geometric artifacts in cone-beam CT [35]. We here use it to measure residual artifacts and the smaller the value, the better the image quality. The ideal entropy value is zero. Ideal residual image is gray (Fig. 20(a)) under a narrow display window, not ones with large white areas (for example areas indicated by arrows in Fig. 20(b), (c), and (d)). We prefer the gray residual image that has a smaller variance. From the quantitative data in Table 9, we know that the wRS method achieves the best or the second best performance in terms of all indices. Combined with the results of graphic comparison, we obtain the same result as that of Table 8.

Table 9

RMSE, entropy and variance of residual images corresponding to images in Fig. 18 and Fig. 20

Target	Measure	TV	ATV	ADM-L₀	ARTV	wRS
Cheese	RMSE	0.0029	0.0037	0.0026	0.0030	0.0025
	Entropy	0.6554	0.8228	0.5362	0.7018	0.5945
	Variance(10^- 5)	0.860	1.309	0.701	0.890	0.606
Walnut	RMSE	0.0027	0.0029	0.0023	0.0026	0.0022
	Entropy	0.5777	0.6525	0.4362	0.5973	0.4660
	Variance(10^- 5)	0.683	0.821	0.486	0.658	0.442

4 Discussions and conclusion

CT reconstruction from limited-angle projections is a challenging problem. The reconstructed images often encounter shading artifact and structure degradation. In this paper, we establish a new reconstruction model based on the wRS to reduce artifacts and preserve structures. However, the objective of wRS model is nonconvex and nonlinear. It is difficult to directly solve this problem. We use a surrogate function to develop an efficient algorithm and compare wRS method with TV, ATV, ADM-L₀, and ARTV methods. Experiments show that the wRS model has strong ability to suppress artifacts and preserve structures. The main reasons can be summarized as three points: 1) The defined wRS term is close to L₀ norm, which is conductive to suppressing shading artifacts. 2) Introduced parameters w_x and w_y indicate image structure reliability along x- and y- direction and can adjust the intensity of the regularization. 3) We develop an efficient algorithm to solve the wRS model. Although wRS method outperforms some methods, there are still some limitations need further consideration.

First, as shown in Algorithm 1, there are multiple parameters to be determined, w_x, w_y, λ, ɛ, and η. For w_x and w_y, we fix one parameter to 1, and then adjust only another one. In simulation experiments, the parameters can be determined according to quantitative data of their respective reconstructed images. For example, in the experiments on the FORBILD head phantom using 90°-data, we study the effect of each parameter with other parameters fixed. We study the effect of parameters λ, η, ɛ, and w_x_. The RMSE and SSIM of images reconstructed using respective parameters are plotted in Fig. 21. We can see that the image quality is not very sensitive to large λ and w_x. However, we need to carefully determine a suitable value that can render us a decent image. Parameters η and ɛ obviously affect image quality. The former is introduced along with the last term in problem (11), so a large value would deliver a smooth image. The latter is introduced as a small constant to avoid zero denominators. A large value may widen the gap between wRS term and its surrogate function. A smaller value may deliver smooth images because of instability and unbalance. There is no ground truth in real CT applications. Parameter selection may depend on the trial and error method, which may lead to unrealistic reconstructions. One of the main disadvantages of this method is that these parameters cannot be determined adaptively. That’s naturally going to be one of the main topics in our next study.

Fig. 21

The behaviors of parameter λ, η, ɛ, and w_x._.

Second, experiments on digital phantoms and real CT data prove that wRS method can effectively suppress shading artifacts and preserve image structures. However, in the real data experiments, CT image is reconstructed from 120°-data in this work. When the scanning angular range is smaller than 120° (for example 90° or smaller), the capability of this method will be significantly weakened. In our opinion, in addition to the prior of sparsity and the degree of confidence of image structures, more useful information should be used to improve the performance of our method. For example, when a prior image is available, relatively complete structures can be used as wise weights to indicate which to suppress and which to preserve, image reconstruction would be easier.

Third, in the experiments on the FORBILD head phantom using 90°-data, the runtime (in sec) per iteration of TV, ATV, ADM-L₀, ARTV and wRS method are about 0.3407, 0.3315 s, 0.3460, 0.5142, and 0.4899. The time of SART is about 0.3106. The time consumption of wRS method is slightly larger than TV, ATV, and ADM-L₀. Except for the time of SART, almost all the time is spent on solving linear Equation (15). The ARTV method also spends most of the time on solving linear equation. We can see that the runtime of ARTV approximates to that of the wRS method. With the development of parallel computing power, GPU implementation of the wRS method can make practical applicationpossible.

In conclusion, we take image gradient magnitudes as weights and introduce constrains on image’s partial derivatives along two orthogonal directions to define wRS. And we propose a reconstruction model regularized by the wRS. Then a surrogate function is used to approximately solve this model. The reconstruction results and quantitative data in simulation experiments, the images and residual images in experiments on real CT data show that wRS method can effectively suppress shading artifacts and preserve image structure.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61701174), General Project of Chongqing Natural Science Foundation (cstc2021jcyj-msxmX0679), Science and Technology Research Program of Chongqing Education Commission of China (KJQN202000808), and Scientific Research Foundation of Chongqing Technology and Business University (2056023).

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Structure-guided computed tomography reconstruction from limited-angle projections

Abstract

Keywords

1 Introduction

2.1 Discretization of CT imaging model

2.2.1 Image reconstruction model regularized by wRS

3 Experiments

Table 5 Scanning parameters in the walnut experiments Parameter Values Distance from source to rotation axis (mm) 280.7 Distance from source to detector (mm) 654.5 Detector units 512 Length of detector (mm) 114.89 Size of reconstructed image 384×384 Angular increment (degree) 0.7031

Footnotes

Acknowledgments

References

Table 5
Scanning parameters in the walnut experiments

Parameter Values

Distance from source to rotation axis (mm) 280.7

Distance from source to detector (mm) 654.5

Detector units 512

Length of detector (mm) 114.89

Size of reconstructed image 384×384

Angular increment (degree) 0.7031