Projection domain decomposition denoising algorithm based on low rank and similarity-based regularization

Abstract

BACKGROUND:

Projection Domain Decomposition (PDD) is a dual energy reconstruction method which implements the decomposition process before image reconstruction. The advantage of PDD is that it can alleviate beam hardening artifacts and metal artifacts effectively as energy spectra estimation is considered in PDD. However, noise amplification occurs during the decomposition process, which significantly impacts the accuracy of effective atomic number and electron density. Therefore, effective noise reduction techniques are required in PDD.

OBJECTIVE:

This study aims to develop a new algorithm capable of minimizing noise while simultaneously preserving edges and fine details.

METHODS:

In this study, a denoising algorithm based on low rank and similarity-based regularization (LRSBR) is presented. This algorithm incorporates the low-rank characteristic of tensors into similarity-based regularization (SBR) framework. This method effectively addresses the issue of instability in edge pixels within the SBR algorithm and enhances the structural consistency of dual-energy images.

RESULTS:

A series of simulation and practical experiments were conducted on a dual-layer dual-energy CT system. Experiments demonstrate that the proposed method outperforms exiting noise removal methods in Peak Signal-to-noise Ratio (PSNR), Root Mean Square Error (RMSE), and Structural Similarity (SSIM). Meanwhile, there has been a notable enhancement in the visual quality of CT images.

CONCLUSIONS:

The proposed algorithm has a significantly improved noise reduction compared to other competing approach in dual-energy CT. Meanwhile, the LRSBR method exhibits outstanding performance in preserving edges and fine structures, making it practical for PDD applications.

Keywords

Dual-energy computed tomography projection domain decomposition Low rank similarity-based regularization

1 Introduction

Dual-energy computed tomography (DECT), as a special type of spectral CT, can reconstruct images of effective atomic number and electron density, thus enhancing the ability to distinguish different substances [1 –3]. Traditional DECT reconstruction methods can be classified into three categories: projection domain decomposition (PDD) [4], image domain decomposition (IDD) [5], and iterative decomposition [6]. PDD imaging utilizes the difference in attenuation between high-energy and low-energy data to achieve separation of material and energy-related information [1]. Compared to IDD and iterative decomposition, the basis material images obtained through PDD typically exhibit a reduced occurrence of beam hardening artifacts. However, the decomposition process is accompanied by noticeable noise amplification [7, 8]. The image denoising in PDD should be performed after decomposition, as the decomposition amplification effect of DECT noise has a significant impact on the basis material image. Therefore, the development of an advanced denoising method is crucial for enhancing the performance of DECT images.

In spectral CT, some algorithms employ single-energy CT image denoising methods to deal each image independently within the energy windows. For instance, Xu et al. incorporated total variation (TV) [9] and dictionary learning (DL) [10] as regularization terms in the iterative reconstruction model for spectral CT. However, due to the excessive noise in spectral CT images, single-energy CT algorithms often fail to achieve satisfactory denoising performance. Since the projection data is collected from the same object across various energy channels in spectral CT, the reconstructed images exhibit identical spatial structures, indicating structural correlation. Therefore, when developing denoising algorithms for multi-energy CT, it is essential to consider the influence of structural correlation.

The utilization of structural correlation is commonly embraced in image denoising and aritfacts removal for spectral CT. Spectral CT denoising algorithms based on structural correlation can be classified into two distinct categories. The first category combines spectral CT images into tensors, thereby utilizing on low-rank properties to establish structural correlation. Gao et al. introduced the Prior Rank, Strength, and Sparsity Model (PRISM) [11], which incorporates low rank and sparsity into spectral CT reconstruction. Zhang et al. extended the DL algorithm to tensor dictionary learning (TDL) [12] through Tucker decomposition and CP decomposition, achieving low-rank and sparse representation of tensor images. Wu et al. proposed the TDL-L0 algorithm [13] by introducing L0-norm constraints in the gradient image domain, giving more emphasis to spatial sparsity and addressing the edge-preserving limitations of the TDL algorithm. The second category consists of denoising algorithms that rely on structural priors. Yu et al. introduced the Priori Image Constrained Compressed Sensing (PICCS) algorithm [14], which employs the full spectrum image processed by the filtered back projection (FBP) algorithm as a priori image. However, denoising results based on compressed sensing often exhibit oversmoothing and blocky artifacts. To mitigate this issue, Kong et al. proposed the PICCS-DL algorithm [15], combining the PICCS algorithm with dictionary learning (DL) to further enhance the sparsity characteristics of spectral CT images. Liu et al. introduced a Total Image Constrained Material Reconstruction (TICMR) algorithm [16] based on non-local total variation (NLTV). This algorithm utilizes these features of the total image to improve the accuracy of material decomposition. Harms et al. developed a denoising algorithm for DECT based on Similarity-Based Regularization (SBR) [17]. This algorithm proposes a regularization term that minimizes the difference between images without and with noise suppression through similarity matrix multiplication. The structural information of the SBR algorithm depends on the edge information of the prior image. Therefore, the edge blurring effect in denoising of basis material images is avoided. However, the above algorithms couldn’t be used in DECT directily. Compared to spectral CT, DECT lacks a sufficient number of energy window images to form a tensor. The PICCS algorithm exhibits some issues, such as blocky artifacts and blurred details. In both the TICMR and SBR algorithms, the absence of similar pixels among edge pixels leads to an excessive concentration of non-local mean weights. The low-rank property of tensors assists in preserving the edge structure of images. Therefore, we incorporate a low-rank condition into the SBR algorithm and propose a novel denoising algorithm.

To exploit the structural correlation in the basis material images of DECT, we propose the Penalty Weighted Least Squares algorithm based on Low Rank and Similarity-Based Regularization (PWLS-LRSBR). In contrast to alternative algorithms that rely on the reconstruction image of projected data as a prior [17 –19], our approach employs decomposition overlay images as structural priors to avoid structural errors stemming from beam hardening artifacts. In the initial step, this method derives similarity weights from the decomposition overlay image and combines it with the basis material images into a tensor. Subsequently, the iterative model, based on the Penalty Weighted Least Squares algorithm, employs similarity-based regularization for denoising and integrates low-rank attribute to enhance the structural stability of edges. Furthermore, we adopt an alternating optimization algorithm to solve the objective function. The proposed algorithm offers two distinct advantages. Firstly, the decomposition overlay prior minimizes artifacts while maintaining a high signal-to-noise ratio, thereby preserving the structural features of the reconstructed images. Secondly, it effectively utilizes the structural characteristics of prior images through similarity-based regularization and low-rank properties, resulting in noise reduction while retaining edge and structural information.

The organization of the rest of this paper is as follows. Section 2 introduces the PWLS-LRSBR method, which is based on the decomposition overlay image. This section includes the DECT noise model, SBR model, PWLS-LRSBR model, and the solving algorithm. Section 3 presents the validation of the PWLS-LRSBR method in comparison with the competing algorithms, using both simulation and experimental data. Finally, Sect. 4 includes discussions and conclusions.

2 Method

2.1 DECT noise model

In order to analyze the image noise in the decomposition domain, it is usually assumed that the projection value P of a pixel is composed of true value and Gaussian noise, as follows: $P = P_{0} + n$ (1) Where P is the projection value, P₀ is the projection value without noise, n is Gaussian noise, with an expected value of 0 and a variance of σ².

The projection equation system of DECT is expressed as: ${\begin{matrix} P_{L} = \int S_{L} (E) exp (- μ_{1} (E) B_{1} - μ_{2} (E) B_{2}) dE \\ P_{H} = \int S_{H} (E) exp (- μ_{1} (E) B_{1} - μ_{2} (E) B_{2}) dE \end{matrix}$ (2) Where P_L and P_H represent the low-energy and high-energy projection values. S_L and S_H represent the normalized spectra of low energy and high energy, respectively. B₁ and B₂ represents the basis material projection of low and high attenuation basis materials, while μ₁ and μ₂ represent the attenuation coefficients under the energy E of low and high attenuation basis materials.

The projection equation system of DECT is complex and difficult to solve. By the integral median theorem, formula (2) is simplified as: ${\begin{matrix} P_{L} = S_{L} (ξ_{1}) e^{- B_{1} μ_{1} (ξ_{1}) - B_{2} μ_{2} (ξ_{1})} \\ P_{H} = S_{H} (ξ_{2}) e^{- B_{1} μ_{1} (ξ_{2}) - B_{2} μ_{2} (ξ_{2})} \end{matrix}$ (3) Where ξ₁ and ξ₂ are two energy values within the energy range (E_min, E_max). The physical meanings of ξ₁ and ξ₂ are the equivalent energy of low-energy and high-energy spectra, respectively.

Solving the formula (3) gives: ${\begin{matrix} B_{1} = \frac{μ_{2} (ξ_{2}) ln P_{L} - μ_{2} (ξ_{1}) ln P_{H}}{μ_{1} (ξ_{2}) μ_{2} (ξ_{1}) - μ_{1} (ξ_{1}) μ_{2} (ξ_{2})} + c_{1} \\ B_{2} = \frac{μ_{1} (ξ_{1}) ln P_{H} - μ_{1} (ξ_{2}) ln P_{L}}{μ_{1} (ξ_{2}) μ_{2} (ξ_{1}) - μ_{1} (ξ_{1}) μ_{2} (ξ_{2})} + c_{2} \end{matrix}$ (4) Where c₁ and c₂ are the constants unrelated to P_L and P_H.

The full differential expressions for B₁ and B₂ can be obtained through the following equation as follows: ${\begin{matrix} {dB}_{1} = \frac{\partial B_{1}}{\partial P_{H}} {dP}_{H} + \frac{\partial B_{1}}{\partial P_{L}} {dP}_{L} = \frac{\frac{μ_{2} (ξ_{2})}{P_{L}} {dP}_{L} - \frac{μ_{2} (ξ_{1})}{P_{H}} {dP}_{H}}{μ_{1} (ξ_{2}) μ_{2} (ξ_{1}) - μ_{1} (ξ_{1}) μ_{2} (ξ_{2})} \\ {dB}_{2} = \frac{\partial B_{2}}{\partial P_{H}} {dP}_{H} + \frac{\partial B_{2}}{\partial P_{L}} {dP}_{L} = \frac{\frac{μ_{1} (ξ_{1})}{P_{H}} {dP}_{H} - \frac{μ_{1} (ξ_{2})}{P_{L}} {dP}_{L}}{μ_{1} (ξ_{2}) μ_{2} (ξ_{1}) - μ_{1} (ξ_{1}) μ_{2} (ξ_{2})} \end{matrix}$ (5) Where μ₁ (ξ₂) μ₂ (ξ₁) - μ₁ (ξ₁) μ₂ (ξ₂) is the denominator in formula (5), defined as disturbance parameterΔ. Disturbance parameter Δ is typically between 10^–4 and 10^–3, which can lead to noise amplification of the decomposition coefficients.

The variance ( $σ_{B_{1}}^{2}$ and $σ_{B_{2}}^{2}$ ) and covariance (Cov_B₁,B₂) of B₁ and B₂ can be calculated: ${\begin{matrix} σ_{B_{1}}^{2} = {(\frac{\partial B_{1}}{\partial P_{H}})}^{2} σ_{P_{H}}^{2} + {(\frac{\partial B_{1}}{\partial P_{L}})}^{2} σ_{P_{L}}^{2} \\ σ_{B_{2}}^{2} = {(\frac{\partial B_{2}}{\partial P_{H}})}^{2} σ_{P_{H}}^{2} + {(\frac{\partial B_{2}}{\partial P_{L}})}^{2} σ_{P_{L}}^{2} \\ {Cov}_{B_{1}, B_{2}} = \frac{\partial B_{1}}{\partial P_{H}} \frac{\partial B_{2}}{\partial P_{H}} σ_{P_{H}}^{2} + \frac{\partial B_{1}}{\partial P_{L}} \frac{\partial B_{2}}{\partial P_{L}} σ_{P_{L}}^{2} \end{matrix}$ (6) Where $σ_{P_{L}}^{2}$ and $σ_{P_{H}}^{2}$ are the noise variances of P_L and P_H. By substituting formula (5) into formula (6), we can obtain: ${\begin{matrix} σ_{B_{1}}^{2} = \frac{1}{Δ^{2}} (\frac{μ_{2}^{2} (ξ_{1})}{P_{H}^{2}} σ_{P_{H}}^{2} + \frac{μ_{2}^{2} (ξ_{2})}{P_{L}^{2}} σ_{P_{L}}^{2}) \\ σ_{B_{2}}^{2} = \frac{1}{Δ^{2}} (\frac{μ_{1}^{2} (ξ_{1})}{P_{H}^{2}} σ_{P_{H}}^{2} + \frac{μ_{1}^{2} (ξ_{2})}{P_{L}^{2}} σ_{P_{L}}^{2}) \\ {Cov}_{B_{1}, B_{2}} = \frac{- 1}{Δ^{2}} (\frac{μ_{1} (ξ_{1}) μ_{2} (ξ_{1})}{P_{H}^{2}} σ_{P_{H}}^{2} + \frac{μ_{1} (ξ_{2}) μ_{2} (ξ_{2})}{P_{L}^{2}} σ_{P_{L}}^{2}) \end{matrix}$ (7)

The correlation coefficient ρ was further found through the variance and covariance of B₁ and B₂: $ρ = \frac{Co v_{B_{1}, B_{2}}}{\sqrt{σ_{p_{H}}^{2} σ_{p_{L}}^{2}}} = - \sqrt{1 - Δ^{2} * c}$ (8) Where c is a constant and Δ is typically between 10^–4 and 10^–3. Therefore, the correlation coefficient ρ closely approximates –1.

Due to the correlation coefficient of basis material projection B₁ and B₂ approaching –1, the reconstructed basis materials images b₁ and b₂ also have a high negative correlation [7]. Based on this, this paper proposes a concept of decomposition overlay image, which is the weighted sum of the basis material images with the highest signal-to-noise ratio, expressed as: $b_{s} = b_{1} / k + b_{2}$ (9) Where b ₁ and b ₂ represent low and high attenuation basis material images, respectively. k is the ratio of noise standard deviations of b ₁ and b ₂.

In CT image denoising, it is common to consider the noise distribution of the image. Therefore, we calculate the noise variance of the decomposition coefficients using a probability density function. Low-energy and high-energy imaging systems are independent of each other, so the noise of P_L and P_H are assumed separately. The two-dimensional joint probability density function of P_L and P_H is shown in formula (10). $f (P_{L}, P_{H}) = \frac{exp {- \frac{1}{2} [\frac{{(P_{L} - \bar{P_{L}})}^{2}}{σ_{P_{L}}^{2}} + \frac{{(P_{H} - \bar{P_{H}})}^{2}}{σ_{P_{H}}^{2}}]}}{2 π \sqrt{σ_{P_{L}}^{2} σ_{P_{H}}^{2}}}$ (10) Where $\bar{P_{L}}$ and $\bar{P_{H}}$ represent the mean of P_L and P_H, respectively. $σ_{P_{L}}^{2}$ and $σ_{P_{H}}^{2}$ represent the variance of P_L and P_H, respectively.

According to the transfer formula of the two-dimensional joint probability density, the joint probability density of the basis material projection can be calculated using formula (11). $g (B_{1}, B_{2}) = f (P_{L}, P_{H}) \cdot | J |$ (11) Where |J| represents the partial derivative determinant of B₁ and B₂ with respect to P_L and P_H.

We choose a simple example to demonstrate the joint probability density function of projection values and decomposition coefficients, as shown in Fig. 1.

Fig. 1

2D joint probability density function image. (a) shows the functions of P_L and P_H, while (b) shows the functions of B₁ and 2 B.

The variance and covariance of B₁ and B₂ can be calculated through the probability density function of B₁ and B₂: ${\begin{matrix} σ_{B_{1}}^{2} = \int \int {(B_{1} - \bar{B_{1}})}^{2} g (B_{1}, B_{2}) {dB}_{1} {dB}_{2} \\ σ_{B_{2}}^{2} = \int \int {(B_{2} - \bar{B_{2}})}^{2} g (B_{1}, B_{2}) {dB}_{1} {dB}_{2} \\ Cov (B_{1}, B_{2}) = \int \int (B_{1} - \bar{B_{1}}) (B_{2} - \bar{B_{2}}) g (B_{1}, B_{2}) {dB}_{1} {dB}_{2} \end{matrix}$ (12) Where $\bar{B_{1}}$ and $\bar{B_{2}}$ are the average values of the basis material projection B₁ and B₂, respectively.

2.2 Reconstruction model based on SBR

In Non Local Mean (NLM) filtering [20], the true value of an image pixel can be estimated by weighted average of pixels of the same or similar materials. $〈 x_{i} 〉 = \sum_{j \in N_{i}} w_{ij} x_{j}$ (13) Where N_i is the pixel with the same or similar material as the i-th pixel, and w_ij is the normalized weight that quantifies the similarity between the i-th and j-th pixel materials (∑_{j∈N_i}w_ij = 1). Formula (13) is transformed into matrix form, expressed as: $〈 x 〉 = Wx$ (14) Where W is the matrix of similar weights.

In DECT, owing to the similar structure between the basis material images and the prior image, they share an identical weight matrix W. Therefore, the Similarity Based Regularization (SBR) algorithm employs the W from the prior image to denoise the basis material images. This algorithm proposes a regularization term that minimizes the difference between images without and with noise suppression through similarity matrix multiplication. The regularization term is as follows: ${∥ Wx - x ∥}_{2}^{2}$ (15)

The statistical iterative reconstruction algorithm is a noise statistical model for projection data, considering the noise variance under different detection units and directions. Wang et al. introduced a regularization term (penalty term) to the statistical iterative reconstruction algorithm, resulting in the formulation of the Penalized Weighted Least Squares (PWLS) algorithm [21]. The PWLS-SBR algorithm integrates a similarity-based regularization item within the PWLS algorithm, and its objective function is formulated as follows: $Φ (x) = \arg \min {{(p - Ax)}^{T} Σ^{- 1} (p - Ax) + β {∥ Wx - x ∥}_{2}^{2}}$ (16) Where the first and second items are fidelity item and penalty item, respectively. β is the penalty parameter controlling the strength of the regularization, ensuring that image details are preserved while removing noise; A represents the system matrix; p represents projection data; x represents the reconstructed image; Σ is a diagonal matrix with diagonal elements $σ_{i}^{2}$ , and σ_i is the noise standard deviation of the i-th projection value p_i.

The similarity matrix W is obtained by calculating the pixel similarity of the prior image y , which is crucial for the noise suppression performance of the PWLS-SBR algorithm. w_ij is composed of pixel similarity terms and pixel distance terms, expressed as: $w_{ij} = exp (- \frac{{(y_{i} - y_{j})}^{2}}{t^{2}}) exp (- \frac{{(i - j)}^{2}}{h^{2}})$ (17) Where t is a similarity parameter that determines the structural resolution of the prior image; h is a distance parameter that represents the search range of similar pixels.

The PWLS-SBR algorithm has demonstrated remarkable achievements in CT denoising, but the structure of denoised images is highly dependent on prior images. Due to the limited count of similar pixels within edge images, the edge’s similarity weights tend to aggregate excessively, leading to unstable denoising results. The low rank property of tensors maintains the structural correlation between images, which is beneficial for the structural stability of edge pixels. This article incorporates the concepts of tensors and low rank into the PWLS-SBR algorithm, and proposes a penalty weighted least squares algorithm based on low rank and similarity-based regularization (PWLS-LRSBR).

2.3 Reconstruction model based on LRSBR

Due to the abundant redundancy between the basis material images of DECT, the relationship between decomposed images can be established through the concept of tensors. The N-dimensional tensor can be defined as $X \in R^{L_{1} \times L_{2} \times \dots \times L_{Nitsc}}$ . It is usual expanded into a two-dimensional matrix to calculate the rank of the tensor. $unfol d_{i} (X) = X_{(i)} \in ℝ^{L_{i} \times (L_{1} \dots L_{i - 1} L_{i + 1} \dots L_{N})}$ (18)

On the contrary, tensors can also be restored through i-th dimensional folding, expressed as: $fol d_{i} X_{(i)} = X$ (19)

The solution of the rank of a tensor is a non convex function, and a common method is to approximate it as the sum of the trace norms of the expanded matrix. $Rank (X) = \sum_{i = 1}^{N} a_{i} {∥ X_{(i)} ∥}_{tr}$ (20) Where N is the dimension of the tensor, a_i is the weight of the i-th dimensional tensor expansion matrix, and ∥ · ∥ _tr is the trace norm of the matrix. ${∥ X_{(i)} ∥}_{tr} = \sum_{j} σ_{j}$ (21) Where σ_j is the j-th largest singular value of the matrix. The singular value decomposition (SVD) algorithm can be used to solve the trace norm.

In order to fully utilize the structural prior of the decomposition overlay image b _s, the basis material images b ₁, b ₂, and b _s are combined into a three-dimensional tensor $X$ . Tensor $X \in R^{L \times N_{xitsc} \times N_{yitsc}}$ is the image of multi-layer basis materials; L represents the count of basis material image layers, typically chosen as three; N_x and N_y are the width and height of the basis material images. $X \in R^{L \times (N_{xitsc} \times N_{yitsc})}$ is the folding of the tensor $X$ in the first dimension. The noise within the basis material images exhibits considerable imbalance. Basis material images with lower noise variance should be assigned a higher weight coefficient, designated as v. $X = [\begin{matrix} v_{1} b_{1} \\ v_{2} b_{2} \\ v_{3} b_{s} \end{matrix}] = [\begin{matrix} v_{1} b_{1} \\ v_{2} b_{2} \\ v_{3} (b_{1} / k + b_{2}) \end{matrix}]$ (22) Where v describes the structural confidence of the basis material images in the tensor, increasing from top to bottom.

In order to further preserve the structure of the reconstructed image, this article proposes a penalty weighted least squares algorithm based on low rank and similarity-based regularization (PWLS-LRSBR). This method incorporates low rank into the PWLS-SBR algorithm and achieves joint reconstruction of tensor $X$ . The objective function is as follows: $\underset{X}{\arg \min} \frac{1}{2} {(P - Aitsc X)}_{Σ^{- 1}}^{2} + β_{1} R_{SBR} (X) + β_{2} R_{*} (X)$ (23) $R_{SBR} (X) = {∥ W X - X ∥}_{2}^{2}$ (24) $R_{*} (X) = {∥ X ∥}_{*}$ (25) Where ∥ · ∥ _* is the kernel norm, and the SVD algorithm is used to shrink the singular values to minimize the kernel norm. β₁ and β₂ are the penalty parameters for SBR and low rank to balance the data-fidelity term and regularization term. A represents the system matrix; $P$ represents the projection data; Σ is a diagonal matrix with diagonal elements $σ_{i}^{2}$ , the noise standard deviation σ_i is calculated using formula (12).

2.4 Optimization algorithm

To solve formula (23), we introduced an intermediate tensor $T$ with the same meaning as $X$ . Additionally, P , T and X are the first dimensional unfolding matrix of tensor images $P, T$ and $X$ . Formula (23) is transformed into: $\underset{T}{\arg \min} \frac{1}{2} {(P - AX)}_{Σ^{- 1}}^{2} + β_{1} R_{SBR} (T) + β_{2} R_{*} (T) + \frac{λ}{2} {∥ X - T ∥}_{2}^{2}$ (26)

To obtain the optimal solution of the PWLS-LRSBR algorithm, the formula (26) is divided into three sub problems to be solved separately. $S 1 : X = \underset{X}{arg min} {(P - AX)}^{T} Σ^{- 1} (P - AX) + \frac{λ}{2} {∥ X - T ∥}_{2}^{2}$ (27) $S 2 : T = \underset{T}{arg min} β_{1} {∥ WT - T ∥}_{2}^{2} + \frac{λ}{2} {∥ X - T ∥}_{2}^{2}$ (28) $S 3 : T = \underset{T}{arg min} β_{2} (\sum_{j} {∥ E_{j} T ∥}_{*})$ (29) Where E_j is the matrix for extracting sub image blocks from the image.

The solution of subproblem (S1) is obtained from the data fidelity term and typically contains noise. The solution of subproblem (S1) is employed as the initial value for optimizing subproblem (S2). Then the solution of subproblem (S2) is used to optimize subproblem (S3). Quality reconstructed images are obtained through iteration.

The subproblem (S1) is solved using a separable parabolic alternative algorithm [22]. $x_{j, n}^{k + 1} = x_{j, n}^{k} - \frac{\sum_{i = 1}^{M} ((\frac{1}{δ_{i}^{2}}) A_{ij} ({[A x^{k}]}_{i} - p_{i, n}))}{\sum_{i = 1}^{M} ((\frac{1}{δ_{i}^{2}}) A_{ij} \sum_{t = 1}^{N} A_{it}) + λ} + \frac{λ (x_{j, n}^{k} - t_{j, n}^{k})}{\sum_{i = 1}^{M} ((\frac{1}{δ_{i}^{2}}) A_{ij} \sum_{t = 1}^{N} A_{it}) + λ}$ (30)

The subproblem (S2) denoises each layer of the tensor image separately, and its objective function is expressed as: $F (T) = β_{1} {∥ WT - T ∥}_{2}^{2} + \frac{λ}{2} {∥ X - T ∥}_{2}^{2}$ (31) Where the matrix W is derived from the decomposition overlay image through the formula (17).

The objective function in formula (31) is convex and differentiable. Therefore, the best solution is for the derivative of the above equation to be 0, expressed as: $(c_{1} {(W - I)}^{T} (W - I) + c_{2} I) T = X$ (32) Where I is the identity matrix, and c₁ and c₂ are the two constants 2β₁/λ and (λ - 1)/λ, respectively. Formula (32) is solved using the Conjugate Gradient (CG) algorithm [23].

Due to the computational complexity of the entire image, the subproblem (S3) divides the image tensor $T$ into n image blocks. These blocks are then sequentially processed through the SVD algorithm. $T_{(j)} = E_{j} T = Unfold (E_{j} T)$ (33) $T_{(j), τ} = U Σ_{τ} V^{T}$ (34) Where the primary structural information resides within the first diagonal element. Therefore, other diagonal elements can be directly assigned a value of 0.

The PWLS-LRSBR algorithm process proposed in this article is shown in Table 1. This algorithm comprises two primary steps: acquiring structural priors and denoising the basis material images. Initially, an appropriate parameter k is chosen to generate a decomposition overlay image with high signal-to-noise ratio. Traditional denoising algorithms, such as K-Singular Value Decomposition (KSVD) [24, 25] and Block-Matching and 3D Filtering (BM3D) [26], are applied to denoise the decomposition overlay image. Meanwhile, the weight matrix W is extracted from the denoised image using formula (17). Subsequently, a tensor is formed by combining the decomposition overlay image with the basis material images. This tensor serves as the initial value for the PWLS-LRSBR reconstruction process. The denoising procedure employs formulas (30), (32), and (34), with iterative alternation between them.

Table 1

PWLS-LRSBR algorithm process

Input: Basis material images b ₁ and b ₂, the number of iterations M
Obtain structural prior and weight matrix:
(1) Select the appropriate k to obtain the decomposition overlay image with the highest signal-to-noise ratio.
(2) Train a global dictionary D using the K-SVD algorithm [24].
(3) Denoise the decomposition overlay image through OMP algorithm [25].
(4) Use formula (17) to obtain the weight matrix W from the denoised image of the decomposition overlay image.
Basis material image denoising:
(1) Combine the basis material images into the initial value of tensors $X$ and $T$ .
(2) Denoising of Tensors $X$ .
For i = 1: M do
(a) Update the image tensor $X$ through formula (30).
(b) Solve SBR denoising for each layer of images using the CG algorithm of formula (32).
(c) Constrain the kernel norm of image tensors through formula (34).
End For

2.5 Parameter selection

2.5.1 Selection of penalty parameters β₁ and β₂

β₁ and β₂ are the penalty parameters used to balance the data-fidelity term and regularization term. The selection of these parameters is widely recognized as a challenge in CT image reconstruction. Therefore, the choice of parameters β₁ and β₂ depends on the noise level of the decomposed DECT image. Practically, these penalty parameters are usually selected based on experience.

2.5.2 Selection of parameters in solving subproblems

The number of iterations for subproblems is also an important factor in achieving a successful result. Therefore, we must choose suitable values in the experiment. The number of iterations for the CG algorithm is set to 100. Similarly, the size of image blocks should also be in an appropriate range. A patch size of 5×5 pixels was used in this paper.

3 Experiments and results

3.1 Experimental data

To evaluate the performance of the PWLS-LRSBR algorithm in DECT projection domain decomposition, we conducted experiments on a digital phantom and a physical phantom.

3.1.1 Digital phantom data

A self-made Shepp-Logan phantom is applied to evaluate the proposed algorithm in this study as shown in Fig. 2. The structure of this model is consistent with that of the Shepp-Logan phantom [27], and new materials are used: Polymethyl methacrylate (PMMA), Polyformaldehyde (POM), and Magnesium (Mg). In the numerical simulation experiment, the initial spectra is obtained from Spectrum GUI (an open-source energy spectra generation program), featuring a tube voltage of 80 kV. The high-energy and low-energy spectra are acquired by attenuating the initial spectra using a double layer scintillator composed of GAGG and LYSO, as shown in Fig. 3. We simulated a fan-beam CT geometry with source to detector distance of 20.0 mm, source to rotation center distance of 30.0 mm, a detector size of 512 with 0.027 mm per detector pixel and 360 projection views. The size of the reconstructed image is 9.26×9.26 mm², and the resolution is 512×512. For each X-ray path, 5×10⁵ photons are assumed emitted from an X-ray source. The projection data with Poisson noises are generated with expectations being the number of photons received in the corresponding noise-free case.

Fig. 2

The image of the self-made Shepp-Logan phantom. The components of phantom are PMMA, POM, and Mg.

Fig. 3

Low-energy and high-energy simulation spectra of daul-layer DECT.

3.1.2 Physical phantom data

Figure 4 shows the physical phantom, which consists of five types of materials: Polypropylene (PP) Polymethyl methacrylate (PMMA), Polyformaldehyde (POM), polytetrafluoroethylene (PTFE), and Magnesium (Mg). Their densities are listed in Table 2. Notably, Material PP, with a diameter of 7.2 mm, serves as the background material. Furthermore, the other four materials are represented in three different sizes with diameters of 1.4 mm, 0.70 mm, and 0.35 mm, enhancing the evaluation of the denoising algorithm’s resolution accuracy. In this system, the distance between the X-ray source and the detector scintillator is set to 20.0 mm, and the distance between the X-ray source and the rotation axis is set to 30.5 mm. 360 projections are uniformly obtained on a fully scanned circular trajectory. The detector pixel is of 1024×1024 matrix with each pixel at 0.0135×0.0135 mm² pitch. In order to further suppress the impact of large noise on the basis material images, the sampled pixels in the reconstruction were merged into 512×512, with a coverage area of 9.9×9.9 mm². The main parameters of dual energy scanning are summarized in Table 3.

Fig. 4

The image of the physical cylindrical phantom. The components of phantom are PP, PMMA, POM, PTFE and Mg.

Table 2

Material Information

Material	Material	Molecular	Electron	Effective
Number	Name	Formula	Density	Atomic Number
1	PP	C3H6	1.03	4.75
2	PMMA	C5H8O2	1.285	5.85
3	POM	CH2O	1.515	6.375
4	PTFE	C2F4	2.112	8.25
5	Mg	—	1.74	12

Table 3

Practical experimental parameters

Parameter	Value
Tube voltage	80 kV
Tube current	115 uA
Additional filter	None
Source to detector distance	20.0 mm
Source to detector distance	30.5 mm
Sampling frequency	0.04 Hz
Exposure time	20 s
Detector resolution	1024×1024 with 0.0135 mm×0.0135 mm pixels
First layer scintillator	GAGG
Second layer scintillator	LYSO
Number of samples	360

3.1.3 Performance evaluation

In this work, several state-of-the-art CT image denoising methods, including DL, PICCS and SBR algorithm, were implemented as competing methods. We compared the proposed and competing method in numerical simulation and practical experiments. All the above methods were performed on PC (Intel (R) Core (TM) i7-10700 CPU @ 2.90 GHz, 32.0 GB RAM) in Matlab (2019b). The FBP reconstructed image is set as the initial image of the iterative method. The number of iterations for numerical simulation is 200, while the number of iterations for real experiments is 100. In the above experiments, the decomposition overlay image was selected as the prior image.

3.1.4 Experimental data acquisition

The attenuation coefficients used for the materials in this study were carried out by the NIST XCOM database. Furthermore, the electron density and effective atomic number of materials have been computed using their respective molecular formulas, as shown in Table 2. According to the ranking of effective atomic numbers, PP and Mg are respectively chosen as the basis materials with low and high-attenuation.

All practical experiments were carried out on the dual-layer DECT system developed by the State Key Laboratory of Precision Measurement Technology and Instruments of Tianjin University. The dual-layer dual energy micro CT system consists of an X-ray source, an air floating rotating translation system, a dual light path detector system, and a control system. Dual-layer DECT [28] employs dual-layer scintillators to differentially attenuate low-energy and high-energy X-ray photons. Due to the consistent geometric spatial characteristics exhibited in its dual-energy images, dual-layer DECT finds wide application in PDD research.

3.2 The advantages of decomposition overlay image

In order to demonstrate the effectiveness and practicality of the proposed DECT denoising algorithm, we utilize the Siddon algorithm [29] to acquire dual-energy projection data. Subsequently, the Levenberg-Marquardt (LM) algorithm [30] is employed to solve the nonlinear equations associated with PDD, resulting in the generation of basis material images. Figure 5 shows the reconstruction images of dual-energy projection data using the Filtered Back Projection (FBP) algorithm.

Fig. 5

Dual-energy reconstruction images of the self-made Shepp-Logan phantom. (a) represents low-energy, (b) represents high-energy.

As shown in Fig. 5, noticeable beam hardening artifacts are evident in the low-energy reconstruction image of the daul-layer DECT, which significantly impacts the structural resolution capability. Nonetheless, high-energy reconstruction of images encounters the challenge of an excessive signal-to-noise ratio, which is due to the limited photon count within the second-layer scintillator of the dual-layer DECT system. Nonetheless, the decomposition overlay image is acquired by conducting a weighted summation of the basis material images, as shown in Fig. 6.

Fig. 6

Decomposition overlay image of the self-made Shepp-Logan phantom.

By comparing Figs. 5 and 6, it can be seen that the decomposition overlay image has the advantages of high signal-to-noise ratio and less artifacts. Decomposition overlay image proves to be a more suitable choice as the structural prior image for DECT denoising. To distinctly observe the advantages of the decomposition overlay map, we conducted a statistical analysis of grayscale values within different material regions, as presented in Fig. 7.

As shown in Fig. 6, it can be clearly seen that the capacity for material discrimination in Fig. 7(c) is superior to that in Figs. 7(a) and (b). In Fig. 7(a), the grayscale values for the materials are widely dispersed, mainly due to the pronounced beam hardening artifacts in the low-energy reconstruction image. In Fig. 7(b), although the material featuring closely concentrated grayscale values, the peak distance between different materials is relatively minimal. This is primarily attributed to the low signal-to-noise ratio of high-energy reconstruction images. Consequently, there are higher signal-to-noise ratio and fewer artifacts in the decomposition overlay image. This emphasizes its suitability as a structural prior image for enhancing material discrimination and improving overall image quality.

Fig. 7

Statistical analysis of material grayscale values. (a) is the results of low-energy reconstruction image, (b) is the results of high-energy reconstruction image, and (c) is the results of decomposition overlay image.

3.3 Performance evaluation on digital phantom

To test the feasibility of the proposed approach, several state-of-the-art CT image denoising methods, including DL, PICCS, SBR, and the proposed LRSBR algorithm, are compared. The DL algorithm, as a traditional denoising approach, tackles significant noise levels without the utilization of structural prior images. On the other hand, both the PICCS and SBR algorithms are denoising algorithms based on structural prior, typically choosing high-energy reconstruction images as the prior images. In order to fairly compare the image quality of these various algorithms, the decomposition overlay image is selected as the structural prior image for all of the above-mentioned algorithms.

Figure 8 shows the results of denoising the basis material images using DL, PICCS, SBR, and the proposed LRSBR algorithm, respectively. The high-level noise in Fig. 8(1) illustrates the problem of noise amplification in PDD. The residual noise observed in Fig. 8(2) demonstrates that DL is difficult to solve the low signal-to-noise ratio of the basis material images. In Fig. 8(3), PICCS still exhibits edge blurring and blocky artifacts owing to the limitations of the total variation algorithm. Figure 8(4) shows the edge instability caused by weight concentration in the SBR algorithm. The reconstruction result in Fig. 8(5) displays exceptional denoising and clear edge structures. Figure 9 presents profiles of the reconstruction results along the red line (as shown in Fig. 8(a1)). The images show that the proposed LRSBR algorithm performded significantly better than other competing algorithms in terms of fine structures and edges.

Fig. 8

Reconstruction results of different algorithms in digital experiments. The first and second rows are the basis material images with low attenuation and high attenuation. From left to right are the results of FBP, DL, PICCS, SBR, and LRSBR, respectively.

Fig. 9

Profiles of the low attenuation basis material images along the curve (as shown in the Fig. 8(a1)) in digital experiments.

To quantify the effectiveness of different algorithms in noise reduction for DECT images, specific image evaluation criteria is emploied. These evaluation criteria encompass Peak Signal to Noise Ratio (PSNR), Root Mean Square Error (RMSE), and Structural Similarity (SSIM). Table 4 presents the quantitative results of these image evaluation criteria for both the competing and proposed algorithms.

Table 4

Quantitative image evaluation standards for competing and proposed algorithms

	DL	PICCS	SBR	LRSBR
PSNR	31.7057	30.5348	34.9240	39.8646
RMSE	0.0260	0.0297	0.0179	0.0102
SSIM	0.9361	0.9392	0.9636	0.9841

From the quantitative image evaluation criteria shown in Table 3, it can be seen that the advantages of LRSBR over other competing algorithms are evident. The DL algorithm and PICCS algorithm exhibits poor structural retention ability and quantitative evaluation criteria due to the impact of an excessive signal-to-noise ratio. The image evaluation criteria of LRSBR (PSNR = 39.8646, RMSE = 0.0102, and SSIM = 0.9841) demonstrates a notable improvement over that of the SBR algorithm (PSNR = 34.9240, RMSE = 0.0179, and SSIM = 0.9636). In the SBR algorithm, the lack of similar pixels in the edge pixels results in the edge image being highly vulnerable to noise. This issue is commonly referred to as the edge instability problem of the SBR algorithm. The proposed LRSBR method makes full use of the edge information from prior images through the low-rank properties of tensors, effectively addressing the issue of edge instability in the SBR algorithm.

3.4 Performance evaluation on physical phantom

In the physical phantom study, we employ the FBP algorithm and the Levenberg-Marquardt algorithm to obtain dual-energy reconstruction images, as shown in Fig. 10. The decomposition overlay image is shown in Fig. 11. In this section, in order to further verify the applicability of the algorithm proposed in this article, FBP, DL, PICCS, SBR, and the proposed LRSBR algorithm were employed to denoise the basis material images of physical phantom.

Fig. 10

Dual-energy reconstruction images of the physical cylindrical phantom. (a) is low-energy reconstruction image, (b) is high energy reconstruction image.

Fig. 11

Denoised decomposition overlay image of the physical cylindrical phantom.

Figure 12 displays the basis material images of different denoising algorithms. In Fig. 12(1), the FBP algorithm’s denoised images reveal noticeable levels of noise in the basis material images. Figure 12(2) displays the images denoised by DL method. Although image noise is greatly suppressed, the edges of the images are blurred. Figure 12(3) presents the images denoised by PICCS method. The fine structures of the denoised image is preserved, but there are still blocky artifacts attributed to the influence of total variation. Figure 12(4) is the denoising result of SBR, which effectively suppresses noise and maintains structural resolution. However, the denoised image exhibits unstable structures on the large gradient edges. Figure 12(5) shows the reconstruction results of LRSBR. The image noise is efficiently suppressed, and the edges and fine structures of the image are well preserved. To further illustrate the performance of these reconstruction methods, the profiles through the red curve (indicated in Fig. 11) are plotted in Fig. 13.

Fig. 12

Reconstruction results of different algorithms in practical experiments. The first and second rows represent basis material images with low and high attenuation, respectively. The red box shows the local magnification of the low attenuation basis material image. From left to right are FBP, DL, PICCS, SBR, and the proposed LRSBR algorithm.

Fig. 13

Profiles of the low attenuation basis material images along the line (as shown in the Fig. 11) in practical experiments.

Compared to DL without prior structural images, PICCS, SBR, and the proposed LRSBR all retain the pore structure present within the physical phantom. This proves that structural prior effectively weakens the adverse impact of substantial noise on smaller-scale structures. The PICCS algorithm, an enhancement of the total variation minimization model, exhibits blocky artifacts despite retaining small structures like pores. The SBR algorithm has a good visual experience, but it eliminates the structure of small gradients in the decomposition overlay image. Moreover, edge structures with large gradients have the problem of concentrated weight, leading to unstable edge structures. The LRSBR algorithm proposed in this article maintains the structural consistency between the decomposition overlay image and the basis material image through low rank of tensor. Unlike the SBR algorithm that maintains structure via pixel grayscale values, the LRSBR algorithm employs image blocks to enforce structural constraints, thereby enhancing edge stability. These results indicate that the PWLS-LRSBR method achieves the superior performance compared with the competing methods, particularly in noise reduction and the preservation of edges.

4 Discussion and conclusion

To address the challenge of noise amplification in PDD, this paper proposes the PWLS-LRSBR algorithm that utilizes the structural priori information from the decomposition overlay image to improve the quality of basis material images in dual-layer DECT. This method is based on the penalty weighted least squares, and applies the structural redundancy information of dual energy CT images to reconstruct basis material images. This is achieved through the utilization of the low rank property of tensors and the incorporation of similarity-based regularization. Through numerical simulation and practical experiments, it has been confirmed that the proposed LRSBR algorithm outperforms DL, PICCS, and SBR algorithms for DECT reconstruction. Compared with competing algorithms, the LRSBR method effectively preserves fine structures and image edges, thereby addressing the instability issue arising from weight concentration in the SBR algorithm.

Although the proposed PWLS-LRSBR algorithm has achieved excellent results, there are still some potential limitations in application. Firstly, in order to enhance the method’s applicability, it is necessary to ensure the matching of the dual-energy projections. In this condition, the proposed LRSBR algorithm is capable of achieving notable edge preservation. Otherwise, it will lead to new artifacts at the edges of the image, consequently impacting the accuracy of the final result. Secondly, the accuracy of spectra estimation is an important factor in PDD, which seriously affects the structural correlation of the basis material images. Owing to more complex factors like initial energy spectra and scintillators, it is difficult to estimate the energy spectra of dual-layer microscopic DECT. These issues diminish the structural coherence of basis material images and subsequently influences the performance of the LRSBR algorithm.

In summary, this paper proposes a denoising algorithm based on low rank and similarity-based regularization, which improves the performance of basis material images in PDD. The proposed algorithm maintains fine structures and image edges, which is of great significance in the denoising research of PDD.

Footnotes

Acknowledgments

This work was support by the National Natural Science Foundation of China (NNSFC) [No. 61771328] and the National Key Research and Development Program of China [No. 2017YFB1103900].

References

, Zhao

, Chen

Z.Q.

, First Dual MeV Energy X-ray CT for Container Inspection: Design, Algorithm, and Preliminary Experimental Results, IEEE Access 6 (2018), 45534–45542.

Thieme

S.F.

, Johnson

T.R.C.

, Lee

, et al. Dual-Energy CT for the Assessment of Contrast Material Distribution in the Pulmonary Parenchyma, American Journal of Roentgenology 193 (2009), 144–149.

Gehrke

, Wirth

K.E.

, Application of conventional- and dual-energy X-ray tomography in process engineering, IEEE Sensors Journal 5 (2005), 183–187.

Stenner

, Berkus

, Kachelriess

, Empirical dual energy calibration (EDEC) for cone-beam computed tomography, Medical Physics 34 (2007), 3630–3641.

Maass

, Baer

, Kachelriess

, Image-based dual energy CT using optimized precorrection functions: A practical new approach of material decomposition in image domain, Medical Physics 36 (2009), 3818–3829.

Maass

, Meyer

, Kachelriess

, Exact dual energy material decomposition from inconsistent rays (MDIR), Medical Physics 38 (2011), 691–700.

Petrongolo

, Zhu

, Noise Suppression for Dual-Energy CT Through Entropy Minimization, IEEE Transactions on Medical Imaging 34 (2015), 2286–2297.

B.J.

, Li

B.H.

, Luo

, et al. Simultaneous Reduction in Noise and Cross-Contamination Artifacts for Dual-Energy X-Ray CT, Biomed Research International 2013 (2013), 8.

, Yu

H.Y.

, Bennett

, et al. Image Reconstruction for Hybrid True-Color Micro-CT, IEEE Transactions on Biomedical Engineering 59 (2012), 1711–1719.

10.

, Yu

H.Y.

, Mou,

X.Q.

, Low-dose x-ray CT reconstruction via dictionary learning, IEEE Transactions on Medical Imaging 31 (2012), 1682–1697.

11.

Gao

, Yu

H.Y.

, Osher

, Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM), Inverse Problems 27 (2011), 22.

12.

Zhang

Y.B.

, Mou

X.Q.

, Wang

, Tensor-Based Dictionary Learning for Spectral CT Reconstruction, IEEE Transactions on Medical Imaging 36 (2017), 142–154.

13.

W.W.

, Zhang

Y.B.

, Wang

, et al. Low-dose spectral CT reconstruction using image gradient l(0)-norm and tensor dictionary, Applied Mathematical Modelling 63 (2018), 538–557.

14.

Z.C.

, Leng

, Li

Z.B.

, Spectral prior image constrained compressed sensing (spectral PICCS) for photon-counting computed tomography, Physics in Medicine and Biology 61 (2016), 6707–6732.

15.

Kong

H.H.

, Lei

X.X.

, Lei

, Spectral CT Reconstruction Based on PICCS and Dictionary Learning, IEEE Access 8 (2020), 133367–133376.

16.

Niu

S.Z.

, Bian

Z.Y.

, Zeng

, Total image constrained diffusion tensor for spectral computed tomography reconstruction, Applied Mathematical Modelling 68 (2019), 487–508.

17.

Harms

, Wang

T.H.

, Petrongolo

, Noise suppression for dual-energy CT via penalized weighted least-square optimization with similarity-based regularization, Medical Physics 43 (2016), 2676–2686.

18.

Shen

, Xing

Y.X.

An Edge-Preserving TotalVariation Denoising Method for DECT Image, in: 60th IEEE Nuclear Science Symposium (NSS)/Medical Imaging Conference (MIC)/20th International Workshop on Room-Temperature Semiconductor X-ray and Gamma-ray Detectors, IEEE, Seoul, South Korea, 2013.

19.

Hao

, Kang

K.J.

, Zhang

, A novel image optimization method for dual-energy computed tomography, Nuclear Instruments & Methods in Physics Research Section a-Accelerators Spectrometers Detectors and Associated Equipment. 722 (2013), 34–42.

20.

Z.B.

, Yu

L.F.

, Trzasko

J.D.

, et al. Adaptive non-local means filtering based on local noise level for CT denoising,in: Conference on Medical Imaging - Physics of Medical Imaging, Spie-Int Soc Optical Engineering, San Diego, CA, 2012.

21.

Wang

Q.X.

, Li

, Lam

K.Y.

, Development of a new meshless –point weighted least-squares (PWLS) method for computational mechanics, Computational Mechanics 35 (2005), 170–181.

22.

Elbakri

I.A.

, Fessler

J.A.

, Statistical image reconstruction for polyenergetic X-ray computed tomography, IEEE Transactions on Medical Imaging 21 (2002), 89–99.

23.

Dai

Y.H.

, Yuan

, A nonlinear conjugate gradient method with a strong global convergence property, SIAM Journal on Optimization 10 (1999), 177–182.

24.

Aharon

, Elad

, Bruckstein

, K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Transactions on Signal Processing 54 (2006), 4311–4322.

25.

Yang

M.R.

, de Hoog,

, Orthogonal Matching Pursuit With Thresholding and its Application in Compressive Sensing, IEEE Transactions on Signal Processing 63 (2015), 5479–5486.

26.

Dabov

, Foi

, Katkovnik

, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Transactions on Image Processing 16 (2007), 2080–2095.

27.

Shepp

L.A.

, Logan,

B.F.

, Reconstructing Interior Head Tissue from X-Ray Transmissions, IEEE Transactions on Nuclear Science 21 (1974), 228–236.

28.

Ren

, Zhen

, Zheng

, Clinical applications and progresses of dual-layer spectral detector CT, Chinese Journal of Medical Imaging Technology 36 (2020), 1555–1558.

29.

Siddon

R.L.

, Fast calculation of the exact radiological path for a three-dimensional CT array, Medical physics 12 (1985), 252–255.

30.

Zhao

, Fan

J.Y.

, On the multi-point Levenberg-Marquardt method for singular nonlinear equations, Computational & Applied Mathematics 36 (2017), 203–223.

Projection domain decomposition denoising algorithm based on low rank and similarity-based regularization

Abstract

BACKGROUND:

OBJECTIVE:

METHODS:

RESULTS:

CONCLUSIONS:

Keywords

1 Introduction

2 Method

2.1 DECT noise model

2.5.1 Selection of penalty parameters β1 and β2

2.5.2 Selection of parameters in solving subproblems

3 Experiments and results

3.1 Experimental data

3.1.1 Digital phantom data

3.1.4 Experimental data acquisition

3.2 The advantages of decomposition overlay image

Footnotes

Acknowledgments

References

2.5.1 Selection of penalty parameters β₁ and β₂