Restarted primal–dual Newton conjugate gradient method for enhanced spatial resolution of reconstructed cone-beam x-ray luminescence computed tomography images

Abstract

Cone-beam x-ray luminescence computed tomography (CB-XLCT) has been proposed as a promising imaging tool, which enables three-dimensional imaging of the distribution of nanophosphors (NPs) in small animals. However, the reconstruction performance is usually unsatisfactory in terms of spatial resolution due to the ill-posedness of the CB-XLCT inverse problem. To alleviate this problem and to achieve high spatial resolution, a reconstruction method consisting of inner and outer iterations based on a restarted strategy is proposed. In this method, the primal–dual Newton conjugate gradient method (pdNCG) is adopted in the inner iterations to get fast reconstruction, which is used for resetting the permission region and increasing the convergence speed of the outer iteration. To assess the performance of the method, both numerical simulation and physical phantom experiments were conducted with a CB-XLCT system. The results demonstrate that compared with conventional reconstruction methods, the proposed re-pdNCG method can accurately and efficiently resolve the adjacent NPs with the least relative error.

1. Introduction

X-ray luminescence computed tomography (XLCT) has been proposed as an emerging hybrid optical/x-ray CT imaging modality since its first demonstration (Carpenter et al 2010, Pratx et al 2010a, 2010b). With the avoidance of autofluorescence and background fluorescence and the help of x-ray excitation position as priors, XLCT may achieve higher spatial resolution and sensitivity compared to other commonly used optical tomography techniques, e.g. fluorescence molecular tomography (FMT) (Guo et al 2019, Meng et al 2019) and bioluminescence tomography (BLT) (Dehghani et al 2018, Gao et al 2018b). In order to maximize the potential of XLCT (Zhang et al 2019), current research efforts are focused on the development of new XLCT imaging systems, more efficient reconstruction methods (Pu et al 2019), and new types of nanophosphors (NPs) with increased photon yield (Naczynski et al 2014).

Currently, there are two primary types of XLCT systems, which are pencil/narrow-beam XLCT (PB-XLCT (Pratx et al 2010a, 2010b) or NB-XLCT (Li et al 2013, 2014)) and cone-beam XLCT (CB-XLCT) (Chen et al 2013, Gao et al 2017, Liu et al 2018), according to the x-ray excitation mode (pencil/narrow-beam or cone-beam). The PB-XLCT and NB-XLCT techniques can achieve high spatial resolution; however, their inherent long data acquisition time is unfavorable for fast imaging of e.g. drug biokinetics. CB-XLCT, on the other hand, which utilizes a cone-beam geometry to cover the whole imaging object, can simplify the scanning process and decrease the imaging time. Thus, it is more suitable for biological applications, especially for fast biomedical imaging (Liu et al 2013, 2014a, 2014b, 2019a).

However, due to the diffusive transport and high scatter of optical photons in biological tissues, the reconstruction of CB-XLCT is an ill-posed problem, which leads to reduced image quality (Gao et al 2018a). Several methods have been proposed to improve the imaging performance. Zhang et al proposed a Bayesian framework-based reconstruction algorithm that can automatically estimate regularization parameters and can preserve the target edge (Zhang et al 2017, 2018). To reduce the imaging views, the sparsity-based L₁-TV method (Liu et al 2019a) and T-FISTA method (Gao et al 2018a) were proposed, and the performance was confirmed by phantom experiments where the edge-to-edge distance (EED) was only 1.7 mm. Besides sparsity, group sparsity was also proposed as a prior for CB-XLCT with the help of CT images (Liu et al 2019b). Though the above algorithms continuously improved the reconstruction performance concerning location accuracy and shape similarity, challenges remain in improving the spatial resolution of CB-XLCT. Furthermore, traditional reconstruction methods are usually time-consuming, especially when the weight matrix is large and Hessian matrix inversion or singular value decomposition (SVD) is in demand. Thus, more effort is needed to develop a fast reconstruction method that allows for increased imaging quality of CB-XLCT.

To improve the imaging quality, a primal–dual Newton conjugate gradient method (pdNCG) was adopted for CB-XLCT reconstruction in our previous study (Gao et al 2019). This method requires only matrix–vector product operations with an inexpensive preconditioner, which can achieve good results efficiently. However, due to the ill-posedness of the CB-XLCT inverse problem, the convergence speed of pdNCG decreases after a fixed number of iterations. To increase the convergence speed, we propose a restarted strategy in the outer iterations by resetting the permission region and the initial value after inner iterations of pdNCG.

To evaluate the performance of the proposed method, numerical simulations and physical phantom experiments were performed on a custom-made CB-XLCT system. In the numerical simulations, a three-dimensional (3-D) digital mouse was simulated, and two luminescent targets with different EEDs (0.1 to 0.3 cm) were embedded into the digital mouse liver. In the physical phantom experiments, two transparent tubes made of polymethyl methacrylate filled with perovskite NPs were immersed in a cylinder phantom as the luminescent targets. Two widely used algorithms, adaptive Tikhonov (Adaptik, an adaptive L2-regularized method) (Cao et al 2013) and Fast Iterative Shrinkage-Thresholding Algorithm (FISTA, a well-known L1-regularized method) (Beck and Teboulle 2009), together with our previously proposed pdNCG method were adopted for comparison. Our results show that the proposed method improves the spatial resolution and decreases the time cost for CB-XLCT image reconstruction.

This paper is organized as follows. In section 2, the forward and inverse problems of CB-XLCT are presented. In section 3, numerical simulations and physical phantom experiments are conducted to validate the reconstruction performance using the proposed algorithm. In section 4, the results are summarized. In section 5, the results are discussed and concluded.

2. Methods

2.1. Photon propagation model of CB-XLCT

In CB-XLCT imaging, according to the Beer–Lambert law, the x-ray intensity at position r can be described as follows (Chen et al 2013):

$$X\left(\mathbf{r}\right)=X\left({\mathbf{r}}_0\right)\exp \left\{-{\int }_{r_0}^r{\mu }_t\left(\tau \right)d\tau \right\}$$ (1)

where X(r₀) is the x-ray intensity at the initial position r₀, and µ_t(τ) is the x-ray attenuation coefficient obtained from reconstructed images. When irradiated by x-rays, x-ray excitable NPs distributed in the imaging object can emit visible or near-infrared (NIR) light. The generated light source density S(r) can be calculated as follows:

$$S\left(\mathbf{r}\right)=\eta X\left(\mathbf{r}\right)\rho \left(\mathbf{r}\right)$$ (2)

where η is the light yield, and ρ(r) is the NP concentration at position r, respectively. The light transportation in scattering media can be modeled by the radiative transfer equation (RTE), but it is challenging to solve the RTE directly due to the mathematical complexity. Due to the weakly absorbing and highly scattering properties of biological tissues in the visible or NIR spectral window, the RTE model can be simplified to the diffusion equation (DE) model with Robin-type boundary condition (Cong et al 2005):

$$\{\begin{array}{c}-\nabla \left[D\left(\mathbf{r}\right)\nabla \Phi \left(\mathbf{r}\right)\right]+{\mu }_a\left(\mathbf{r}\right)\Phi \left(\mathbf{r}\right)=X\left(\mathbf{r}\right)\hspace{1em}\hspace{0.17em}\left(\mathbf{r}\in \Omega \right)\\ \Phi \left(\mathbf{r}\right)+2\alpha D\left(\mathbf{r}\right)\left[\nu \nabla \Phi \left(\mathbf{r}\right)\right]=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathbf{r}\in \partial \Omega \right)\end{array}$$ (3)

where D(r) is the diffusion coefficient, which can be calculated by $D\left(\mathbf{r}\right)={\left(3\left({\mu }_{\text{a}}\left(\mathbf{r}\right)+{\mu }_s^{\text{'}}\left(\mathbf{r}\right)\right)\right)}^{-1}$, and µ_a and ${\mu }_s^{\text{'}}$ are the absorption and reduced scattering coefficients of the tissue, respectively. Φ(r) is the photon fluence at position r, Ω is the domain of the imaging object, ∂Ω denotes the boundary of Ω, α is the optical reflective index mismatch, and ν is the outward unit normal vector on ∂Ω.

Based on the finite-element method (FEM), equation (3) can be transformed into the following matrix:

$$\mathit{A}\mathit{\rho }={\mathbf{\Phi }}_{\text{meas}}$$ (4)

where A is a weight matrix used to construct a linear relationship between the unknown NP distribution ρ in the biological tissues and the known photon flux measurements on the object surface Φ_meas.

2.2. CB-XLCT reconstruction based on restarted pdNCG

In CB-XLCT, reconstructing the NP distribution is equivalent to finding an approximate solution to equation (4). Due to the diffusive transport of optical photons in biological tissues, and the limited number of measurements, the inverse problem of CB-XLCT is ill-posed and underdetermined. Thus, equation (4) is usually solved with an L_q regularization term to obtain stable and robust results, as follows:

$$\text{minimize}\hspace{0.17em}f\left(\mathbf{\rho }\right)={\Vert \mathit{A}\mathbf{\rho }-{\mathbf{\Phi }}_{\text{meas}}\Vert }^2+\lambda {\Vert \mathit{\rho }\Vert }_q$$ (5)

where λ is the regularization parameter. If q = 1, a sparsity regularization is employed, and if q = 2, equation (5) will be a Tikhonov regularization problem.

In our previous study (Gao et al 2019), a sparsity-based pdNCG method was proposed to improve the CB-XLCT imaging performance. A pseudo-Huber function is first introduced to replace the nondifferentiable L₁-norm in equation (5).

$$\text{minimize}\hspace{0.17em}f\left(\mathit{\rho }\right)=\phi \left(\mathit{\rho }\right)+\lambda \sum _{i=1}^n\left({\left({\mu }^2+{\rho }_i^2\right)}^{\frac{1}{2}}-\mu \right)$$ (6)

where

$$\phi \left(\mathit{\rho }\right)={\Vert \mathit{A}\rho -{\mathbf{\Phi }}_{\text{meas}}\Vert }^2$$ (7)

µ is a smoothing parameter that controls the approximation, i.e. for µ → 0, the second term in equation (6) tends to the L₁-norm.

Then, equation (6) is rewritten in the primal–dual form to derive the pseudo-Huber function.

$$\underset{g\in g_{\infty }\le 1}{\text{minimize}}f\left(\mathit{\rho }\right)=\phi \left(\mathit{\rho }\right)+\lambda \sum _{i=1}^n\left(\mu {\left(1-g_i^2\right)}^{\frac{1}{2}}-\mu \right)+\lambda {\mathit{g}}^T\mathit{\rho }\}$$ (8)

where g is the dual variable.

A detailed pseudo-code to solve equation (8) is given below.

1. Loop: For k = 1,2,…, until ${\Vert {\mathbf{d}}^k\Vert }_{{\rho }^k}\le$, where ε > 0> 0

2. Obtain d^k by solving approximately the system

   $$H\left({\mathit{\rho }}^k,{\mathbf{g}}^k\right)\mathbf{d}=-\nabla f\left(\mathit{\rho }\right)$$ (9)
   using preconditioned conjugate gradients (PCG), where

   $$H\left(\mathit{\rho },\mathbf{g}\right)=\lambda \mathbf{D}\left(\mathbf{I}-\mathbf{D}\text{diag}\left(\mathit{\rho }\right)\text{diag}\left(\mathbf{g}\right)\right)+{\nabla }^2\phi \left(\mathit{\rho }\right)$$ (10)

   where $D_i={\left({\mu }^2+{\rho }_i^2\right)}^{-\frac{1}{2}}$
3. Obtain △ g^k by calculating

   $$\Delta {\mathbf{g}}^k=\mathbf{D}\left(\mathbf{I}-\mathbf{D}\text{diag}\left(\mathit{\rho }\right)\text{diag}\left(\mathbf{g}\right)\right){\mathbf{d}}^k-\left({\mathbf{g}}^k-\mathbf{D}{\mathit{\rho }}^k\right)$$ (11)
4. Set $\Delta {\tilde{\text{g}}}^{k+1}={\text{g}}^k+\Delta {\text{g}}^k$ and calculate

   $${\mathbf{g}}^{k+1}=P_{\cdot \infty \le 1}\left(\Delta \stackrel{k+1}{\mathbf{g}}\right)$$ (12)

   where $P_{.\infty \le 1}(\cdot )$ is the orthogonal projection in the l_∞ ball
5. Find the lowest integer j ≥ 0 such that the function f (ρ) is sufficiently decreased along d^k:

   $$f\left({\mathit{\rho }}^k+c_2^j{\mathbf{d}}^k\right)⩽f\left({\mathit{\rho }}^k\right)-c_1{c}_2^j{\Vert {\mathbf{d}}^k\Vert }_{x^k}^2$$ (13)

   where 0 < c₁ < 0.5, 0 < c₂ < 1, and set $\alpha =c_2^j$

6. Set ${\mathit{\rho }}^{k+1}={\mathit{\rho }}^k+\alpha {\mathit{d}}^k$

More detailed information about this method can be found in (Dassios et al 2015).

Though the pdNCG method improves the image quality of CB-XLCT, the convergence speed of pdNCG decreases with an increasing number of iterations. To alleviate this, we employed a two-level iterative step algorithm with a restarted strategy (Shi et al 2013). In the inner iteration, equation (5) is computed by the pdNCG algorithm to obtain ρ^k, and the negative values in ρ^k are set to zero to get ${\rho }_{\ge 0}^k$. In the outer iteration, a new optimization matrix is constructed with the removal of the columns in A corresponding to the zero values in ρ^k to reset the permission region, where the NPs may distribute. With this technique, some parts of the imaging object are excluded for reconstruction. Thus, the optimization function can be transformed into

$${\mathit{\rho }}^{k+1}=\underset{{\mathit{\rho }}^k⩾0}{\min }\left\{{\Vert {\mathbf{A}}_{re}^k{\mathit{\rho }}^k-{\mathbf{\Phi }}_{\text{meas}}\Vert }^2+\lambda {\Vert {\mathit{\rho }}^k\Vert }_1\right\}$$ (14)

For every inner iteration, the initial value for pdNCG is set to ${\rho }_{\ge 0}^k$, which will increase the convergence speed of pdNCG. With the proper initial value and a permission region, the ill-posedness of the inverse problem can be alleviated, and the reconstruction time can be reduced by the restarted strategy. The implementation of re-pdNCG is summarized in figure 1.

Figure 1.

Flow chart of the re-pdNCG method.

Both an L2-regularized method (Adaptik) and an L1-regularized method (FISTA) were implemented for comparison. Considering the negative elements of ρ are physically impossible, a non-negative constraint was added to both the Adaptik and FISTA methods. The regularization parameters and iteration numbers were set according to Gao et al (2018a). The parameters of the re-pdNCG method were chosen empirically by comparing the reconstruction results to the actual shape and position of the luminescent targets. With this experimentation, we found the optimal parameters k_inner, k_outer, λ, µ to be 5, 10, 10^–1 and 10^–5, respectively.

3. Experimental setup

3.1. Numerical simulations

Numerical simulations were first implemented to test the performance of the proposed method. A digital mouse was placed on the rotation stage, and the rotational axis was defined as the z-axis and the bottom plane was set as Z = 0 cm. The mouse was 2.6 cm in height and contained the primary organs (heart, lungs, liver, spleen, bone, and stomach) (see figure 3(a)). Two cylindrical tumors with a height of 3 mm and a diameter of 4 mm were placed in the liver (see figures 2(a) and (b)). The scattering and absorption coefficients were assigned to the organs according to Alexandrakis et al (2005). In the simulation, the cone-beam x-ray source was assumed to be monochromatic with an energy of 24 keV, which corresponds to the average effect of a polychromatic x-ray source of around 60 kV. The x-ray attenuation coefficient was set to be 0.5 cm⁻¹, according to Zhang et al (2017).

Figure 3.

Schematic diagram (a) and photo (b) of the CB-XLCT imaging system. A stage is positioned in the center of the hybrid imaging system.

Figure 2.

Schematic of the 3-D digital mouse model used for numerical simulations. (a) Digital mouse with main organs. The investigated region is 2.6 cm in height. Different organs are represented by different colors. (b) 3-D display of the tumor locations. The black circles indicate the central slices of the tumor.

To evaluate the imaging performance with the re-pdNCG method, three cases of experiments have been conducted, where the two tumors were separated with an EED of 0.3 cm, 0.2 cm, and 0.1 cm (cases 1–3). The mouse model was rotated by 360°, with an angular increment of 90° (with a total of four projections acquired). Then, to make the simulation more realistic, zero-mean white Gaussian noise was added to generate noisy boundary measurements with a signal-to-noise ratio (SNR) set to 20 dB (Gao et al 2018a).

3.2. Phantom experiments

In this work we used an X-RAD SmART Small Animal IGRT research system for performance evaluation. As shown in figure 3, a microfocus x-ray source (Comet MXR225/22, Switzerland) with maximal voltage of 225 kV and maximal power of 640 W was used to irradiate the imaging object. The photon signals from the incorporated NPs were acquired by a high sensitively electron-multiplying charge-coupled device (EMCCD) camera (iXon Ultra 897, USA) with a camera lens (Nikkor 50 mm f/1.8D), which was placed at a right angle to the x-ray beam. A CMOS flat panel detector (PerkinElmer XRD 0822 xP3, USA) was used to acquire transmitted x-rays. All the phantoms were fixed on the stage, and the x-ray source, x-ray detector, and EMCCD were rotated by 360° with the gantry.

The x-ray tube current and voltage were set to 40 mA and 60 kV, respectively. The EM gain, integrating time and binning of EMCCD were set to 260, 1 s, and 1 × 1, respectively. For luminescent imaging, four images were collected with a 90° interval. For CT imaging, 200 images were collected with a 1° interval.

Two cases of phantom experiments were conducted. In both cases, the phantom consisted of a cylinder (diameter of 2.8 cm) containing water and intralipid with µ_a = 0.02 cm⁻¹ and ${\mu }_s^{\text{'}}=10.0\hspace{0.17em}{\text{cm}}^{-1}$ (Gao et al 2018a). Two small glass tubes filled with x-ray-excitable perovskite materials (CsPbI₃) were immersed in the phantom. The luminescence property of CsPbI₃ was investigated, as shown in figure 4. The emission spectrum of CsPbI₃ NPs with an excitation tube current of 40 mA is shown in figure 4(a), and the luminescence intensity light output vs x-ray tube current is shown in figure 4(b). It can be seen from figure 4(b) that there is strong linearity between light output and x-ray tube current.

Figure 4.

X-ray luminescence spectrum of CsPbI₃ NPs. (a) Emission spectrum of CsPbI₃ NPs with an excitation tube current of 40 mA. (b) Luminescence intensity vs x-ray tube current.

The outside diameter of the tube was 0.3 cm. The EEDs of the two tubes were set to 0.22 cm (case 1) and 0.16 cm (case 2). Figure 5 summarizes the CT results of the phantom experiments. Figures 5(a) and (b) are the representative x-ray projections of the phantom in case 1 and case 2, respectively. The region between the green and red lines was selected for the XLCT reconstruction, which was 2.6 cm in height. Figures 5(c) and (d) are the CT slices indicated by the blue lines shown in figures 5(a) and (b), respectively.

Figure 5.

Illustration of the phantom experiments. (a) and (b) are the representative x-ray projections of the phantom in case 1 and case 2, respectively. (c) and (d) are the CT slices indicated by the blue lines shown in 5(a) and (b), respectively.

3.3. Quantitative evaluation

Images reconstructed with the different methods were compared in terms of location error (LE), shape recovery, and spatial resolution with the LE, Dice similarity coefficient (DICE), and spatial resolution index (SPI), respectively.

LE is defined as the Euclidean distance between the reconstructed and true positions of the targets:

$$\text{LE}={\Vert {\mathbf{p}}_r-{\mathbf{p}}_t\Vert }_2$$ (15)

where p_r and p_t denote the centers of the reconstructed and true targets, respectively.

DICE denotes the similarity between the reconstructed and true luminescent areas to assess the quality of morphological reconstructions:

$$\text{DICE}=\frac{2\left|\mathbf{RO}{\mathbf{I}}_r\cap \mathbf{RO}{\mathbf{I}}_t\right|}{\left|\mathbf{RO}{\mathbf{I}}_r\right|+\left|\mathbf{RO}{\mathbf{I}}_t\right|}$$ (16)

where ROI_r and ROI_t denote the reconstructed and true luminescent areas, respectively. The DICE index increases with increased similarity between the reconstructed and true targets.

SPI is a spatial resolution quantitative index to analyze the ability of the algorithms to resolve the targets:

$$\text{SPI}=\frac{{\rho }_{\text{max}}^l-{\rho }_{\text{valley}}^l}{{\rho }_{\text{max}}^l-{\rho }_{\min }^l}$$ (17)

where ρ ^l denotes the value of the profile along a given line that connects the two centers on the reconstructed cross-section. ${\rho }_{\max }^l$, ${\rho }_{\text{valley}}^l$ and ${\rho }_{\min }^l$ are the maximal, valley, and minimal values between the two peak values, respectively. The higher the SPI value is, the more clearly the two targets are separated.

4. Results

4.1. Numerical simulations

The reconstructed 2-D slices obtained with the Adaptik, FISTA, pdNCG and proposed re-pdNCG algorithms are shown in figure 6. The two white circles in each slice denote the real positions of the luminescent targets. All the slices shown in figure 6 are taken from a height of Z = 1.3 cm. The different columns in figure 6 represent the reconstruction results obtained using the Adaptik, FISTA, pdNCG and re-pdNCG methods and the rows represent different EEDs.

Figure 6.

Reconstructed luminescent targets with different EEDs from simulation. The columns represent the tomographic slice reconstructed by Adaptik, FISTA, pdNCG, and re-pdNCG algorithms, respectively, and the rows represent the different EEDs between the two targets. The slices are those indicated with black circles in figure 2. The white circles represent the real positions of the luminescent targets. All the reconstructed images were normalized for better comparison.

The results indicate that all three methods can achieve high contrast and high spatial resolution when the EED is 0.3 cm. As the EED reduces to 0.2 cm, the Adaptik method cannot resolve the two adjacent targets, which is most likely due to the fact that the L2-norm regularization method results in an over-smoothing effect, as shown in figure 6(e). With sparsity as prior, the L1-norm-regularization-based FISTA and pdNCG methods can separate the two targets much more clearly than the Adaptik method, but the spatial resolution is lower than that obtained with the proposed method, as shown in figure 6(h). When the EED is 0.1 cm, it can be seen from figures 6(i)–(k) that both the L2-norm and the L1-norm regularization method are unable to distinguish the targets. In contrast, the proposed re-pdNCG method can easily recover the distributions of the NPs as shown in figure 6(l).

Figure 7 shows the 3-D rendering of the simulated XLCT results. The blue objects represent the recovered targets. As expected, the proposed re-pdNCG method can distinguish the two targets clearly even when the EED reduces to 0.1 cm, while this was not feasible with the Adaptik, FISTA and pdNCG methods.

Figure 7.

3-D results of the reconstructed XLCT images with different EEDs in simulations. The columns represent the tomographic slice reconstructed by Adaptik, FISTA, pdNCG, and re-pdNCG algorithms, respectively, and the rows represent the different EEDs between the two targets. The blue objects represent the recovered targets.

Table 1 shows the quantitative analysis results of the reconstructions (EED = 0.1 cm) obtained with the three methods. The proposed re-pdNCG method achieves the lowest LE, highest DICE, and highest SPI of the three methods, which indicates that the re-pdNCG method reconstructs the target with the best localization accuracy, shape recovery, and spatial resolution.

Table 1.

Quantitative evaluation of simulations with EED = 0.1 cm.

	LE (mm)		DICE
	Tube 1	Tube 2	Tube 1	Tube 2	SPI
Adaptik	1.08	3.72	0.67	0.28	0.13
FISTA	0.60	4.21	0.60	0.24	0.21
pdNCG	0.54	4.03	0.63	0.30	0.18
re-pdNCG	0.48	0.65	0.72	0.58	0.88

4.2. Phantom experiments

Figure 8 shows the XLCT reconstruction results of the phantom experiment of case 1, which had an EED of 0.22 cm. All the images are normalized to the maximum intensity value. Figures 8(a)–(d) are the XLCT slices (indicated by the blue line slice in figure 5(a)) reconstructed with the Adaptik, FISTA, pdNCG and re-pdNCG methods, respectively. Figures 8(e)–(h) are the XLCT/CT fusion results, and figures 8(i)–(l) are the 3-D rendering results. The L2-norm-based Adaptik method shows an over-smoothed recovery, and it cannot separate the targets (see figures 8(a) and (e)). Though the L1-norm-based FISTA method can achieve better shape recovery than Adaptik, the spatial resolution is low (see figures 8(b) and (e)). In contrast, both the pdNCG and the proposed re-pdNCG method can separate the two targets clearly, and the re-pdNCG method achieves relatively lower LE and better shape recovery (see figures 8(g) and (h)). It can be further confirmed that the proposed method outperforms the others by the 3-D results (see figures 8(g)–(i)).

Figure 8.

Reconstructed luminescent targets of phantom experiment case 1. The columns represent the tomographic slice reconstructed by Adaptik, FISTA, pdNCG and re-pdNCG algorithms, respectively, and the rows represent the reconstructed XLCT, XLCT/CT fusion, and 3-D results, respectively. The red circles in the first two rows depict the boundaries of the phantom. The blue objects represent the recovered targets, which can be obtained using an isosurface value equal to 10% of the maximum value of the volume data.

Figure 9 gives the XLCT reconstruction results of the phantom experiment of case 2, which had an EED of 0.16 cm. All the images are normalized to the maximum intensity value. Figures 9(a)–(d) are the XLCT slices (indicated by the red-line slice in figure 5(b)) reconstructed with the Adaptik, FISTA, pdNCG and re-pdNCG methods, respectively. Figures 9(e)–(h) are the XLCT/CT fusion results, and figures 9(i)–(l) are the 3-D rendering results. Similar to the results of case 1, both the L2-norm-based Adaptik method and the L1-norm-based FISTA method are unable to achieve high image quality considering the localization accuracy, target shape, and spatial resolution (table 2). Both the pdNCG and re-pdNCG methods can get better results than Adaptik and FISTA, and the re-pdNCG method can obtain the least LE, which is demonstrated by both the tomographic (see figures 9(g) and (h)) and 3-D results (see figures 9(k) and (l)).

Figure 9.

Reconstructed luminescent targets of phantom experiment case 2. The columns represent the tomographic slice reconstructed by Adaptik, FISTA, and re-pdNCG algorithms, respectively, and the rows represent the reconstructed XLCT, XLCT/CT fusion, and 3-D results, respectively. The red circles in the first two rows depict the boundaries of the phantom. The blue objects represent the recovered targets, which can be obtained using an isosurface value equal to 10% of the maximum value of the volume data.

Table 2.

Quantitative evaluation of phantom experiments.

		LE (mm)		DICE
Case #	Method	Tube 1	Tube 2	Tube 1	Tube 2	SPI
1	Adaptik	2.98	3.50	0.11	0.08	0.02
	FISTA	3.62	2.81	0.15	0.17	0.20
	pdNCG	0.66	1.23	0.50	0.33	1.00
	re-pdNCG	0.28	1.19	0.54	0.41	0.91
2	Adaptik	2.83	2.70	0.07	0.14	0.12
	FISTA	2.04	3.93	0.07	0.13	0.02
	pdNCG	1.06	0.49	0.36	0.64	1.00
	re-pdNCG	0.53	0.97	0.54	0.61	0.96

Table 2 shows the quantitative analysis results of the phantom experimental reconstructions obtained with the four methods. Similar to the numerical simulations, the pdNCG and re-pdNCG methods yield lower LE, and higher DICE and SPI, than the Adaptik and FISTA methods. The average LE, DICE and SPI obtained by pdNCG and re-pdNCG are close, indicating that the two methods can achcieve similar imaging performance in our phantom experiments.

Table 3 shows the runtime and iteration number (outer iteration for re-pdNCG) for different methods. It indicates that the re-pdNCG method costs the least computation time, which means it is more efficient than the other methods.

Table 3.

Runtime and iteration number for different methods.

Case #	Method	Runtime (s)	Iteration number
1	Adaptik	113.9	15
	FISTA	191.3	500
	pdNCG	80.3	30
	re-pdNCG	23.1	10
2	Adaptik	111.3	15
	FISTA	194.5	500
	pdNCG	83.2	30
	re-pdNCG	21.4	10

5. Discussion and conclusion

As a promising hybrid imaging technique, CB-XLCT has its unique advantages for non-invasive biomedical imaging. However, the CB-XLCT reconstruction is a severely ill-posed problem, leading to inferior image quality. In this work, we propose a restarted strategy for CB-XLCT reconstruction, named re-pdNCG, which aims to mitigate the ill-posedness of the inverse problem to achieve improved spatial resolution in the reconstructed images while keeping the runtime low. The proposed algorithm is composed of inner and outer iterations. In the inner iterations, the pdNCG method was used to solve the L1-norm regularization optimization problem, which introduced the pseudo-Huber function to replace the nondifferentiable L1-norm. As the pdNCG method requires only matrix–vector product operations, the runtime is low. The solution of every inner iteration can be viewed as a rough permission region, which was used to construct a new weight matrix by the restarted strategy in the outer iteration. Owing to the reset permission region, the ill-posedness was alleviated. Moreover, the inner iteration results provided initial values for the newly constructed objective function that can improve the convergence speed of the pdNCG method.

The simulation (cases 1–2) and phantom experiment results reveal that two adjacent NPs can be accurately resolved with the pdNCG and re-pdNCG methods when the EED of the two tubes is larger than 1 mm. However, this is not the case with the widely used Adaptive and FISTA methods, where the adjacent targets could not be distinguished. The close quantitative indexes obtained by the pdNCG and re-pdNCG methods in table 2 indicate that the two methods can achieve similar performance when the EED is larger than 1 mm. However, as the EED of the two tubes decreases to 1 mm (case 3 in simulation), only the re-pdNCG method can resolve the two targets (see figures 6(l) and 7(l)), which demonstrates the ability of the re-pdNCG method to improve the spatial resolution of CB-XLCT. Furthermore, the re-pdNCG method outperforms the pdNCG method in all cases in terms of convergence speed (see table 3).

There are a few issues to be addressed. Firstly, it should be noted that the regularization parameters and the iteration numbers are crucial to the imaging performance. In this paper, the settings for the proposed method were set manually by excluding suboptimal parameters in proper ranges according to the results. For the Adaptive and FISTA methods, the parameters were selected according to our previous study (Gao et al 2018a). Automatic parameter selection will be our future work. Secondly, the imaging process can be simplified by reducing the luminescent projections and collecting x-ray transmission and luminescent data simultaneously. With a limited-view imaging strategy or by adding a mirror (Yang et al 2018), the imaging views can be decreased. Thirdly, more priors may be applied to get better results, such as group sparsity (Jiang et al 2019) or smooth priors like total variation (TV) (Liu et al 2019a).

In conclusion, we propose a restarted strategy for CB-XLCT reconstruction. With the proposed re-pdNCG reconstruction method, high imaging quality and fast reconstruction speed can be achieved. The method outperformed conventional techniques and represents an important step in the wide application of CB-XLCT in biomedical imaging. We mention that some research for the in-vivo imaging is still on-going, which includes the optimization of in-vivo stability and biocompatibility of the NPs, along with the cytotoxicity study of the nanophosphors. Future work will be focused on in-vivo imaging with this method.