1.

INTRODUCTION

Comparing with the conventional computed tomography (CT), photon-counting CT (PCCT) can obtain multiple measurements of the scanned object at multi-energy bins and provide abundant energy-dependent material-specific information. Due to the energy discrimination capability, PCCT can effectively improve contrast-to-noise ratio, increase the dose efficiency and reduce electronic noise.^{1, 2}

However, the collected PCCT measurements in the narrow energy bins are corrupted by serious quantum noise,³ and the PCCT image quality degrades obviously due to the limited photons. To solve this problem, many statistical iteration reconstruction (SIR) methods have been proposed in the past decades. The main idea of the SIR is to construct a reconstruction model with data fidelity and regularization terms, where the first term incorporates the statistical property of X-ray photons and the second term provides the prior information of the desired PCCT images. For example, Rigie et al. introduced a total nuclear variation regularization to leverage similar gradient information and improve the image quality.⁴ Kim et al. developed a patch-based low-rank regularization to maintain the image structures and reduce noise.⁵ Semerci et al. combined a tensor nuclear norm and a total variation regularization to suppress noise.⁶ Zhang et al. proposed to deliver inner spectrum correlation information and constructed a tensor-based dictionary learning strategy.⁷ Niu et al. considered the self-similarity of the spectral CT images and proposed a non-local low-rank and sparse matrix decomposition method.³ Wu et al. proposed to encourage the similarity of spectral CT images by utilizing a cube-based tensor regularization.⁸ Recently, Zeng et al. proposed a full-spectrum-knowledge-aware tensor by imposing the global correlation, piecewise smooth and latent full-spectrum properties of PCCT images.⁹ These methods have been shown great potential in preserving image details and suppressing noise. However, there remain some challenges in practice. First, the SIR methods usually utilize fan-beam geometry and the computation costs will be a burden for cone-beam geometry. Second, SIR methods are sensitive for the hyper-parameters, and appropriate parameters selection is needed for different clinical applications.

Recently, deep learning (DL) technology has been widely adopted in CT imaging field. In spectral CT imaging field, Lu et al. utilized DL-based method for material decomposition.¹⁰ Fang et al. proposed to remove ring artifacts for PCCT data by using DL-based method.¹¹ Wu et al. employed a DL-based method for reconstructing PCCT images.¹² It is shown that the DL-based methods achieved competing results compared with the SIR methods. However, the current DL-based methods need a large quantity of paired data (i.e., noisy and high-quality data) to obtain a desired model by supervised training strategy. Moreover, collecting large-scale spectral CT data is time-consuming, and the PCCT data in clinics is hard to be obtained. By the way, a number of energy-integrating detector (EID) data, which is easily to be obtained, is not yet included in training the DL-based method for PCCT imaging.

Therefore, we present an unsupervised DL-based method in the image domain by utilizing the prior information of the paired EID dataset (i.e., low-dose images/high-quality ones). Specifically, we first initialize supervised and unsupervised networks for EID and PCCT images, respectively. Then, the supervised network is trained on well paired EID dataset and serves as the prior knowledge to regularize the unsupervised PCCT network training. Moreover, a data-fidelity term for characterizing the PCCT image characteristics is constructed as the self-supervised loss. Finally, with the prior knowledge and self-supervised terms, we can train the network for PCCT images following an unsupervised learning strategy. We call the presented DL-based model as full-spectrum-knowledge-aware DL-based network, shorten as “FSANet”. We evaluate the presented FSANet and other reconstruction methods on synthesized clinical data. Experimental results demonstrate that the presented FSANet outperforms the competing methods in terms of qualitative and quantitative metrics.

2.

METHODS

Considering the spatial and energy dimensions of the spectral data, the PCCT imaging model can be expressed as follows:

where y = {y_n, n ≤ N} and X* = are the measurements and desired PCCT images along the multi-energy bins, and N is the total number of the energy bins. A is the linear projection operator for PCCT imaging, and ε denotes the noise in the projection domain.

The images reconstructed from Y suffer from noise and artifacts. To improve image quality, DL-based methods are feeded with the PCCT images and produce the denoised ones. It can be expressed as follows:

where represents the network of the DL-based method with parameters θ_DL. are the estimated PCCT images by the network. X = (A^TA)⁻¹ A^TY) are the network inputs which are directly reconstructed by the filtered back projection (FBP) algorithm.¹³ Followed by the supervised strategy, the network parameters θ_DL are optimized by minimizing the loss function between the target images X^target and which can be expressed as follows:

where L is the user defined loss function and X^target are the FBP-reconstructed images from the high-dose measurements.

In order to utilize the prior information of the network pre-trained on paired EID dataset, we present a full-spectrum-knowledge-aware (FSA) loss function, as follows:

where denotes a pre-trained network for EID images with parameters θ_pre. is the network for PCCT images with parameters θ. T (·) is an operator to transform the PCCT images to the EID images. || · ||₁ is L₁ norm. Moreover, we also construct a self-supervised loss to encourage the data fidelity of the PCCT images, as follows:

In summary, the total loss function of the presented model is expressed as follows:

where α is a hyper-parameter of the loss function. It can be seen that the presented model engages an unsupervised learning strategy where only noisy PCCT images is involved in the training stage. Fig. 1 illustrates the presented FSANet method for PCCT image recovery. Finally, we utilize Adam¹⁴ to optimize the parameters in the network.

Figure 1.

Illustration of the presented FSANet method for PCCT image recovery. The pipeline with black arrows denote the data flow of the PCCT images. The pipeline with orange arrows denote calculations of loss functions. The supervised and unsupervised networks have the same architecture. It should be noted that the optimization for the parameters of the unsupervised network follows an unsupervised strategy, and only noisy PCCT images are involved during its training period.

3.

RESULTS

3.2

Qualitative Analysis

Fig. 2 illustrates the visual comparisons of the presented and compared methods on Case 1. The images at normal dose are chosen as the ground truth. It can be observed that FBP algorithm suffers from noise. TDL effectively removes the noise-induced artifacts and improves the image quality, but losses the image resolution. On the contrary, the presented FSANet produces more remarkable results closing to the Supervised Net and the ground truth in terms of noise reduction and structure preservation. Moreover, zoomed in regions-of-interest (ROIs) by the blue boxes in Fig. 2 is selected for better visual inspection. It can be seen that TDL smooths the structure edges, but the Supervised Net and presented FSANet methods maintain the image details.

Figure 2.

Results of the presented and compared methods on Case 1. The display windows from Bin 1 to 5 are [0.005, 0.008], [0.0028, 0.005], [0.002, 0.004], [0.0015, 0.0035] and [0.0015,0.0028] mm⁻¹, respectively. Zoomed ROIs indicated by the blue boxes are displayed for better visualization.

Fig. 3 shows the results of different methods on Case 2. Similar to Case 1, FBP induces noise to the images, TDL is prone to produce blurry results, and the presented FSANet avoids over smoothing and preserves structure details. The zoomed in ROIs indicated by the red boxes in Fig. 3 also demonstrate the advanced performance of the presented FSANet method. Fig. 4(a) and (b) show the profiles indicated by the green and orange lines in the Fig. 2 and Fig. 3, respectively. From the results, we can observe that the presented FSANet produces the closet results to the ground truth compared with the FBP and TDL methods.

Figure 3.

Results of the presented and compared methods on Case 2. The display windows from Bin 1 to 5 are [0.0015, 0.0025], [0.0009, 0.0018], [0.0007, 0.0015], [0.0006, 0.0012] and [0.0005, 0.0010] mm⁻¹, respectively. Zoomed ROIs indicated by the red boxes are displayed for better visualization.

Figure 4.

Profiles of the green and orange lines in Bin 1 for Case 1 and 2, respectively. (a) are the profiles of the results from different methods on Case 1, respectively. (b) are the profiles of the results from different methods on Case 2, respectively.

4.

DISCUSSION AND CONCLUSION

DL-based methods have been shown promising performance in conventional CT imaging, and it has also inspired the application in PCCT imaging. However, most of them are supervise-based and need a large quantity of paired training dataset, which is hard to be obtained for PCCT. To address this intrinsic limitation, in this work, we presented a DL-based PCCT denoising method with an unsupervised learning strategy, called “FSANet”.

Specifically, we first trained a denoising network on paired EID dataset and served it as a prior for PCCT images. Then, we constructed this prior network as a training loss to regularize the PCCT network learning in an unsupervised manner. Moreover, a self-supervised loss of the noisy PCCT images is introduced to promote data-fidelity of the PCCT images. Finally, with the mentioned two loss terms, we obtained the presented FSANet method. Simulation experiments demonstrated the feasibility and effectiveness of the presented FSANet method. In the future, clinical patient studies would be involved to further demonstrate the denosing performance of the presented FSANet method.