Improved deep convolutional dictionary learning with no noise parameter for low-dose CT image processing

Abstract

Background:

Low-Dose computed tomography (LDCT) reduces radiation damage to patients, however, the reconstructed images contain severe noise, which affects doctors’ diagnosis of the disease. The convolutional dictionary learning has the advantage of the shift-invariant property. The deep convolutional dictionary learning algorithm (DCDicL) combines deep learning and convolutional dictionary learning, which has great suppression effects on Gaussian noise. However, applying DCDicL to LDCT images cannot get satisfactory results.

Objective:

To address this challenge, this study proposes and tests an improved deep convolutional dictionary learning algorithm for LDCT image processing and denoising.

Methods:

First, we use a modified DCDicL algorithm to improve the input network and make it do not need to input noise intensity parameter. Second, we use DenseNet121 to replace the shallow convolutional network to learn the prior on the convolutional dictionary, which can obtain more accurate convolutional dictionary. Last, in the loss function, we add MSSIM to enhance the detail retention ability of the model.

Results:

The experimental results on the Mayo dataset show that the proposed model obtained an average value of 35.2975 dB in PSNR, which is 0.2954 –1.0573 dB higher than the mainstream LDCT algorithm, indicating the excellent denoising performance.

Conclusion:

The study demonstrates that the proposed new algorithm can effectively improve the quality of LDCT images acquired in the clinical practice.

Keywords

Low-dose computed tomography DenseNet deep learning convolutional dictionary learning.

1 Introduction

With the development of medical technology and the improvement of people’s health awareness, LDCT technology has entered the spotlight gradually. During CT scanning, the higher the radiation dose, the greater the damage to the body, which has caused widespread health concerns among researchers [1 –3]. LDCT technology has achieved a large number of research achievements in the medical field and has a wide range of applications. At present, the most commonly used method to reduce the radiation energy is to reduce the X-ray current intensity. Although LDCT technology reduces radiation damage to patients, its reconstructed images contain severe mottle noise and streak artifacts, and the lower the X-ray energy, the stronger the noise, which seriously affects the doctor’s diagnosis of the disease [4]. Researchers carry out denoising processing in three aspects: projection domain noise reduction, iterative reconstruction noise reduction, and post-processing algorithm noise reduction. Among them, the post-processing algorithm has the advantages of easy portability and not requiring projection data, etc.

In this paper, we solve the problem of severe noise of LDCT images from the perspective of the post-processing algorithm. The post-processing algorithm does not need to acquire the projection data and denoising the reconstructed images directly. However, there is no accurate noise distribution in the reconstructed images, which is a difficult point in the denoising process of LDCT images. In general, image-denoising algorithms are divided into traditional methods and deep learning methods. Traditional methods often have clear physical meaning and strict mathematical reasoning, such as K-SVD [5], partial differential equations (PDE) [6, 7], NLM [8], BM3D [9] algorithms, etc. The development of computer hardware and the excellent ability of convolutional neural networks (CNN) to extract features have enabled deep learning algorithms to achieve dazzling results in image processing. Numerous deep learning algorithms outperform traditional methods, such as ResNet [10], U-Net [11], DenseNet [12], etc., but they have the drawback of unclear physical meaning, frequently described as “black box”. Therefore, some researchers proposed combining traditional methods with CNN to solve the problems in image processing. Meyer Scetbon et al. [13] proposed the LKSVD algorithm to build a bridge between traditional methods and deep learning. They used a neural network approach to calculate the parameter λ in the K-SVD algorithm, which improve the denoising performance. Hongyi Zheng et al. [14] analyzed the current situation of combining traditional methods with deep learning, then they made full use of the ability of deep learning to extract features, meanwhile strictly adhering to the solution process of the convolutional dictionary learning algorithm and proposed the deep convolutional dictionary learning algorithm (DCDicL). The denoising performance of the DCDicL method is improved substantially. When processing the LDCT images, it is necessary to improve the traditional methods, such as Zhoubo Li et al. [15] improved the 3D NLM algorithm, which processes LDCT images using the method of local noise estimation adaptively; Tingting Zhao et al. [16] derived a method based on data statistics, using the minimum mean square error criterion of Wiener filter to avoid the standard deviation of input LDCT image noise, which improved the BM3D algorithm and achieved excellent performance in the subsequent emphysema quantification task; Lina Jia et al. [17] found that the K-SVD algorithm could not accurately perform sparse representation of LDCT images using fixed-size dictionary atoms, and proposed a sparse representation denoising algorithm of Grouped Dictionaries with Adaptive Atom Sizes, which had great denoising performance on parts of LDCT images, such as edges and texture details; Yang Chen et al. [18] divided the LDCT images into frequencies, then trained three discriminative dictionaries in the high-frequency region to eliminate the artifacts with orientation and scale information, which improved the quality of the LDCT images effectively; Yi Liu et al. [19] proposed a new PDE method for LDCT image processing, combining image gradient, local variance and residual image to improve the diffusion function, which suppressed the mottle noise and streak artifacts effectively.

Deep learning methods have achieved excellent performance in the field of LDCT image noise reduction, the WavResNet [20] achieved second place in the 2016 AAPM Low-Dose CT Challenge. After that, many deep learning methods for processing LDCT images, the WGAN [21] and REDCNN [22] algorithms improved LDCT image quality significantly; Edge enhancement-based EDCNN [23] and quadratic autoencoder (Q-AE) networks [24] were proposed; Recently, vision transformers based CT former [25] was applied to LDCT image noise reduction. Deep residual convolutional sparse coding networks (DRCSC) [26] were proposed for processing LDCT images, which consist of three components: input feature maps prepare, recursive manner for feature maps learning by convolutional sparse coding, and high-frequency information recover. DRCSC can restrain artifacts and noise for LDCT imaging. Driven by a large amount of data, the processing results of deep learning methods have been more significantly improved in both subjective visual effects and objective evaluation metrics, compared with model-driven traditional algorithms.

Inspired by the excellent denoising performance achieved by the DCDicL algorithm [14]. In this paper, we propose an improved deep convolutional dictionary learning (IDCDicL) algorithm with no noise parameter, which uses the DenseNet121 to replace the shallow convolutional network, learning prior on convolutional dictionary more accurately, thus, generating more accurate convolutional dictionary; Meanwhile, because the noise parameter of LDCT image is difficult to estimate, we modify the input network to avoid input noise parameter, thus improve the convenience, and add convolutional layers in the input network to expand the perceptual field, thus extract more features of LDCT image; Finally, adding the MSSIM loss in loss function to improve the ability of the IDCDicL to protect the texture details of LDCT images. Experimental results show that the IDCDicL algorithm effectively suppresses mottle noise and streak artifacts in LDCT images while protecting important details, meeting and exceeding the denoising performance of some state-of-the-art LDCT network algorithms.

2 Related works

2.1 Convolutional dictionary learning

Dictionary learning is a patch-based image method, which lacks the shift-invariant property. Convolutional dictionary learning [27] solves the shift-invariant problem by replacing the matrix multiplication with the convolution operation, the mathematical expression is: $\begin{matrix} \underset{D, {M_{i}}}{arg min =} \frac{1}{N} \sum_{i = 1}^{N} \frac{1}{2} | | D \otimes M_{i} - X_{i} | |_{2}^{2} & s . t . λ_{A} ψ (M_{i}), λ_{D} γ (D) \end{matrix}$ (1) where D is the convolutional dictionary, which is a set of filters. M_i is the convolutional coefficient matrix, which can be solved using convolutional sparse coding (CSC) [28]. ψ (M_i) and γ (D) are priors on the convolutional coefficient matrix and the convolutional dictionary. Convolutional dictionary learning methods have more applications in image processing [29, 30].

2.2 Residual network

The representative works on residual networks mainly include ResNet [10] proposed by Kaiming He and DenseNet [12] network proposed by Gao Huang. The residual learning proposed by ResNet, which overcomes the network degradation problem that occurs in deep convolutional networks, and identity mapping attenuates the problem of gradient disappearance in the back-propagation of the network, thus enhancing the ability of the network to extract features. DenseNet network goes further, each layer of the network is closely connected with other layers of the network, which achieves feature transfer between the higher layer network and the lower layer network. This structure enhances feature reuse and enhances the ability of the network to extract features further. The main difference between these two networks is the model framework and network connection way, as shown in Fig. 1.

Fig. 1

Different connection methods of the ResNet network and DenseNet network.

3 Methods

3.1 Weaknesses of the DCDicL method for processing LDCT images

The DCDicL method achieves excellent performance for processing Gaussian noise. It adheres to the mathematical derivation of convolutional sparse representation, meanwhile using the powerful feature extraction ability of deep learning. The DCDicL algorithm learns the prior on the convolutional coefficient matrix and convolutional dictionary with deep learning, which can overcome the drawback of manually selecting prior information. At present, the DCDicL algorithm is one of the best denoising algorithms to deal with Gaussian noise. However, applying it to LDCT images has three problems. 1): Because the goal of DCDicL is to denoising Gaussian noise, the noise intensity is needed in the input network, and the noise intensity seriously affects the effectiveness. If the noise intensity is set too small, the noise in the image cannot be completely denoised, causing the problem of noise residue; If the noise intensity is set too large, it leads to over-smoothing processing results and even loss of image texture and structural details. Using the pre-trained DCDicL model to process one image with a noise standard deviation of 25 from the WED dataset [31], the processing results and the regions of interest (ROI) are shown in Fig. 2. When the noise intensity is set to 15, the ROI has the problem of noise residue, when the noise intensity is set to 35, the texture information of the ram’s horn in the ROI is over-smoothing. When the noise intensity is set to 25, the ROI achieves a great denoising performance, which indicates the important influence of the noise intensity parameter on the DCDicL algorithm. 2): The subnetwork for learning the prior on the convolutional dictionary in the DCDicL is a shallow convolutional network, and the prior on the convolutional dictionary has an important impact on the LDCT image processing. LDCT images have complex organ tissue structure and rich texture details, selecting a shallow convolutional network does not extract enough prior on the convolutional dictionary, which affects the final denoising performance; 3): DCDicL uses the L1 loss as the parameters update criterion, which tends to over-smoothing in processing results, thus cannot conducive to the protection of image structure and detail information.

Fig. 2

The processing results of DCDicL with set different noise intensity.

3.2 The proposed model

Inspired by the excellent denoising performance of DCDicL, we improve the input network architecture to delete the noise intensity parameter, which enhances the convenience of the algorithm. Meanwhile, increasing the number of convolutional layers in the input network to improve the denoising performance; Then, using the DenseNet121 to replace the shallow convolutional network that learns the prior on the convolutional dictionary, thus obtaining a more accurate adaptive convolutional dictionary; Adding the MSSIM loss in the loss function, which can enhance the ability of the model to protect the structure and texture details of LDCT images.

In the process of solving the objective function, we improve the neural network structure to find a more accurate convolutional sparse matrix M and convolutional dictionary D, the mathematical equation of the objective function is expressed as: $min_{{D_{i}, M_{i}}} \frac{1}{N} \sum_{i = 1}^{N} \frac{1}{2} | | D_{i} \otimes M_{i} - X_{i} | |_{2}^{2} + λ_{A} ψ (M_{i}) + λ_{D} γ (D_{i})$ (2) where D_i is the convolutional dictionary. Unlike traditional convolution dictionary that uses the common convolution dictionary for all images, DCDicL sets adaptive convolution dictionaries according to the image characteristics, which has better denoising performance. The solution to Equation (2) can be divided into two parts: $min \frac{1}{N} \sum_{i = 1}^{N} L (D_{i} \otimes M_{i}, X_{i}^{gt})$ (3) $s . t . {D_{i}, M_{i}} = arg min_{D, M} \frac{1}{2 δ_{i}^{2}} | | D \otimes M - X | |_{2}^{2} + λ_{A} ψ (M) + λ_{D} γ (D)$ (4) where δ is the noise standard deviation, which has an important effect on denoising performance. Equation (3) is used as the loss function to optimize the parameters. Equation (4) is the objective function to solve the convolution dictionary and the convolution coefficient matrix, which is decomposed into data terms and prior terms. The objective functions are obtained by adding two auxiliary variables M^p and D^p. The new objective functions are obtained:

equation 5a $min_{M^{p}, M} \frac{1}{2 δ^{2}} | | D \otimes M - X | |_{2}^{2} + λ_{M^{p}} ψ (M^{p}), s . t . M = M^{p}$ (5a)

equation 5b $min_{D^{p}, D} \frac{1}{2 δ^{2}} | | D \otimes M - X | |_{2}^{2} + λ_{D^{p}} γ (D^{p}), s . t . D = D^{p}$ (5b)

equation 6a Equation (5) is solved using the half-quadratic splitting algorithm [32], which is converted to the following objective functions: $min_{M, M^{p}} \frac{1}{2 δ^{2}} | | D \otimes M - X | |_{2}^{2} + λ_{M^{p}} ψ (M^{p}) + \frac{μ_{M}}{2} | | M - M^{p} | |_{2}^{2}$ (6a)

equation 6b $min_{D, D^{p}} \frac{1}{2 δ^{2}} | | D^{p} \otimes M - X | |_{2}^{2} + λ_{D} γ (D) + \frac{μ_{D}}{2} | | D - D^{p} | |_{2}^{2}$ (6b)

equation 7a Solving Equation (6) using the iterative algorithm: $\begin{matrix} M_{t} = {Solve}_{M} (X, D_{(t - 1)}, M_{t - 1}^{p}, α_{M^{p}}) \\ \begin{matrix} = arg min_{M} \frac{1}{2} \end{matrix} | | D_{(t - 1)} \otimes M - X | |_{2}^{2} + \frac{α_{M^{p}}}{2} | | M - M_{t - 1}^{p} | |_{2}^{2} \end{matrix}$ (7a)

equation 7b $M_{t}^{p} = {Net}_{M} (M_{t}, β M^{p})$ (7b)

equation 7c $\begin{matrix} D_{t} = {Solve}_{D} (X, D_{t - 1}^{p}, M_{t}^{p}, α_{D^{p}}) \\ \begin{matrix} = arg min_{D} \frac{1}{2} \end{matrix} | | D \otimes M_{t} - X | |_{2}^{2} + \frac{α_{D^{p}}}{2} | | D - D_{t - 1} | |_{2}^{2} \end{matrix}$ (7c)

equation 7d $D_{t}^{p} = {Net}_{D} (D_{t}, β D^{p})$ (7d) where {αM, αD, βM, βD} are the parameters, which related to the noise intensity and {μM, μD, λM, λD}. For the data terms, the convolutional coefficient matrix M in Equation (7a) is solved using Fast Fourier Transform according to the literature [28], the convolutional dictionary D in Equation (7c) can be solved using the least squares method. For the prior term M^p, D^p, and the parameters {αM, αD, βM, βD}, the solutions were performed using the deep convolutional neural network.

In Section 3.1, we present three weaknesses of DCDicL. Next, the paper will describe how to solve these three weaknesses and improve the quality of LDCT images.

The DCDicL algorithm needs the noise standard deviation in the input network HyperNet and HeadNet. In HyperNet, the noise standard deviation is taken as input, then four constraint parameters are obtained after two convolutional layers and the SoftPlus layer; In HeadNet, the noise standard deviation and noise image are taken as input, then the initial convolutional coefficient matrix is obtained after two convolutional layers. We consider that the noise intensity is the feature of the noise image, which can be learned from the noisy image directly. So we deepen the number of convolutional layers of the input network to obtain the required parameters from the noise image. In HyperNet, we take the noise image as the input to obtain the four constraint parameters through four convolutional layers, as shown in Fig. 3(a); In HeadNet, only the noise image is input, then through four convolutional layers to obtain the initial convolutional coefficient matrix, as shown in Fig. 3(b). By changing the input network and increasing the number of convolutional layers, which improves the feature extraction ability of the network, meanwhile, the problem of the DCDicL needs the noise standard deviation is solved.

Fig. 3

Improved subnetworks HyperNet and HeadNet.

DCDicL uses a shallow convolutional network NetD to learn the priori on convolutional dictionary. LDCT images contain severe mottle noise and streak artifacts, complex tissue and organ structures, in addition, LDCT images have poor image clarity and low contrast. The use of a shallow convolutional network cannot learn the priori on convolutional dictionary effectively, which affects the denoising performance. Hence, we take the advanced DenseNet to learn the prior on convolutional dictionary. Using the feature reuse property of DenseNet to enhance the ability to extract features, which can learn accurate prior on convolutional dictionary to improve the denoising performance. The improved subnetwork is shown in Fig. 4.

Fig. 4

The improved subnetwork Net_D.

equation 8 The L1 loss as the parameter update criterion in the loss function of DCDicL, which tends to make the denoising performance over-smoothing and not conducive to detail preservation. For LDCT images, which are rich in texture details, using L1 loss will lose structural information such as texture details. Therefore, we add the MSSIM loss to the loss function to enhance the protection of texture structure information. The improved loss function is:

$L = α L_{1} + (1 - α) L_{MSSIM}$ (8)

Through the above methods, we obtain the improved model IDCDicL, which improves the convenience of the algorithm without inputting noise parameter; Improving the network structure for learning the prior on convolutional dictionary, which can obtain the convolutional dictionary more accurately; Adding the MSSIM loss in the loss function to improve the denoising performance of the model for LDCT images.

The NetX learns the prior on coefficient matrix M, which consists of 7 blocks, each block consists of residual units. The first 3 blocks down-sample the feature maps through stride convolution, and the last 3 blocks up-sample the feature maps by transposed convolution. The subnetwork structure of NetX is shown in Fig. 5. LDCT images are pre-processed by the HeadNet, and the noise-reduced CT images are output after multiple Stages. The network structure of IDCDicL is shown in Fig. 6. To make it easier to understand the proposed algorithm, we open source the code, the trained model and the images used in the paper, which can be downloaded athttps://github.com/LIUyi827728/IDCDicL.

Fig. 5

The subnetwork Net_X.

Fig. 6

Flow chart of the proposed algorithm

4 Results and discussion

To verify the effectiveness and robustness of the proposed algorithm, we performed experiments on the CT dataset of 10 patients published from the Mayo [33] and clinical LDCT images authorized by hospitals, respectively. We selected state of the art LDCT denoising methods: WGAN [21], REDCNN [22], CT former [25], EDCNN [23], and DRCSC [26] as comparison experiments. We found in experiments that training the network directly using LDCT images is prone to non-convergence and model training failure, so we used transfer learning to assist the training of the model. We used the WED dataset [31] to obtain the pre-trained model of the network. Then the training was continued using representative 762 pairs of LDCT images from Mayo dataset, using 35 pairs of images in L004 folder as the validation set and 50 pairs of images in L081 folder as the test set. A random block of 128*128 size was taken from the image for training, meanwhile, random data enhancement was applied to the image. The Adam optimizer [34] was selected to update the parameters with a learning rate of 1e-4 for the WED dataset and 1e-6 for the Mayo dataset, with the learning rate decreasing to the previous 0.5 after every 2e5 iterations. The used computer CPU was i7-9700k @ 3.60 GHz and the GPU was NVIDIA GeForce RTX 2080 SUPER. The integrated development environment is PyCharm, the deep learning framework is PyTorch, and the programming language is Python. The parameter number is 48.24 million, and the processing time of a size of 512*512 LDCT image is 1.13 seconds. The display window of all images is [–160, 240]HU.

Using the peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) [35], feature similarity (FSIM) [36], and gradient magnitude similarity deviation (GMSD) [37] as objective evaluation metrics. PSNR evaluates the denoising capability of the algorithm, the less the noise, the bigger its value. The MSSIM, FSIM and GMSD evaluate the ability to image structure, feature, and gradient protection ability of the algorithm, respectively. It should be noted that the smaller the value of GMSD, the better the algorithm protects the gradient information of the image. All four objective evaluation metrics are required to provide noise-free image as reference image.

4.1 Mayo LDCT image denoising processing

A pre-trained model is obtained by training the WED dataset, then training the model using the Mayo dataset. When the stable denoising performance is obtained on the validation set, it is considered to obtain the optimal training parameters. 50 pairs of LDCT images in the L081 folder are selected as the test set, and the average objective indexes of the test set are shown in Table 1.

Table 1
Average values of evaluation metrics of the test set images from the L081 folder

LDCT WGAN REDCNN CTformer EDCNN DRCSC IDCDicL

PSNR 31.5243 34.2402 34.3726 34.7556 34.7596 35.0021 35.2975

MSSIM 0.9125 0.9388 0.9424 0.9429 0.9452 0.9431 0.9484

FSIM 0.9652 0.9775 0.9767 0.9768 0.9777 0.9740 0.9778

GMSD 0.0638 0.0409 0.0411 0.0418 0.0404 0.0465 0.0405

	LDCT	WGAN	REDCNN	CTformer	EDCNN	DRCSC	IDCDicL
PSNR	31.5243	34.2402	34.3726	34.7556	34.7596	35.0021	35.2975
MSSIM	0.9125	0.9388	0.9424	0.9429	0.9452	0.9431	0.9484
FSIM	0.9652	0.9775	0.9767	0.9768	0.9777	0.9740	0.9778
GMSD	0.0638	0.0409	0.0411	0.0418	0.0404	0.0465	0.0405

In Table 1, the proposed algorithm achieves the best results in three objective evaluation metrics, and the PSNR metric is improved substantially. Compared with the state-of-the-art denoising algorithm EDCNN, the proposed algorithm is 0.0001 worse on GMSD, which indicates that the proposed algorithm not only denoising LDCT image substantially, but also has good protection of edge structure and texture details. In the next section, we will analyze the effect of the proposed algorithm from the perspective of image vision.

4.1.1 Analysis of CT image of abdominal cavity lesions

In Fig. 7, compared with normal dose CT (NDCT) image, there are severe noise and artifacts in LDCT images, which affect the physician’s observation of the lesions, tissues and organs. All six algorithms have great denoising performance on the abdominal images. The WGAN algorithm has good protection in image structure and details, but the noise reduction is not complete, and the problem of noise residue appears, as shown by the red arrow in the ROI in Fig. 8(c1); The REDCNN algorithm has the problem of over-smoothing, as shown in Fig. 8(d1), the edge of the lesion is blurred; The CTformer algorithm has a good overall effect, but there are subtle artifacts, as shown in Fig. 8(e1) shown by the red arrow; The EDCNN has an excellent ability to protect edge and texture details, and its noise reduction also has a good effect, however, its edge enhancement feature is easy to retain the streak artifacts as details, as shown by the red arrow in Fig. 8(f1); The DRCSC algorithm has significant denoising performance, but also has the problem of weak smoothing, the edge detail of the tissue is lost; The processing result of the proposed method is closest to NDCT images, with obvious denoising effect and not prone to over-smoothing, as shown in Fig. 8(h1). In addition, the processing result with clear lesion edges, no noise interference in the liver section, and good maintenance of tissue edge details, as indicated by the green arrow in Fig. 8(h2). In the objective evaluation metrics of the abdominal images in Table 2, the optimal values of the proposed algorithm were obtained for all four metrics, which also verified the effectiveness of the proposed algorithm.

Fig. 7

Results of abdominal CT image processing.

Fig. 8

ROI effect of the abdominal image.

Table 2

Objective evaluation indicators of abdominal images

	LDCT	WGAN	REDCNN	CTformer	EDCNN	DRCSC	IDCDicL
PSNR	31.5598	34.4342	34.5859	34.9352	34.8923	35.5756	35.4694
MSSIM	0.9169	0.9430	0.9459	0.9471	0.9485	0.9467	0.9517
FSIM	0.9656	0.9785	0.9777	0.9776	0.9785	0.9743	0.9788
GMSD	0.0629	0.0404	0.0403	0.0413	0.0398	0.0462	0.0393

4.1.2 Analysis of thoracic CT image

The proposed algorithm also has great performance in processing thoracic LDCT images, the overall processing result is shown in Fig. 9. The region of heart plat and the spine region containing streak artifacts and edges of the thoracic LDCT image are selected as the ROI, which is focused on the analysis of the thoracic image processing effect. The WGAN algorithm processes the thoracic LDCT to preserve texture and gradient information better, but there is severe noise residue, as shown in the red arrow in Fig. 10(c1); The REDCNN shows a weak over-smoothing phenomenon in the heart flat region; The CTformer algorithm has a small number of irregular artifacts; The EDCNN has weak artifacts at the spinal muscles; The DRCSC algorithm has great performance in processing thoracic LDCT image; The proposed algorithm has an excellent performance in noise reduction and protecting edge textures. The visual effect is closest to NDCT images, with obvious noise reduction in the weak-noise heart region and strong-noise spine edges are also clearer, as indicated by the green arrow in Fig. 10(h2). The objective evaluation metrics in Table 3, the PSNR of the IDCDicL has a large improvement compared with other algorithms, MSSIM also has the optimal value, and the WGAN processing thoracic LDCT images have the best metrics in FSIM and GMSD, but its effect of noise reduction is not very well. With comprehensive visual effects and objective evaluation values, the proposed algorithm processing thoracic CT image has the best effect.

Fig. 9

Thoracic image processing results.

Fig. 10

Thoracic image ROI effect.

Table 3

Objective evaluation indicators of thoracic images

	LDCT	WGAN	REDCNN	CTformer	EDCNN	DRCSC	IDCDicL
PSNR	33.4555	35.4895	35.0658	35.8052	35.5902	35.8005	36.1567
MSSIM	0.9560	0.9671	0.9662	0.9690	0.9690	0.9678	0.9707
FSIM	0.9804	0.9839	0.9823	0.9834	0.9833	0.9827	0.9834
GMSD	0.0432	0.0345	0.0370	0.0346	0.0350	0.0370	0.0351

4.2 Clinical LDCT image analysis

To further verify the robustness and practicality of the proposed algorithm, we conducted experiments on clinical LDCT images. We selected pelvic image and lesion image for analysis. Due to the small number of clinical LDCT images and the absence of corresponding NDCT images, we use the trained model of the Mayo dataset for processing clinical images. Other comparison algorithms also use the trained model of the Mayo dataset.

4.2.1 Pelvic clinical LDCT image analysis

The overall processing effect of pelvic clinical LDCT is shown in Fig. 11. The region of organ tissue with complex structure, the region of the spine, and the region of flat muscle fat tissue were selected to analyze the denoising performance. In the LDCT image, regular streak artifacts and mottle noise can be seen. All six denoising algorithms achieve great performance for pelvic clinical LDCT image. The WGAN has a good protection effect on gradient information, however, noise residuals still appear, as shown by the red arrow in Fig. 12(b1); The REDCNN has a weak blur at the tissue edge, as shown by the blue arrow in Fig. 12(c1); The CTformer also has a small amount of noise residue, as shown by the red arrow in Fig. 12(d1); The EDCNN has strong edge protection, but excessive edge enhancement occurs in Fig. 12(e2); The DRCSC has a great visual effect in the ROIs, but have weak smooth from the perspective of the whole image, as shown in Fig. 11(f); The visual effect of the proposed model processing pelvic LDCT image is better than other algorithms obviously, the edge of tissue is clear in Fig. 12(g1), the edge of bone is clear in (g2), the fat tissue has less noise in (g3).

Fig. 11

Pelvic clinical LDCT image processing results.

Fig. 12

The ROI effect of pelvic clinical image.

4.2.2 Lesion clinical LDCT image analysis

A lesion clinical LDCT image is selected to further verify the effectiveness of the proposed algorithm. In the overall visual effect in Fig. 13, the denoising performance of the WGAN, REDCNN, CTformer and EDCNN algorithms all have the problem of denoising in the flat region inside the tissue. The WGAN has the most severe noise residue with a small number of streak artifacts, as shown in Fig. 14(b2); The CTformer has irregular artifacts, which affect the visual effect, as shown clearly in Fig. 13(b); In the lower left part of Fig. 13(f), the DRCSC occurs noise residue slightly; The overall visual effect of the proposed model is excellent, in Fig. 14(g1), the edge of organ tissue is clear, and the noise reduction in the flat region is significant, in Fig. 14(g2), the edge of the lesion is clear, the noise in the surrounding flat region is suppressed, and the lesion is clearly distinguished, which proves the effectiveness of the proposed algorithm.

Fig. 13

Lesion clinical LDCT image processing results.

Fig. 14

The ROI effect of lesion clinical image.

4.3 Ablation experimental analysis

The IDCDicL algorithm suppresses the mottle noise and streak artifacts in LDCT image effectively, improves the image quality substantially. The model does not require noise intensity parameter in the input network, which facilitates the practical application. The ablation experiments in this section discuss the role of the improvement module.

4.3.1 Comparison of IDCDicL and DCDicL algorithm

The DCDicL algorithm has excellent performance in suppressing Gaussian noise, which is one of the most advanced denoising algorithms. Inspired by DCDicL, we propose IDCDicL to process LDCT images, because the noise statistical information in LDCT images is difficult to determine and the noise intensity cannot be estimated, we improve the network structure to avoid the input noise standard deviation, then improve the network structure for learning the prior on convolutional dictionary, and finally modify the loss function. The pelvic image in L081 was selected for comparison experiment, and the result is shown in Fig. 15. The visual effects and evaluation metric indicate that the IDCDicL algorithm has a better denoising performance on LDCT images.

Fig. 15

Comparison results of IDCDicL and DCDicL algorithm.

4.3.2 Effection of the improvement module on the algorithm

The effect on the noise reduction results in the proposed model, which can be obtained by removing the DenseNet module and MSSIM loss in the proposed model. In Table 4, removing the DenseNet module results in a substantial reduction of all metrics compared to the IDCDicL, proving that the DenseNet module has an important role in the network model; Removing the MSSIM loss, the FSIM and GMSD metrics decreased more compared to the IDCDicL, which indicates that the MSSIM loss mainly improves the capability of edge protection and detail information; In addition, We retained the original input network, make the difference between LDCT and NDCT images, solve the noise standard deviation, and keep the DenseNet and MSSIM losses for experiments, which shows that modifying the input network and removing the noise parameters do not weaken the denoising performance. The results of different model structures for processing the Mayo dataset L081 folder images are shown in Table 4. This section focuses on the effect of the DenseNet module, MSSIM loss, and noise parameter on the IDCDicL. In addition, the number of convolutional layers in the HeadNet network, the number of layers in the DenseNet network, and the number of stages in the model all have different effects on the IDCDicL. The proposed model has experimented with several times, the final result is 4 convolutional layers in HeadNet, 121 layers in DenseNet, and 4 stages in the model.

Table 4
Average indicators for the improvement module (mean ± std)

PSNR MSSIM FSIM GMSD

DCDicL 34.3958±1.2612 0.9372±0.0144 0.9695±0.0054 0.0535±0.0078

IDCDicL-NoDenNet 34.7223±1.2573 0.9377±0.0169 0.9691±0.0080 0.0537±0.0119

IDCDicL-NoMSSIM 35.1713±1.2595 0.9471±0.0168 0.9764±0.0054 0.0431±0.0081

IDCDicL-Sigma 35.2350±1.2562 0.9479±0.0147 0.9775±0.0052 0.0419±0.0079

IDCDicL 35.2975±1.2512 0.9484±0.0136 0.9778±0.0050 0.0405±0.0074

	PSNR	MSSIM	FSIM	GMSD
DCDicL	34.3958±1.2612	0.9372±0.0144	0.9695±0.0054	0.0535±0.0078
IDCDicL-NoDenNet	34.7223±1.2573	0.9377±0.0169	0.9691±0.0080	0.0537±0.0119
IDCDicL-NoMSSIM	35.1713±1.2595	0.9471±0.0168	0.9764±0.0054	0.0431±0.0081
IDCDicL-Sigma	35.2350±1.2562	0.9479±0.0147	0.9775±0.0052	0.0419±0.0079
IDCDicL	35.2975±1.2512	0.9484±0.0136	0.9778±0.0050	0.0405±0.0074

5 Conclusion

In this paper, we propose an improved deep convolutional dictionary learning algorithm to reduce mottle noise and streak artifacts in LDCT images, while preserving structural and detailed information such as edges and textures in the images, improving the quality of LDCT images effectively. Different from Gaussian noise, noise in LDCT image is more complex and has no statistical characteristics, which increases the difficulty of noise removal. Convolutional dictionary learning uses a set of convolutional dictionaries to reconstruct images, which has the advantage of the shift-invariant property, and the property of multidimensional convolutional kernel has reasonable physical meaning. Combined with the powerful feature extraction capability of deep learning, the more accurate convolutional sparse matrix and convolutional dictionary are solved to reconstruct high-quality LDCT images.

In DCDicL, we make three improvements: Increasing the number of convolutional layers and eliminating the noise intensity parameter in the input network, which solve the problem of difficult estimation of noise intensity of LDCT images; Then, using DenseNet121 instead of a shallow convolutional network to learn the prior on convolutional dictionary, because LDCT images are more complex compared to natural images; Finally, the MSSIM loss is added to the loss function to enhance the algorithm’s protection of structural details. Compared with other great LDCT denoising deep learning algorithms, the proposed model effectively improves the ability of the algorithm to suppress noise and protect the texture details of LDCT images. The experimental results show that the IDCDicL has better results in both the Mayo dataset and clinical LDCT images, which proves the effectiveness and robustness. The proposed model still has some shortcomings: The initial training of the network is difficult, and the training time is longer compared to the EDCNN; The convolutional dictionary is a set of filters, and the reason for the longer training time may be that the initialization of the convolutional dictionary is zero. Setting a suitable initial value for the convolutional dictionary may accelerate the training process of the network; Whether choosing different sizes of convolutional dictionaries to reconstruct LDCT images may improve the image quality, which is also an issue that can be explored.

Footnotes

Acknowledgment

This work was supported in part by the National Nature Science Foundation of China under Grant (61801438), in part by the Science and Technology Innovation Project of Colleges and Universities of Shanxi Province (2020L0282), in part by the Open Fund Project of Key Laboratory of Computer Network and Information Integration, Ministry of Education (K93-9-2022-02).

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