Separation‐based model for low‐dose CT image denoising

Low-dose computed tomography (LDCT) image often contains mottle noise and streak artefacts, which seriously interfere with clinical diagnosis. In this study, the separation-based (SEPB) method is proposed for mottle noise and streak artefacts suppression and structure preservation. In it, the LDCT image is decomposed into the structural image with residual mottle noise and the streak artefacts image with residual structural details by the image decomposition structural-preserving image smoothing method. The structural image is filtered by the K-singular value decomposition algorithm to remove the residual mottle noise, and the structural details in the streak artefacts image are extracted by the morphological component analysis theory. The extracted structural details are added to the filtered structural image to get the LDCT result image. Meanwhile, in the process of extracting the structural details, the streak artefacts dictionary learned from the streak artefacts image is corrected by the local intuitional fuzzy entropy to remove its structural atoms. The experiments are conducted on the modified Shepp–Logan phantom, the pelvis phantom and the clinical abdominal data to evaluate the proposed SEPB method. Compared to several comparative denoising methods, the experimental results show that the SEPB method has better performance in subjective visual effect and objective indicators.


Introduction
Compared with other radiological examinations, X-ray computed tomography (CT) technology provides high-resolution medical sectional anatomy images, which is widely applied in clinical medical diagnosis field. However, the X-ray radiation in CT imaging is harmful to the patients, especially for those patients undergoing repetitive CT scans, which may induce genetic disease and cancer [1,2]. Therefore, it is necessary to reduce the X-ray radiation dose. Of all the methods to reduce the radiation dose, the simplest one is to reduce the tube current, but the corresponding reconstructed image, i.e. the low-dose CT (LDCT) image, is polluted by mottle noise and streak artefacts.
To improve the quality of LDCT image, many methods have been proposed. In general, these methods can be divided into three categories: ① projection data filtering methods; ② iterative reconstruction methods; and ③ post-processing methods.
Projection data filtering method refers to filtering the projection data before the reconstruction process. There are many filtering methods used to filter the projection data, mainly including multiscale least-square filter [3], fuzzy-median filter [4], bilateral filter [5], modified Rudin-Osher-Fatemi (ROF) filter [6], edgepreserving Huber penalty filter [7], and total generalised variation filter [8]. However, the raw projection data are not always accessible, which limit the widespread use of this kind of methods. Iterative reconstruction method aims to solve a regularised cost function which combines the noise characteristics of projection data and the image prior information to improve the LDCT image. Many priors have been used in the past decade, involving total variation (TV) [9], non-local means (NLMs) [10], dictionary learning [11], low rank [12,13], and artificial neural network [14]. It is well known that the iterative reconstruction method can obtain a reconstructed image of good quality, but the disadvantage of this method is the high computational complexity.
The three categories, i.e. post-processing methods, which directly process the reconstructed LDCT images and do not rely on the raw projection data. Various post-processing approaches have been proposed so far, such as NLMs methods [15][16][17], partial differential equation methods [18,19], which suffer from blocky effect seriously, block-matching 3D (BM3D) methods [20,21] and sparse representation methods [22] with the time complexity. In recent years, many deep learning approaches have also been used in LDCT images denoising, such as an approach of the deep convolutional neural network [23] and a residual encoder-decoder convolutional network algorithm [24]. These methods show better performance in the task of LDCT image denoising problem. Moreover, it is a good idea to try to use DBNs [25] and CNN-MLP [26] for LDCT image denoising in the future.
The NLM algorithm [27] is a classical denoising algorithm that can well preserve the structural information of the natural image while removing noise, and it also has good performance in LDCT images denoising [15][16][17]. As an improved algorithm of the NLM algorithm, the structural-preserving image smoothing (SPIS) algorithm [28] can achieve the separation of the texture part and the structural part of the natural image well. Therefore, this paper will use the SPIS model to achieve the separation of the structural part in the LDCT image from the streak artefacts part, due to the fact that the streak artefacts of the LDCT image and the textures of the natural image have similar structural features. In practice, after separation, the structural part contains residual mottle noise and the streak artefacts part contains structural details. Therefore, it is necessary to remove the mottle noise in the structural part and extract the structural details in the streak artefacts part. We know that the K-singular value decomposition (K-SVD) algorithm [29], which belongs to the sparse representation theory, is a classical algorithm and has been widely concerned since it was introduced for its good performance in significantly improving the quality of images. Therefore, the K-SVD algorithm is used to remove the residual mottle noise in the structural part to obtain the clean structural part. In addition, in this paper, the morphological component analysis (MCA) theory [30] is utilised to extract the structural details in the streak artefacts part. Finally, add the extractive structural details in the streak artefacts part to the K-SVD filtered structural part to get the LDCT result image. In conclusion, based on the SPIS model, combining the K-SVD model and MCA theory, this paper proposes a post-processing method for LDCT image denoising, the separation-based (SEPB) model.
In this paper, to extract the structural details in the streak artefacts part of LDCT image, the online dictionary learning method [31] is used to learn the structural dictionary from the structural part filtered by K-SVD model and learn the streak artefacts dictionary from the streak artefacts part, and the streak artefacts dictionary is corrected by the local intuitionistic fuzzy entropy [32] to eliminate the structural atoms. Next, the structural dictionary and the corrective streak artefacts dictionary are used to form the combined dictionary. Then, the structural details of the streak artefacts part are spare coded by the combined dictionary.
The remaining part of the paper is organised as follows. In Section 2, the SPIS model, the K-SVD algorithm and the MCA theory are introduced, then the SEPB model is proposed. Section 3 gives and discusses the experimental results. The conclusion is given in Section 4.

SPIS model
As an improved model of NLM, the SPIS model can separate an image into the structural part and textural part. Given an image Y, for each pixel y ∈ Y, extract d features corresponding to y. Then the d features for all pixels y ∈ Y are stored in the matrix F, and the matrix F is called the feature image of Y. Suppose that x and y are the two small patches centred on pixels x and y, respectively, x, y ∈ Y . In [28], we know that the similarity of x and y can be described by the similarity of the corresponding region covariance descriptors in the feature image F. The calculation formula for the region covariance descriptor C y is given as follows: where n is the number of pixels in the small patch y, z k (k = 1, …, n) is the d-dimensional feature vector composed of d features of the pixel k ∈ y, μ is the mean of these feature vectors, and T is the transpose operator. The SPIS model decomposes a degraded LDCT image Y into its structural image S and its streak artefacts image T, that is The structural component of the pixel y as where N(y, r) denotes the squared neighbourhood centred at y and of size (2r + 1) × (2r + 1) pixels, C = C x + C y , μ x , μ y and C x , C y are the means and covariances of the small patches centred on the pixels x and y of the image Y, respectively. The d-dimensional feature vector is the same as [28]. The streak artefacts component of the pixel y as Moreover, the Cholesky decomposition of the covariance matrices is used to transform the covariance matrices into a Euclidean vector space. The SPIS model can effectively extract the structural part of the image from the textural part. In view of this property, the SPIS model is used in this paper to separate the structural part of the LDCT image from the streak artefacts part.

K-SVD model
The K-SVD model is a sparse representation model based on the learned dictionary. Let Y and X ∈ R d × m be the given LDCT image and the result image, respectively. y i, j = R i, j Y is the small image patch of size d × d, which is extracted from Y and centred on the pixel y. The dictionary D for sparse representation is obtained by solving the following problem: where D ∈ R d × k is the dictionary matrix, k is the number of atoms of D, and the regularisation parameter μ i, j > 0. The first term of expression (4) is a fidelity term, and it is expected that the sparse representation result is as close as possible to y i, j . The second term of expression (4) is a regularisation term, and it is expected to be as sparse as possible. Expression (4) is solved by the method of alternative optimisation. First step: fix the dictionary D, solve coding coefficients α i, j ; second step: update the dictionary D according to α i, j . Repeat the above two steps until convergence and get the dictionary D.
Based on the dictionary D, the result image X can be obtained by solving the following equation: and X is obtained: where λ is a positive weight parameter and I is the unit matrix.

MCA theory
The MCA theory is capable of separating the different components of an image. For any image X, assume that it is composed of different morphological components X i (i = 1, 2, …, k). For each component X i , it can be sparsely represented by its corresponding base dictionary D i . However, if it is sparsely represented by the base dictionaries of the other morphological components, X i cannot be sparsely represented or the representation may not be the most sparse.
The morphological component X i (i = 1, 2, …, k) is obtained by solving the following optimisation problem (6) using the orthogonal matching pursuit algorithm: where σ is the noise standard deviation (STD). D = D 1 , D 2 , …, D k is a combined dictionary composed of different base dictionaries D i (i = 1, 2, …, k). The difference of the combination dictionary D obviously affects the separate morphological component X i , so it is crucial to construct an accurate combined dictionary.

Proposed model
LDCT images often contain mottle noise and streak artefacts. In order to get clean CT images without mottle noise and streak artefacts, in this paper, we propose the SEPB LDCT image denoising method, i.e. SEPB method. The flowchart of the proposed SEPB LDCT image denoising method is given in Fig. 1. Each step of the proposed SEPB model will be introduced in detail in the following sections.

Definitions of intuitionistic fuzzy set and local intuitionistic fuzzy entropy
Definition 1: Let X be a given domain, then an intuitionistic fuzzy set A in X is defined as follows: are the membership function and non-membership function of x in A, respectively, and

Local intuitionistic fuzzy entropy:
Given an LDCT image X, A represents the structural areas of X, and it is an intuitionistic fuzzy set. For each pixel x ∈ X, μ A (x) represents the membership function of x in the structural areas A, v A (x) represents the nonmembership function, and π A (x) represents the hesitation function. The expressions are as follows: is the normalised gradient of the image X.
According to the above definitions, the expression of local intuitionistic fuzzy entropy of the local neighbourhood R in X is given as follows: where n × n is the number of pixels in R. In the SEPB model, n is set to 3.

Separating the structural part and the streak artefacts part of the LDCT image:
The SPIS model can well realise the separation of the structural part and the textural part of natural images. Moreover, the streak artefacts of LDCT images and the textures of natural images have great similarities in visual characteristics, so this paper uses SPIS model to separate the structural part and the streak artefacts part of LDCT images first, then the structural part and the streak artefacts part are further processed according to their respective features. Fig. 2 shows the result images of modified Shepp-Logan phantom separated by the SPIS model. Fig. 2a shows the LDCT image; Figs. 2a1 and a2 are the structural part and the streak artefacts part, respectively. It can be clearly seen that the SPIS model is able to separate the LDCT image effectively. In Fig. 2a1, the structural part image contains obvious edge structures while with little mottle noise. In Fig. 2a2, the streak artefacts part contains abundant streak artefacts with almost no edge structures, so it is reasonable to use the SPIS model to separate LDCT images.

Filter processing of the structural part of LDCT image:
Observing the structural part of the modified Shepp-Logan phantom (Fig. 2a1), we can see that some mottle noise still remains. In order to remove residual mottle noise thoroughly, filter processing is needed for the structural part. We know that the noise cannot be sparsely represented by the K-SVD dictionary, but the structure can be sparsely represented by it, so the K-SVD model is used to filter the structural part to remove the residual mottle noise in it. Fig. 3 shows the structural part of the LDCT image ( Fig. 2a1) with or without the K-SVD filter processing. The mottle noise of the structural part filtered by the K-SVD model (Fig. 3a1) is significantly less than that of not processed by the K-SVD model (Fig. 3a2). It verifies that the mottle noise in the structural part of the LDCT image can be effectively removed through the processing of K-SVD model.

Extract the structural details in the streak artefacts part of the LDCT image:
In this paper, we assume that an LDCT image contains three parts: the structural part, the streak artefacts part, and the mottle noise part. Through the above two Sections 3.2.1 and 3.2.2, the LDCT image is separated into the structural part without mottle noise and the streak artefacts part with structural details. So it is necessary to extract the structural details from the streak artefacts part. According to the MCA theory, that is, the streak artefacts part contains two morphological components: the structural morphological component and the streak artefacts morphological component, and the structural morphological component is expected to be extracted from the streak artefacts morphological component. The MCA theory is used to realise the extraction.
From Fig. 2a2, it can be seen that the streak artefacts part contains a large amount of streak artefacts and little edge structures. The structural part filtered by the K-SVD model (Fig.  3a1) contains only edge structures. Based on this fact, the online dictionary learning method is used to learn the streak artefacts dictionary D T from the streak artefacts part and learn the structural dictionary D S′ from the structural part filtered by K-SVD model. However, for the pelvis phantom, due to the influence of a small quantity of edge structures in the streak artefacts part, the streak artefacts dictionary learned from the streak artefacts part always contains structural atoms, as shown in Fig. 4a. Therefore, the structural atoms of the streak artefacts dictionary need to be removed to obtain the corrective streak artefacts dictionary D T .
Combine the streak artefacts dictionary D T with the structural dictionary D S′ to form a combined dictionary D = D T , D S′ . The structural details in the streak artefacts part of LDCT image is obtained by solving (6) based on the dictionary D.
As mentioned above, the accuracy of the combined dictionary D plays an important role. So, the streak artefacts dictionary is corrected by removing the structural atoms of it and get the accurate streak artefacts dictionary D T which only contains the streak artefact atoms. In addition, from Fig. 4a, the streak artefacts dictionary of pelvis phantom, we can see that the streak artefact atoms contain weaker edge information, but the structural atoms contain stronger edge information. Due to their different structural characteristics, the corresponding local intuitionistic fuzzy entropy is different. The streak artefacts atom's local intuitionistic fuzzy entropy is lower, but the structural atom's local intuitionistic fuzzy entropy is higher, and the details will be discussed in Section 4.2.2. So, in this paper, the streak artefacts dictionary is corrected by the local intuitionistic fuzzy entropy to get the corrective streak artefacts dictionary without the structural atoms. Fig. 5 shows the extracted structural details from the streak artefacts part of the pelvis phantom. It can be seen that the MCA theory can effectively extract the structural details from the streak artefacts part of LDCT image.

Combining the structural details in the streak artefacts part of LDCT image with the structural part of LDCT image filtered by K-SVD model:
Finally, the structural details in the streak artefacts part of LDCT image is added to the structural part filtered by K-SVD model to get the LDCT result image X without mottle noise and streak artefacts. This process can be expressed mathematically as S′ + T S = X .
So far, the whole procedure of the proposed SEPB model has been presented, and it is summarised in Algorithm 1.
Algorithm 1: SEPB LDCT image denoising model Input: the LDCT image Y. Output: the denoised image X .
(1) Separation: Process Y by the SPIS model, which corresponds to (1), and obtain the structural part S and the streak artefacts part T; (2) The structural part S and the streak artefacts part T are processed, respectively: (a) the structural part S is filtered using the K-SVD model, which corresponds to (5), and obtain the S′; (b) learn the structural dictionary D S′ from S′ and the streak artefacts dictionary D T from T using the online dictionary learning method, and D T is corrected by the local intuitionistic fuzzy entropy, which corresponds to (7), and obtain D T , then extract the structural details T S from T by means of the MCA theory, which corresponds to (6).
(3) Combination: Add S′ and T S to get the result image X .

Experiments results and analysis
Experiments were carried out on the modified Shepp-Logan phantom, the pelvis phantom, and the clinical abdominal data to verify the validity of the proposed SEPB model in this paper, and the new model is compared with several other denoising algorithms, including NLM model, SPIS method, relative total variation (RTV) model [33], BM3D model [34], K-SVD algorithm, weight buclear norm minimization (WNNM) model [35] and nonlocally centralized sparse representation (NCSR) model [36]. However, it is very regreTable that the type of noise for low-dose CT images is quite complex, including the quantum noise, the noise caused by inherent defects of hardware systems and the noise introduced during the image reconstruction process. In order to observe the denoising effect of the SEPB model on the LDCT images with different mottle noise and streak artefact intensities, the experiments were carried out on two different intensities. The setting method is given as follows.
It is proposed in [37] that the projection data after system calibration and logarithmic transformation follow the nonstationary Gaussian distribution: where x i and σ i 2 are the mean and variance of the projection data received by the i detector, respectively, f i and η are the parameters related to the system. In this paper, LDCT images of different mottle noise and streak artefact intensities are reconstructed by simulated projection data of different noise intensities, and the simulated projection data of different noise intensities are obtained by setting different values of η and f i . In the case of fixed η, the higher the value of f i , the more serious the mottle noise and streak artefacts contained in the reconstructed LDCT image. Set η = 22, 000 in the modified Shepp-Logan phantom, η = 62, 000 in the pelvis phantom, both of them take f i = 100 and f i = 200. Figs. 6a1-d1, a2-d2 show the simulated projection data with different noise intensities and the corresponding phantoms with different mottle noise and streak artefact intensities.
Moreover, Figs. 6e1 and e2 are the clinical abdominal data, which are obtained by scanning a 53-year-old male patient at high and low doses on a SIEMENS Sensation Cardiac 64-slice CT scanner with a tube voltage of 120 kVp. They are acquired in digital imaging and communications in medicine and the slice thickness of them is 1.5 mm. In this protocol, the high dose is for the 260 mAs protocol and the low dose is for the 50 mAs protocol. Fig. 6e1 is the HDCT abdominal image and Fig. 6e2) is the LDCT abdominal image. Both are in the size of 512 × 512 pixels.

Objective evaluation indicators
This paper uses three objective evaluation indicators to measure the performance of denoising algorithms, namely peak signal-to-noise ratio (PSNR), mean structural similarity (MSSIM) [38] and STD [16]. The higher the PSNR, the smaller the degree of image distortion, and the higher the MSSIM, the better the corresponding denoising algorithm can maintain the structural information of the original image. Moreover, if the STD values of regions of interest (ROI) in the HDCT image and the processed LDCT image are close, the corresponding algorithm is efficient. The expressions of PSNR, MSSIM and STD are given as follows: (1) PSNR PSNR = 10 ⋅ log 10 255 2 MSE , where Y is the result image, X is the reference image, and M × N is the number of pixels of X or Y.
where Y is the result image and X is the reference image. x i and y i are the small patches of size k × k of X and Y, respectively, mn is the number of the small patches in X or Y. μ x i and μ y i are the mean values of x i and y i , respectively, and σ x i 2 and σ y i 2 are the variances of x i and y i , respectively. cov(x i , y i ) is the covariance of x i and y i , and c 1 , c 2 are the constants.
(3) STD where STD Ω represents the STD of the region Ω, x i j p and x¯Ω p are the pixel intensity and the average of all pixels in Ω, respectively. Ω is the pixel number in Ω.

Parameter analysis
The parameters for the proposed SEPB model are listed in Table 1.

Size of the similar patch:
The quality of the streak artefacts image and the structural image, which are obtained by separating the LDCT image using the SPIS model, is of great significance. Therefore, it is necessary to obtain the streak artefacts image and the structural image with good quality. Through experiments, it is found that the size of the image similar patch of the SPIS model significantly influences the quality of the separated results of LDCT image. Fig. 7 is the streak artefact images of the LDCT image separated by the SPIS model and the corresponding streak artefact dictionaries learned from them. The size of image similar patch is 3 × 3 in Figs. 7a1 and a2, 5 × 5 in Figs. 7b1 and b2, and 7 × 7 in Figs. 7c1 and c2, respectively. It is observed that the streak artefacts in Figs. 7a1 are not particularly adequate, and the streak artefact atoms of the streak artefacts dictionary (Fig. 7a2) are not obvious; Fig. 7b1 contains relatively more streak artefacts and less edge structures, and the small amount of edge structures hardly interfere in the accuracy of the learned streak artefacts dictionary. As shown in Fig. 7b2 it has no structural atoms and only contains streak artefact atoms. Fig. 7c1 is of obvious edge structures, and they will affect the accuracy of the learned streak artefacts dictionary. As we can see, Fig. 7c2 contains obvious structural atoms in addition to the streak artefact atoms. Therefore, the size of image similar patch of the SPIS algorithm in the proposed SEPB model is set as 5 × 5.

Local intuitionistic fuzzy entropy threshold:
From the modified Shepp-Logan phantom, we can learn the streak artefacts dictionary that only contains streak artefact atoms, as shown in Fig.  7b2. However, from the pelvis phantom, we cannot learn the clean streak artefacts dictionary, as shown in Fig. 4a, which contains obvious structural atoms. So, in this section, we will discuss the correction of the streak artefacts dictionary, which removes the   structural atoms of the streak artefacts dictionary to get a more accurate streak artefacts dictionary. Fig. 4a is the streak artefacts dictionary learned from the streak artefacts image η = 62000, f i = 100 of pelvis phantom. In order to remove the structural atoms of the streak artefacts dictionary, the local intuitionistic fuzzy entropy of the atom is introduced. Fig. 4b shows the local intuitionistic fuzzy entropy of atoms in the streak artefacts dictionary, and it has a monotonous change trend. Moreover, it is found through experiments that the local intuitionistic fuzzy entropy of the structural atom is higher, and the local intuitionistic fuzzy entropy of the streak artefacts atom is lower, so an appropriate threshold value of the local intuitionistic fuzzy entropy can effectively remove the structural atoms of the streak artefacts dictionary.
In order to find an appropriate threshold, Figs. 4a1-a4 show the different corrective versions of the streak artefacts dictionary of the pelvis phantom using different local intuitionistic fuzzy entropy thresholds (0.95, 0.85, 0.75 and 0.76). It can be seen that the corrective streak artefacts dictionary with the local intuitionistic fuzzy entropy thresholds of 0.95 and 0.85 contain not only the streak artefact atoms but also the structural atoms. However, the corrective streak artefacts dictionary with the local intuitionistic fuzzy entropy threshold of 0.75 contains only streak artefact atoms and does not contain structural atoms. In addition, if the local intuitionistic fuzzy entropy threshold is increased to 0.76, the corrective streak artefacts dictionary contains the structural atoms again. Therefore, the threshold of local intuitionistic fuzzy entropy for the pelvis phantom η = 62000, f i = 100 is set as 0.75. Using the same method, the threshold of the local intuitionistic fuzzy entropy of the pelvis phantom η = 62000, f i = 100 is set as 0.79. Fig. 8 shows the result images of the modified Shepp-Logan phantom η = 22000, f i = 100 , which are processed by the proposed SEPB model and the other comparative denoising algorithms. The first column is the clean CT image, the LDCT image, the NLM processed LDCT image, the SPIS processed LDCT image, the RTV processed LDCT image, the BM3D processed LDCT image, the K-SVD processed LDCT image, the WNNM processed LDCT image, the NCSR processed LDCT image and the SEPB processed LDCT image. The second and third columns are the partial  (Fig. 8f1), K-SVD algorithm (Fig. 8g1), WNNM algorithm (Fig. 8h2) and NCSR algorithm (Fig. 8i1) obviously contain streak artefacts (shown by orange arrows). Moreover, SPIS algorithm (Fig. 8d2) and RTV algorithm (Fig. 8e2) contain residual mottle noise (shown by red arrow). In contrast, SEPB model (Fig. 8j) contains no residual mottle noise or streak artefacts. Therefore, the SEPB model proposed in this paper is effective to remove mottle noise and streak artefacts.

Experiments on modified Shepp-Logan phantom:
In addition to comparing the visual effects of the various comparative algorithms, Fig. 9, Table 2 and Fig. 10 show the objective evaluation indicators of them. Fig. 9 lists the PSNR and SSIM values of the regions of interest (ROIs, marked in the red square) in the modified Shepp-Logan phantom η = 22, 000, f i = 100 processed by various algorithms. In it, except for ROI1, of which, the maximum PSNR value is the K-SVD algorithm, almost all the largest PSNR and SSIM values of the remaining ROIs are the SEPB model. Figs. 12 shows the results of the pelvis phantom processed by various comparative algorithms, including NLM method, SPIS model, RTV model, BM3D model, K-SVD method, WNNM model, NCSR model and the proposed SEPB model. To compare the denoising algorithms more distinctly, in the clean CT image of the pelvis phantom (Fig. 11), take two small areas (marked by red rectangles) to display enlarged images. Compared to the LDCT image (Figs. 12b1 and b2), the NLM method (Fig. 12c2), the RTV model (Fig.  12e2), the BM3D model (Fig. 12f2), the K-SVD model (Fig. 12g1), the WNNM model (Fig. 12h2) and the NCSR model (Fig. 12i2) contain residual streak artefacts (marked by orange arrows); the SPIS model (Fig. 12d1) does not contain obvious streak artefacts, but contains residual mottle noise (marked by the red arrow). In contrast, SEPB can effectively remove mottle noise and streak artefacts, and the corresponding result images are closer to the HDCT image (Figs. 12a1 and a2).

Experiments on pelvis phantom:
Moreover, Table 2 lists the PSNR values of the pelvis phantom result image processed by various algorithms, and the maximum PSNR value is also the SEPB model.

Experiments on clinical abdominal data:
To further highlight the performance of SEPB, the clinical abdominal result images are shown in Fig. 13. There into, Fig. 13a is the HDCT image, and Fig. 13b is the corresponding LDCT image. From  Fig. 13b, we know that mottle noise and streak artefacts lead to fuzzy tumour boundaries under low dose CT scanning condition. Fig. 13j is the restored image using the proposed SEPB model, it has clearer tumour boundaries and the image structure is closer to  the HDCT image (Fig. 13a). The denoised images by NLM, SPIS and RTV are shown in Figs. 13c-e, respectively, in which serious residual mottle noise exists. In Figs. 13f-h, the processed LDCT images by BM3D, K-SVD and WNNM lead to some strange streak artefacts (marked by orange arrows). In comparison to Figs. 13c-e, Fig. 13i can suppress mottle noise but blurs the tumour boundaries (marked by the green arrow). Fig. 10 lists the STD values of the regions of interest (ROIs, marked in the red square) in the clinical abdominal result images processed by various algorithms. In it, the STD values of the SEPB model in both ROIs are closer to those of the HDCT image, respectively.
According to the evaluation indicators and the visual effects, it can be verified that the SEPB model we proposed has better ability to remove mottle noise and streak artefacts, and preserve structural information. However, as the proposed model needs three steps: separating, then processing, and finally combining, the proposed method is time consuming.

Conclusion
This paper proposes a post-processing approach to improve the quality of LDCT images, namely the SEPB model. In the SEPB model, firstly, the LDCT image is separated by the SPIS algorithm to get the structural image with residual mottle noise and the streak artefacts image with residual structural details. Next, the structural image is filtered by the K-SVD algorithm to remove the residual mottle noise of it, and the structural details of the streak artefacts image are extracted by the MCA theory. Finally, the LDCT result image with mottle noise and streak artefacts removed, and normal tissue structures preserved is obtained by combining the K-SVD filtered structural image with the structure details extracted from the streak artefacts image. Moreover, in order to extract the structural details in the streak artefacts image accurately, the SEPB model uses the local intuitionistic fuzzy entropy to get rid of the structural atoms of streak artefacts dictionary and obtain the corrective streak artefacts which only contains the streak artefact atoms.
Compared with the other comparative algorithms, the SEPB model not only can effectively remove mottle noise, streak artefacts and preserve normal tissue structures of the LDCT images, but also has higher objective indicators. In the SEPB model, the structural image needs to be filtered by the K-SVD algorithm. However, it is known that the K-SVD algorithm is time consuming, which leads to high calculation complexity of the proposed SEPB model. In our future work, it is necessary to reduce the calculation complexity of the proposed SEPB model by parallel technology. In addition, for different LDCT images, the threshold of the local intuitionistic fuzzy entropy used for correcting the streak artefacts dictionary is set by visual observation, so the robustness of the SEPB model needs to be enhanced. In this case, setting the adaptive threshold of the local intuitionistic fuzzy entropy to remove the structural atoms of the streak artefacts dictionary is also a direction of our future research.