Abstract

Ring artifacts seriously deteriorate the quality of CT images. Intensity-dependence of detector responses will result in intensity-dependent ring artifacts and time-dependence of CT hardware systems will result in time-dependent ring artifacts. However, only the intensity-dependent ring artifacts are taken into consideration in most post-processing methods. Therefore, the purpose of this study is to propose a general post-processing method, which has a significant removal effect on the intensity-dependent ring artifacts and the time-dependent ring artifacts. First in the proposed method, transform raw CT images into polar coordinate images, and the ring artifacts will manifest as stripe artifacts. Secondly, obtain structure images by smoothing the polar coordinate images and acquire texture images containing some details and stripe artifacts by subtracting the structure images from the polar coordinate images. Third, extract the stripe artifacts from the texture images using mean extraction and texture classification, and obtain the extracted ring artifacts by transforming the extracted stripe artifacts from polar coordinates into Cartesian coordinates. Finally, obtain corrected CT images by subtracting the extracted ring artifacts from the raw CT images, and iterate the corrected CT images in above steps until the ring artifacts extracted in the last iteration are weak enough. Simulation and real data show that the proposed method can remove the intensity-dependent ring artifacts and the time-dependent ring artifacts effectively while preserving image details and spatial resolution. In particular, real data prove that the method is suitable for new CT systems such as the photon counting CT.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

X-ray computed tomography (CT) has been widely used in many fields such as disease detection [14], image-guided radiation therapy (IGRT) [5,6], and industrial detection. The flat-field correction method in CT can get high quality projection images by suppressing the non-uniformity of incident cone beam and the inconsistency of detector responses [7,8]. However, in practice, raw CT images often have ring artifacts in the form of concentric circles, which may interfere with the diagnosis of diseases and the detection of industrial parts [9]. The causes of these ring artifacts are more complicated than expected, and can be summarized as the intensity-dependence of the detector responses and the time-dependence of the CT hardware systems (including the time-dependence of incident cone beam [10,11] and the time-dependence of the detector responses [12,13]). These two factors make the flat-field correction method not ideal. For energy integrated CT systems, the ring artifacts seriously reduce the accuracy of clinical diagnosis and the precision of industrial-part detection; for photon counting CT systems, the ring artifacts further deteriorate the accuracy of material identification, making it impossible to take advantage of their unique advantages. Therefore, it is urgent to develop an effective and stable ring artifact removal method.

Because ring artifacts are directly caused by the CT hardware systems, there are roughly four methods to remove the ring artifacts by adjusting the CT hardware systems. The first method is to replace the usual two-point flat field correction by a multi-point, piecewise linear flat field correction [14]. The second method requires a series of flat-field images acquired before, during, and after the CT scan to capture the dynamic characteristics of the CT hardware systems [15]. The third method is to move the sample or detector systems during an acquisition in defined horizontal and vertical steps to overcome the shortcomings of the hardware systems [16,17]. The fourth method requires multiple flat-field images under different beam filters with varying thicknesses to identify the clustered projection image artifacts that lead to the ring artifacts [18]. However, these methods have high operating costs on hardware level and are difficult to implement. In addition, due to the instability or time-dependence of the CT hardware systems, it is possible to retain some ring artifacts in the corrected CT images [19]. In recent years, pre-processing methods and post-processing methods have become research hotspots because they do not have any operations in hardware level and can also achieve good performance [4,9,2023].

Pre-processing methods are mainly based on the raw projection sinograms [24,25]. The main idea of these methods is that the ring artifacts manifest as stripe artifacts on the raw projection sinograms, reducing the difficulty of detection and elimination. Originally, Kowalski removed the stripe artifacts in the raw projection sinograms through simple low-pass filters, but this method will lose high-frequency details and deteriorate image quality [26]. To preserve more high-frequency image details, Raven discovered that the stripe artifacts in the polar angle direction are located in the center of image after Fourier transform and can be further removed by low-pass filters [27]. Furthermore, Munch et al. combined wavelet decomposition and a Fourier low-pass filter to distinguish the stripe artifacts and the image details more accurately [28]. Boin et al. proposed a method for removing the stripe artifacts in the sinograms using a moving average filter during the reconstruction process [29]. Subsequently, Ashrafuzzaman et al. proposed a variable window moving average (VWMA) filter and a weighted moving average (WMA) filter to remove the stripe artifacts in the sinograms [30]. Titarenko et al. proposed a method based on the assumption of smoothness of the raw projection sinograms. This method uses the notion of a regularized solution from the theory of ill-posed problems and is applied to the sinograms before image reconstruction [31]. Miqueles et al. proposed a fast algorithm for ring artefact reduction in tomography by generalizing the method of Titarenko et al. Compared with the original method, this method is fast due to the use of the conjugate gradient method with an explicit solution [32]. Kim et al. proposed a method to remove stripe artifacts from sinograms by estimating the sensitivity of each detector element and equalizing them in sinograms [33]. Nieuwenhove et al. computed the computed through principal component analysis of a set of flat fields, and used a linear combination of the most important eigen flat fields to individually normalize each X-ray projection [15]. Titarenko et al. wrote the basic ring artifact suppression algorithm as a 1D filter and combined with standard filtering used in filtered back-projection algorithms [34]. Vågberg et al. proposed a method to correct ring artifacts by obtaining changes in scintillator thickness [35]. Vo et al. divided the streak artifacts into four categories and correcting them separately, and achieved good results in pre-processing methods [36]. Croton et al. proposed a pixel-wise detector calibration using hundreds of data points for each position on the detector, rather than the standard two-point flat-field calibration [37]. Pre-processing methods work on the raw projections, and it is difficult to intuitively select parameters, which limits the applicability of these methods. Besides, commercial cone-beam systems often go with their own reconstruction software, which may only output reconstructed images to end-users. This greatly limits the application of pre-processing methods. Post-processing methods work on the reconstruction images with no need for the raw projection and have better adaptability and flexibility that pre-processing methods. In addition, only the post-processing methods involve the transformation of Cartesian coordinates to polar coordinates and possibility of reducing the spatial resolution of the CT images. Therefore, pre-processing methods and post-processing methods are generally considered to be two different methods [36]. We focus on post-processing methods in this paper, and the proposed method is not tested against those seemingly comparable pre-processing methods.

Post-processing methods are mainly based on the idea of transforming the raw CT images from Cartesian coordinates to polar coordinates [38]. The ring artifacts in the raw CT images manifest as the stripe artifacts in polar coordinates, greatly reducing the difficulty of detection and elimination. Sijbars et al. proposed a method for removing the stripe artifacts based on morphological operators [39]. In polar coordinates, this method uses a sliding window to remove artifacts in the areas with serious stripe artifacts, but the selection of parameters is very important to the results. Subsequently, Wei et al. achieved a more rigorous distinction of artifacts and image details in synchrotron radiation CT images by combing wavelet decomposition and a Fourier low-pass filter [40]. According to the assumption that the gray-value features of the ring artifacts are local extrema, Kyriakou et al. [41] and Prell et al. [42] used a median filter to remove the stripe artifacts in polar coordinates. Since the stripe artifacts have obvious structure features, the variational methods have natural advantages in removing the stripe artifacts. Bouali et al. used unidirectional variational model (UV) to remove the stripe artifacts in images acquired by the moderate resolution imaging spectroradiometer (MODIS) [43]. Wu et al. used the variational method to remove shading artifacts in CT images without prior texture information [44]. Xu et al. used relative total variation (RTV) methods to separate texture and structure in natural images [45]. According to the sparse distribution of the stripe artifacts, Yan et al. proposed a variational method that can effectively remove the stripe artifacts, but the coordinate transformation process deteriorated the spatial resolution of the images [46]. Furthermore, Liang et al. applied the relative total variance (RTV) method in polar coordinates, and proposed a general iterative image domain ring artifact removal method [47]. This method can be widely used in clinical CT images without deteriorating the image details. However, this method may not be able to completely remove strong ring artifacts (this method deteriorates the spatial resolution when transforming the stripe artifacts into the ring artifacts, resulting in remaining some ring artifacts in the corrected CT images). Chao et al. proposed a radial basis function neural network (RBFNN) to remove ring artifacts. However, for the best important step recognition of artifacts, the method of manual recognition after artifact enhancement are used in this method. If there are strips or gaps in the artifacts with no obvious features, the manual recognition method is likely to have a certain error rate [48]. Fang et al. used deep learning methods for ring artifacts removal respectively in image domain, projection domain and the polar coordinate system. However, this method requires extensive training data, and the ring artefacts in the training data should be similar to the artefacts in experimental data [49]. Generally, most of post-processing methods default that the stripe artifacts in polar coordinates have constant gray values in the polar angle direction. That is, only the ring artifacts caused by the intensity-dependent detector responses, which can be called the intensity-dependent artifacts in this paper, are taken into consideration. However, in practice, the ring artifacts caused by the time-dependence of the CT hardware systems, which can be called the time-dependent artifacts in this paper, may cause gray values of the stripe artifacts to be not constant. For the above two kinds of ring artifacts, we propose a general image domain iterative post-processing method based on the relative total variance (RTV) in this paper. This method can effectively remove the intensity-dependent ring artifacts and the time-dependent ring artifacts in two parts while preserving the spatial resolution and the image details.

Both numerical simulation and real data are used to evaluate the proposed method. In the numerical simulation, we add some mixed ring artifacts (the mixed ring artifacts are composed of the intensity-dependent ring artifacts and the time-dependent ring artifacts) in a Shepp Logan phantom. The real data from real photon counting CT system (the photon counting CT has become a research hotspot) and real synchrotron data sets are used to evaluate the practical effectiveness of the proposed method for new CT systems. Considering that the method of Liang et al. can be regarded as a representative of post-processing variational method, and the method of Wei et al. (based on Wavelet-Fourier filter) can be regarded as a representative of post-processing filtering method, so these two algorithms are comparison algorithms in this article. The subjective evaluations are mainly based on image comparisons, and the objective evaluations are mainly based on signal to noise ratio (SNR), contrast noise ratio (CNR), and full width at half maxima (FWHM). The results show that the proposed method has achieved good results in the simulation and the real data, and can be widely used in clinical medicine and industrial detection.

2. Methods and materials

2.1 Workflow

As is widely known, the ring artifacts manifest as the stripe artifacts in polar coordinates, greatly reducing the difficulty of detection and elimination. Therefore, in this paper, we can transform the raw CT images into polar coordinate images. Most images can be expressed as “structure + texture”. Xu et al. have obtained the structure images with good edge preservation from natural images using the RTV method [37]. For a polar coordinate image, both the stripe artifacts and the image details have obvious texture characteristics, and they constitute the texture image. Therefore, the main idea of the proposed method is to obtain the texture image using the RTV algorithm, and then extract the stripe artifacts from the texture image. The framework for removing the ring artifacts are shown in Fig. 1. The steps of the proposed method are as follows:

  • 1. Transform the raw CT image with ring artifacts into a polar coordinate image using interpolation [41].
  • 2. Obtain a structure image by smoothing the polar coordinate image using an edge-preserving smoothing method such as the RTV algorithm (this step is described in detail in Section 2.2).
  • 3. Generate a texture image by subtracting the structure image from the polar coordinate image. The texture image can hardly contain all image details and stripe artifacts, which is a disadvantage of this step.
  • 4. Extract the stripe artifacts from the texture image (this step is described in detail in Section 2.3).
  • 5. Extract the ring artifacts by transforming the extracted stripe artifacts into Cartesian coordinate system. The spatial resolution will be lost when the extracted stripe artifacts are transformed into the ring artifacts, which is a disadvantage of this step.
  • 6. Obtain a corrected CT image with less ring artifacts by subtracting the extracted ring artifacts obtained in Step 5 from the raw CT image. Because the signal of the extracted ring artifacts is weak enough, the loss of spatial resolution in Step 5 has little effect on the corrected CT image. Compared with the raw CT image, the corrected CT image hardly loses spatial resolution and has less ring artifacts.
  • 7. Update the raw CT image using the corrected CT image and repeat Steps 1-6 to remove ring artifacts gradually until the ring artifacts extracted in the last iteration are weak enough (the stopping criterion of iterations is described in detail in Section 2.4). The purpose of this step is to remove the remaining ring artifacts in the corrected CT image to overcome the disadvantages in Steps 3 and 5.

 

Fig. 1. The framework for removing the ring artifacts.

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The most critical points in the proposed method are as follows: texture image acquisition based on the RTV algorithm, stripe artifact extraction and iterative stopping criterion.

2.2 Texture image acquisition based on the RTV algorithm

The polar coordinate image can be expressed as “structure + texture”, and we can obtain the structure image by smoothing the polar coordinate image. In order to obtain the structure image accurately, it is critical to select a good edge-preserving smoothing algorithm. In this paper, we use the RTV algorithm as the smoothing algorithm, which does not require a priori information. The relative total variation is used to capture the nature of structure and texture, and the non-uniform anisotropic texture can be effectively removed. More details about the RTV algorithm can be found in the work of Xu et al. [45]. From the perspective of practical application, the RTV algorithm can be simply described by:

$$S = tsmooth(I,\lambda ,\sigma ,\max Tter,\varepsilon ),$$
where S is the output image (structure image). I is the input image (polar coordinate image).$\lambda $ is the smoothing weight and should be selected according to the richness of ring artifacts and details in the polar coordinate image (the smaller the smoothing weight, the more details in the structure image). $\sigma $ is the maximum size of texture element (in order to completely remove the stripe artifacts, $\sigma \textrm{ = }4$ is taken in this paper according to the width of most stripe artifacts in the polar coordinate image). $\max Tter$ is the number of iterations (we select $\max Tter\textrm{ = }4$, which is the default value in the work of Xu et al.). $\varepsilon $ is the parameter that controls the sharpness of the output image (we select $\varepsilon \textrm{ = }0.02$, which is the default value in the work of Xu et al.) [45]. In fact, most of parameters are default value in the work of Xu et al. We only modify the value of $\sigma $ according to the width of most stripe artifacts in the polar coordinate image. $\lambda $ is the most important parameter in Eq. (1), and we need to adjust $\lambda $ according to the specific polar coordinate images.

We can obtain the texture image by subtracting the structure image from the polar coordinate image. In order to avoid the possibility of losing image details when extracting the stripe artifacts from the texture image, the key of this step is that the texture image should not contain any structure information. Therefore, for the RTV algorithm, we should select a small smoothing weight. At this time, the texture image contains only some image details and stripe artifacts. We can only remove some ring artifacts in a single extraction. The remaining ring artifacts can be extracted in subsequent iterations. However, it is notable that if the weight is too small, it may increase the time consumption of the whole workflow. Therefore, we need to adjust $\lambda $ according to the specific image to ensure that the texture image contain no structure information and the workflow is not too time-consuming.

2.3 Stripe artifact extraction

The texture image is composed of the stripe artifacts and the image details. The key step of the proposed method is to extract the stripe artifacts from the texture image. Generally, for stable CT systems, their polar coordinate images may only contain the intensity-dependent stripe artifacts. For nonstable CT systems, their polar coordinate images may contain the mixed stripe artifacts (the mixed stripe artifacts are composed of the intensity-dependent stripe artifacts and the time-dependent stripe artifacts). Therefore, we can discuss the intensity-dependent stripe artifacts and the time-dependent stripe artifacts respectively.

2.3.1 Extraction of intensity-dependent stripe artifacts

Firstly, we discuss the intensity-dependent stripe artifacts whose features are simple enough. When the smoothing weight of the RTV algorithm is small, for the image details, the sum of gray values of all pixels in the polar angle direction can be regarded as zero. It is well known that the intensity-dependent stripe artifacts have same gray values in the polar angle direction of the texture image. Since the texture image can be divided into intensity-dependent stripe artifacts and the image details, the intensity-dependent stripe artifacts can be estimated as the average gray value of all pixels in the polar angle direction of the texture image [47], as shown in Fig. 2. If necessary, we can take the range of 80% in the middle size of the gray value points for mean value calculation in order to exclude the influence of dead pixels. At this time, our method is similar to the method of Liang et al. and can be considered as an improvement of the whole framework for removing the intensity-dependent ring artifacts [47]. The method updates the raw CT image using the corrected CT image, which avoids the loss of spatial resolution when the stripe artifacts are transformed into ring artifacts, and has a good intensity-dependent ring artifact removal effect.

 

Fig. 2. The process for extracting the intensity-dependent stripe artifacts.

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2.3.2 Extraction of time-dependent stripe artifacts

Secondly, we discuss the time-dependent stripe artifacts whose features are complex. The time-dependent ring artifacts in CT images may be caused by many factors. For example, in synchrotron tomography, the thermal instability of the optics, thermal fluctuations in the cooling system, the instability of the bending magnet on the APS storage ring, and small time-dependent perturbations of the CCD sensor may cause the time-dependent ring artifacts [11]. In photon counting CT, the photon counting detector usually suffer from the local charge trap effects, which usually depend on the polarization time and the exposure time [19]. In addition, because the manufacturing techniques of the photon counting detector are not fully mature enough, some pixels often show gain instability. In conclusion, both the instability of X-ray sources and the instability of the detector responses may lead to time-related ring artifacts, and the characteristics of time-dependent artifacts may be related to specific CT hardware systems. From the work of Brombal et al. [19] and the data obtained from our photon counting CT system, it can be found that the actual time-dependent stripe artifacts are discontinuous in the polar angle direction (the gray value is not constant). We hope to provide a more accurate and detailed description and analysis, but the time dependence of different CT hardware systems is difficult to predict. According to the discontinuity characteristics of stripe artifacts, we can divide the pixels into different categories. The pixels in each category have similar gray values in the in the texture image, and should be spatially continuous. It is difficult to estimate the exact category number of time-dependent stripe artifacts because of the complexity of the time-dependent stripe. Therefore, for simplicity, we assume that the stripe artifacts can be divided into two categories, and other categories of the stripe artifacts (if they do exist) can be extracted in subsequent iterations. This paper requires that pixels of the same category have the highest gray-value similarity and the smallest differences. The classification function can be described by:

$$\begin{array}{l} \arg \min {\kern 1pt} {\kern 1pt} {\kern 1pt} F({p,q} ){\kern 1pt} {\kern 1pt} = \sum\limits_{i = p}^q {{{({{R_i} - E{C_1}} )}^2}} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {\kern 1pt} \sum\limits_{j = 1}^{(p - 1)} {{{({{R_j} - E{C_2}} )}^2} + {\kern 1pt} \sum\limits_{j = (q + 1)}^M {{{({{R_j} - E{C_2}} )}^2}} } {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {R_i} \in {C_1},{R_j} \in {C_2}, \end{array}$$
where ${R_i}$ is the $i\textrm{ - }th$ pixel in the polar angle direction. ${C_1}$ represents the first category of pixels, that is, the pixels from the $p\textrm{ - }th$ pixel to the $q\textrm{ - }th$ pixel in the polar angle direction. $E{C_1}$ represents the mean value of all pixels in the first category. ${C_2}$ represents the second category of pixels, that is, all pixels except the first category of pixels in the polar angle direction (from the first pixel to the $(p - 1)th$ pixel and the $({q + 1} )th$ pixel to the $M\textrm{ - }th$ in the polar angle direction). $E{C_2}$ represents the mean value of all pixels in the second category; $M$ represents the total number of pixels in the polar angle direction. All pixels of the two categories are continuous.

In order to preserve the image details, the number of the pixels in each category should be greater than the set threshold. Here we must analyze the effect of this threshold on the results. We obtain the raw structure image by smoothing the polar coordinate image using the RTV algorithm. The structure image contains almost no image details, makes the details in the texture image obvious as shown in Fig. 2. If the threshold is too small, the classification function may mistakenly divide the image details into one category and other pixels into another category, resulting in the loss of image details. In this paper, we select the threshold as 10% of total pixels. The threshold is large enough to avoid the loss of small (within 10% of the total pixels) details. In fact, the threshold can be modified to other values according to the specific texture image, such as 5% (as long as the loss of image details can be avoided, the threshold has little effect on the results). Figure 3 shows the three-dimensional (3D) graph of the classification function $F({p,q} ){\kern 1pt} {\kern 1pt}$ for a certain stripe artifact in the polar angle direction (if $q > = p$ or $(q - p) < = 0.1\ast M$, we set the value of the classification function $F({p,q} ){\kern 1pt} {\kern 1pt}$ to the maximum value).

 

Fig. 3. The 3D graph of the classification function $F({p,q} ){\kern 1pt} {\kern 1pt}$ changes with p and q for a certain stripe artifact.

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The time-dependent stripe artifacts can be estimated as the mean value of all pixels in their corresponding categories. The number of the pixels in each category should be greater than the set threshold. The corrected texture image can be obtained by subtracting the extracted stripe artifacts from the texture image. Considering the complexity of the time-dependent stripe artifacts, it is difficult to remove them completely in a single extraction. We can increase the number of iterations according to the complexity of the time-dependent stripe artifacts. Finally, we can extract the final time-dependent stripe artifacts by subtracting the final corrected texture image from the texture image. The process for extracting the time-dependent stripe artifacts is shown in Fig. 4, and we can find that there are more than two categories of stripe artifacts in the final time-dependent stripe artifacts. Therefore, we can extract complex (multi-category) time-dependent stripe artifacts by dividing the pixels into two categories. Dividing pixels into three or more categories may increase the complexity of the method, but it does not help in extracting time-dependent stripe artifacts.

 

Fig. 4. The process for extracting the time-dependent stripe artifacts.

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For the intensity-dependent stripe artifacts and the time-dependent stripe artifacts, the premise of this step is that the texture image does not contain any structural information. Therefore, when obtaining the texture image, we should select a small smoothing weight for the RTV algorithm.

2.4 Stopping criterion of iterations

The main idea of the method is to remove some ring artifacts during each iteration, and gradually removes the ring artifacts in CT images in subsequent iterations. If the ring artifacts extracted in the last iteration are weak enough compared to the ring artifacts extracted in the first iteration, the workflow is terminated. We can use L1 norm to quantize the ring artifacts and the stopping criterion can be described by:

$${C_s} = {||{RIN{G_K}} ||_1}/{||{RIN{G_1}} ||_1},$$
where $RIN{G_K}$ is the ring artifacts extracted in the $K - th$ iteration, and $RIN{G_1}$ is the ring artifacts extracted in the first iteration.

In practice, the raw CT images may contain the intensity-dependent ring artifacts for the stable CT hardware systems, and may contain mixed ring artifacts for the unstable CT hardware systems. In this case, we can divide the proposed method into two parts. The first part of the proposed method only contains the process for extracting the intensity-dependent stripe artifacts. The second part only contains the process for extracting the time-dependent stripe artifacts. We can remove intensity-dependent ring artifacts through the first part of the proposed method. In addition, we can remove the remaining time-dependent ring artifacts in the corrected images through the second part of the proposed method (the corrected images are obtained by the first part of the proposed method). Since the features of the intensity-dependent ring artifacts are simple enough, ${C_s}$ can be set to 0.008 in the first part of the proposed method. However, because the features of the time-dependent ring artifacts are complex, ${C_s}$ can be determined according to the severity of the time-dependent ring artifacts.

2.5 Experiment and evaluation method

In order to verify the effectiveness of the proposed method, we used the proposed method to remove the ring artifacts in the Shepp-Logan phantom and an aluminum part [50]. Reconstruction parameters are shown in Table 1:

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Table 1. Reconstruction parameters.

Considering that the first part of the proposed method is similar to the work of Liang et al., we use the method of Liang et al. as the comparison method, and the parameters and the stopping criterion are the same as the first part of the proposed method. Image qualities were compared by subjective observation, CNR, FWHM, and SNR. The weaker the ring artifacts, the higher the CNR and SNR. The better the image maintains the spatial resolution, the closer the FWHM is to the ideal value.

3. Results

Due to space limitations, we just show simulation results with mixed artifacts and the real data from real photon counting CT system and synchrotron data sets. The photon counting CT system is mainly composed of CdTe photon counting detector XCounter Hydra FX50 and X-ray source Comet MXR-451HP/11. The synchrotron data sets used in this paper are from Zenodo https://doi.org/10.5281/zenodo.1443568.

3.1 Simulation results

First, in order to verify the effectiveness of the proposed method for removing mixed ring artifacts, we randomly changed detector responses in the Shepp-Logan phantom 360° projections and some detector responses are time-dependent. We reconstructed the CT images with the mixed ring artifacts using the FDK algorithm. We used the methods of Wei et al., Liang et al., and the proposed method to obtain the corrected images. For the mixed ring artifacts, we removed the intensity-dependent ring artifacts through the first part of the proposed method, and removed the remaining time-dependent ring artifacts through the second part of the proposed method. The comparisons are shown in Fig. 5. Through visual observations, we can compare the effect of the three methods for removing mixed ring artifacts. The method of Wei et al. suppressed the intensity of some ring artifacts, but introduced some additional artifacts and destroyed some details. The method of Liang et al. suppressed the intensity of some ring artifacts, but introduced some additional ring artifacts. The first part of the proposed method was performed closely to the method of Liang et al. However, the second part of the proposed method effectively removed the remaining ring artifacts while preserving the spatial resolution and the image details. Therefore, the proposed method (including the first part and the second part) is superior to the methods of Wei et al. and Liang et al. for the Shepp-Logan phantom with the mixed ring artifacts.

 

Fig. 5. The comparisons of the corrected images of the Shepp-Logan phantom with the mixed artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al. (d) The corrected image obtained by the first part of the proposed method. (e) The corrected image obtained by the second part of the proposed method. (f) The reference image; the images in the second row are zoomed-in views of the images in the first row; all above display windows are [0.98, 1.05]. The images in third row are the ring artifacts of the images in the first row, and the display windows are [-0.001, 0.005].

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3.2 Real data results

In order to verify the practical effectiveness of the proposed method, the images of an aluminum part were obtained by the photon counting CT system, and we used the methods of Wei et al., Liang et al., and the proposed method to obtain the corrected images. In order to verify the anti-noise performance of the proposed method, the images have some noise. We can find that there are serious real mixed ring artifacts in the uncorrected CT images. In this case, we removed the intensity-dependent ring artifacts through the first part of the proposed method, and removed the remaining time-dependent ring artifacts through the second part of the proposed method. The comparisons are shown in Fig. 6. Through visual observations, we can compare the effect of three methods for removing the real mixed ring artifacts. The method of Wei et al. suppressed the intensity of some ring artifacts, but some ring artifacts still remained. The method of Liang et al. suppressed the intensity of some ring artifacts, but introduced some additional ring artifacts. The first part of the proposed method performed closely to the method of Liang et al. However, the second part of the proposed method effectively removed the remaining ring artifacts while preserving the spatial resolution and the image details. Therefore, the proposed method (including the first part and the second part) is superior to the methods of Wei et al. and Liang et al. for the real data. At the same time, it proves that the proposed method can be applied to new CT systems whose manufacturing techniques are not fully mature enough.

 

Fig. 6. The comparisons of the corrected images of the aluminum part with the real mixed ring artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al. (d) The corrected image obtained by the first part of the proposed method. (e) The corrected image obtained by the second part of the proposed method. The images in the second row are zoomed-in views of the images in the first row; all above display windows are [-0.2,1.2]. The images in third row are the ring artifacts of the images in the first row (if we assume Fig. 7(e) as the reference image), and the display windows are [-0.05,0.25].

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Furthermore, we used the methods of Wei et al., Liang et al., and the proposed method to correct the real data from synchrotron data sets. There are ring artifacts in the uncorrected CT images. In this case, we removed the intensity-dependent ring artifacts through the first part of the proposed method, and removed the remaining time-dependent ring artifacts through the second part of the proposed method. Since the original images is very large, we only look at some parts for details comparison. The comparisons are shown in Fig. 7. Through the visual observations, we can compare the effect of three methods for removing the real ring artifacts. The method of Wei et al. suppressed the intensity of some ring artifacts, but some ring artifacts remained such as the position marked as A. From the position marked as B in the second row of Fig. 7, it can be seen that the hole is smaller than the normal, so some details may be lost in Fig. 7(b). The method of Liang et al. suppressed the intensity of some ring artifacts, but introduced some additional ring artifacts. The first part of the proposed method performed closely to the method of Liang et al. However, the second part of the proposed method effectively removed the remaining ring artifacts while preserving the spatial resolution and the image details. Therefore, the proposed method (including the first part and the second part) is superior to the methods of Wei et al. and Liang et al. for the real data from synchrotron data sets. At the same time, it proves that the proposed method can be applied to new CT systems whose manufacturing techniques are not fully mature enough.

 

Fig. 7. The comparisons of the corrected images of the real synchrotron data sets with ring artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al.; (d) the corrected image obtained by the first part of the proposed method; (e) the corrected image obtained by the second part of the proposed method; the images in the second row are zoomed-in views of the images in the first row. The images in third row are the ring artifacts of the images in the first row (if we assume Fig. 7(e) as the reference image), and all above display windows are [-0.0015,0.0005].

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3.3 Numerical evaluations

Because the data from synchrotron data sets have no obvious structure, we only evaluate the results of simulated data from Shepp-Logan phantom and real data from the photon counting CT system. We use FWHM as a spatial resolution metric and the simulated data are used to verify the effect the proposed method numerically. In the simulated Shepp Logan phantom, we fitted the line profiles around the edge marked as line C in the second row of Fig. 5 with a logistic error function and computed the FWHM and shows the result in Table 2. The SNR at the position A, the CNR at position B in Figs. 5 and 6 were calculated and shows the result in Table 2.

Tables Icon

Table 2. Numerical evaluations of the simulation and experimental data.

From the data in Table 2, we can find the corrected images obtained by the proposed method have significant improvements in image quality compared to the uncorrected images and the corrected images obtained by the methods from Wei et al. and Liang et al. Particularly, when the time-dependent ring artifacts are serious, the first part of the proposed method (for removing the intensity-dependent ring artifacts) still performs slightly better than the method of Liang et al., which further shows the superiority of the proposed method. The proposed method can remove the ring artifacts and maintain the spatial resolution as much as possible.

4. Discussion

4.1 Influence of the smoothing weight on the removal of intensity-dependent ring artifacts

When using the RTV algorithm to smooth the polar coordinate images, the smoothing weight has great influence on the generation of the structure images. In order to verify the influence of the smoothing weight on the intensity-dependent ring artifact removal, the Shepp-Logan phantom with the mixed ring artifacts shown in Fig. 5(a) were corrected with the smoothing weights $\lambda \textrm{ = 5}.0e - 6$ and $\lambda \textrm{ = 5}.0e - 7$, respectively. We consider the effect of removing intensity-dependent artifacts in the first part of the method, and consider the effect of removing time-dependent artifacts in the second part of the method. In this paper, we calculated the evaluation of image detail maintenance using the structural similarity index (SSIM) [47]. The numbers of iterations required to reach the stopping criterion are 29 and 198, respectively. The variation of SSIM with the methods, the smoothing weights and the number of iterations is shown in Fig. 8. The results show that the SSIM increases gradually with the number of iterations, and both the method of Liang et al. and the first part of the proposed method are robust in some degree (different smoothing parameters have little effect on the final SSIM). For the Shepp-Logan phantom with the mixed ring artifacts shown in Fig. 5(a), the first part of the proposed method achieves better results than the method of Liang et al.

 

Fig. 8. The variation of SSIM with the methods, the smoothing weights and the number of iterations.

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4.2 Influence of the smoothing weight on the removal of time-dependent ring artifacts

In order to verify the influence of the smoothing weight on the removal of the time-dependent ring artifacts, the Shepp-Logan phantom with the time-dependent ring artifacts shown in Fig. 5(e) were corrected with the smoothing weights $\lambda \textrm{ = }3.0 \times {10^{\textrm{ - }6}}$, $\lambda \textrm{ = 5}.0 \times {10^{\textrm{ - }6}}$ and $\lambda \textrm{ = 7}.0 \times {10^{\textrm{ - }6}}$, respectively. The number of iterations required to reach the stopping criterion are 91, 68, and 44 respectively. When the stopping criterion of iterations was reached, the workflow would be stopped. The variation of SSIM with the smoothing weights and the number of iterations is shown in Fig. 9. The results show that the SSIM increases gradually with the number of iterations, and different smoothing parameters have little effect on the final SSIM. This proves that the second part of the proposed method is robust in some degree, and can effectively remove the time-dependent ring artifacts.

 

Fig. 9. The variation of SSIM with the methods, the smoothing weights and the number of iterations.

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4.3 Advantages and prospects of research

The proposed method shows good performance in simulation and real data. There are two main advantages of the proposed method listed as follows. Firstly, for the intensity-dependent ring artifacts, the first part of the proposed method overcomes the shortcoming of the method if Liang et al. by iterating the corrected CT image again. Secondly, the second part of the method can effectively remove the time-dependent ring artifacts while preserving the spatial resolution and the image details, greatly improving the applicability of the proposed method. However, for multi-material samples, when attenuation characteristics of materials vary greatly and the time-dependent ring artifacts are very serious, the proposed method may face some difficulties: for the RTV algorithm, selecting a large smoothing weight may lead to a loss of image details, and selecting a small smoothing weight may be unable to completely remove the time-dependent ring artifacts. Pre-processing methods work on the raw projections, and post-processing methods work on the reconstruction images. For the multi-material CT images, the method for removing serious time-dependent ring artifacts, the combination of pre-processing methods and post-processing methods may be a good solution to completely remove ring artifacts, because it is hard to obtain a perfect algorithm suppressing all ring artifacts.

4.4 Computation load of the proposed method

The computation load of the first step of the proposed method is basically the same as that of the method of Liang et al. However, we must point out that the second part of the proposed method is more time-consuming than the method of Liang et al. and the first part of the proposed method. Even though the second part of the proposed method is more computationally intense than the other methods here considered, we believe that there are some methods to reduce the computation load. We can accelerate the computation with CPU multi-thread and GPU multi-thread. Furthermore, we can reduce the computation load of the texture classification (the only time-consuming step) by image scaling via the following steps. First, scale the texture image to reduce its width. Second, generate low-resolution classification function 3D graphs and find the temporary minimum points. Third, search the real minimum points of classification function in the neighborhoods of the temporary minimum points. Finally, extract the stripe artifacts according to the real minimum points. If we reduce the width of the texture image to 1/2 of the original, the computation load can be reduced to 1/4 of the original cost time. The optimized method takes 22x longer than the method of Liang et al., but have better performance in removing ring artifacts.

5. Conclusion

In this study, we propose a robust iterative post-processing method to remove the intensity-dependent ring artifacts and the time-dependent ring artifacts. The proposed method is evaluated on a Shepp Logan phantom with the mixed ring artifacts, an industrial aluminum part with real ring artifacts, and real synchrotron data sets. The simulation and real data show that the proposed method can effectively remove intensity-dependent and time-dependent ring artifacts while preserving the image details and the spatial resolution. In real data, this method increased the SNR from 5.88 to 14.01 and CNR from 3.89 to 11.26 of the regions of interest. In particular, this method is particularly suitable for new CT systems (such as the photon counting CT) with immature manufacturing technologies but unique advantages, and can be widely used in clinical, medical, and industrial detection.

Funding

Ministry of Industry and Information Technology of the People's Republic of China (MJ-2017-F-05); Fundamental Research Funds for the Central Universities (31020190504006); Technology Field Fund of Basic Strengthening Plan (2019-JCJQ-JJ-391); Shanxi Provincial Key Research and Development Project (2020GY-145).

Disclosures

The authors declare no conflicts of interest.

References

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2. A. Cuadros, X. Ma, and G. R. Arce, “Compressive spectral X-ray tomography based on spatial and spectral coded illumination,” Opt. Express 27(8), 10745–10764 (2019). [CrossRef]  

3. X. Zhao, P. Chen, J. Wei, and Z. Qu, “Spectral CT imaging method based on blind separation of polychromatic projections with Poisson prior,” Opt. Express 28(9), 12780–12794 (2020). [CrossRef]  

4. Z. Wang, J. Li, and M. Enoh, “Removing ring artifacts in CBCT images via generative adversarial networks with unidirectional relative total variation loss,” Neural. Comput. Appl. 31(9), 5147–5158 (2019). [CrossRef]  

5. H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014). [CrossRef]  

6. X. Liang, S. Gong, Q. Zhou, Z. Zhang, Y. Xie, and T. Niu, “SU-F-J-211: Scatter Correction for Clinical Cone-Beam CT System Using An Optimized Stationary Beam Blocker with a Single Scan,” Med. Phys. 43(6), 3457 (2016). [CrossRef]  

7. W. Yuan, “Extended Applications of Image Flat-Field Correction Method,” Acta. Photonica. Sin. 36(9), 1587–1590 (2007).

8. X. Tang, R. Ning, R. Yu, and D. Conover, “Cone beam volume CT image artifacts caused by defective cells in x-ray flat panel imagers and the artifact removal using a wavelet-analysis-based algorithm,” Med. Phys. 28(5), 812–825 (2001). [CrossRef]  

9. F. Sadi, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter,” Comput. Biol. Med. 40(1), 109–118 (2010). [CrossRef]  

10. R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008). [CrossRef]  

11. V. Titarenko, S. Titarenko, P. J. Withers, D. C. Francesco, and X. Xiao, “Improved tomographic reconstructions using adaptive time dependent intensity normalization,” J. Synchrotron Radiat. 17(5), 689–699 (2010). [CrossRef]  

12. V. Astromskas, E. N. Gimenez, A. Lohstroh, and N. Tartoni, “Evaluation of Polarization Effects of, Collection Schottky CdTe Medipix3RX Hybrid Pixel Detector,” IEEE Trans. Nucl. Sci. 63(1), 252–258 (2016). [CrossRef]  

13. P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017). [CrossRef]  

14. J. Lifton and T. Liu, “Ring artefact reduction via multi-point piecewise linear flat field correction for X-ray computed tomography,” Opt. Express 27(3), 3217–3228 (2019). [CrossRef]  

15. N. V. Van, B. J. De, C. F. De, L. Mancini, F. Marone, and J. Sijbers, “Dynamic intensity normalization using eigen flat fields in X-ray imaging,” Opt. Express 23(21), 27975–27989 (2015). [CrossRef]  

16. G. R. Davis and J. C. Elliott, “X-ray microtomography scanner using time-delay integration for elimination of ring artefacts in the reconstructed image,” Nucl. Instrum. Methods Phys. Res., Sect. A 394(1-2), 157–162 (1997). [CrossRef]  

17. W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001). [CrossRef]  

18. C. Altunbas, C. J. Lai, Y. Zhong, and C. C Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014). [CrossRef]  

19. L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018). [CrossRef]  

20. X. Tang, R. Ning, R. Yu, and D. L. Conover, “2D wavelet-analysis-based calibration technique for flat panel imaging detectors: Application in cone beam volume CT,” Proc. SPIE 3659, 806–816 (1999). [CrossRef]  

21. M. Boin and A. Haibel, “Compensation of ring artefacts in synchrotron tomographic images,” Opt. Express 14(25), 12071–12075 (2006). [CrossRef]  

22. R. A. Ketcham, “New algorithms for ring artifact removal,” Proc. SPIE 6318(38), 63180O (2006). [CrossRef]  

23. P. Wu, T. Mao, S. Xie, K. Sheng, and T. Niu, “WE-G-207-09: A Practical Bowtie Ring Artifact Correction Algorithm for Cone-Beam CT,” Med. Phys. 42(6Part41), 3698 (2015). [CrossRef]  

24. E. M. A. Anas, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in CT imaging through detection and correction of stripes in the sinogram,” Phys. Med. Biol. 55(22), 6911–6930 (2010). [CrossRef]  

25. A. N. M. Ashrafuzzaman, S. Y. Lee, and M. K. Hasan, “A self-adaptive approach for the detection and correction of stripes in the sinogram: suppression of ring artifacts in CT imaging,” EURASIP J. Adv. Signal. Process. 2011(1), 183547 (2011). [CrossRef]  

26. G. Kowalski, “Suppression of Ring Artefacts in CT Fan-Beam Scanners,” IEEE Trans Nucl. Sci. 25(5), 1111–1116 (1978). [CrossRef]  

27. C. Raven, “Numerical removal of ring artifacts in microtomography,” Rev. Sci. Instrum. 69(8), 2978–2980 (1998). [CrossRef]  

28. B. Munch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet — Fourier filtering,” Opt. Express 17(10), 8567–8591 (2009). [CrossRef]  

29. M. Boin and A. Haibel, “Compensation of ring artefacts in synchrotron tomographic images,” Opt. Express 14(25), 12071–12075 (2006). [CrossRef]  

30. C. Altunbas, C. Lai, Y. Zhong, and C. C. Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014). [CrossRef]  

31. S. Titarenko, P. J. Withers, and A. Yagola, “An analytical formula for ring artefact suppression in X-ray tomography,” Appl. Math. Lett. 23(12), 1489–1495 (2010). [CrossRef]  

32. E. X. Miqueles, J. Rinkel, F. O’Dowd, and J. S. V. Bermúdez, “Generalized Titarenko’s algorithm for ring artefacts reduction,” J. Synchrotron Radiat. 21(6), 1333–1346 (2014). [CrossRef]  

33. Y. Kim, J. Baek, and D. Hwang, “Ring artifact correction using detector line-ratios in computed tomography,” Opt. Express 22(11), 13380–13392 (2014). [CrossRef]  

34. V. Titarenko, “1D Filter for Ring Artifact Suppression,” J. Synchrotron Radiat. 23(6), 800–804 (2016). [CrossRef]  

35. W. Vågberg, J. C. Larsson, and H. M. Hertz, “Removal of ring artifacts in microtomography by characterization of scintillator variations,” Opt. Express 25(19), 23191–23198 (2017). [CrossRef]  

36. N. T. Vo, R. C. Atwood, and M. Drakopoulos, “Superior techniques for eliminating ring artifacts in x-ray microtomography,” Opt. Express 26(22), 28396–28412 (2018). [CrossRef]  

37. L. Croton, G. Ruben, K. S. Morgan, D. M. Pagnin, and M. Kitchen, “Ring artifact suppression in X-ray computed tomography using a simple, pixel-wise response correction,” Opt. Express 27(10), 14231–14245 (2019). [CrossRef]  

38. W. Chen, D. Prell, Y. Kyriakou, and W. A. Kalender, “Accelerating Ring Artifact Correction for Flat-Detector CT using the CUDA Framework,” Proc. SPIE 7622, 76223A (2010). [CrossRef]  

39. J. Sijbers and A. Postnov, “Reduction of ring artefacts in high resolution micro-CT reconstructions,” Phys. Med. Biol. 49(14), N247–N253 (2004). [CrossRef]  

40. Z. Wei, S. Wiebe, and D. Chapman, “Ring artifacts removal from synchrotron CT image slices,” J. Instrum. 8(06), C06006 (2013). [CrossRef]  

41. Y. Kyriakou, D. Prell, and W. A. Kalender, “Ring artifact correction for high-resolution micro CT,” Phys. Med. Biol. 54(17), N385–N391 (2009). [CrossRef]  

42. D. Prell, Y. Kyriakou, and W. A. Kalender, “Comparison of ring artifact correction methods for flat-detector CT,” Phys. Med. Biol. 54(12), 3881–3895 (2009). [CrossRef]  

43. M. Bouali and S. Ladjal, “Toward Optimal Destriping of MODIS Data Using a Unidirectional Variational Model,” IEEE Trans. Geosci. Remote. Sens. 49(8), 2924–2935 (2011). [CrossRef]  

44. P. Wu, X. Sun, H. Hu, T. Mao, W. Zhao, K. Sheng, A. A. Cheung, and T. Niu, “Iterative CT shading correction with no prior information,” Phys. Med. Biol. 60(21), 8437–8455 (2015). [CrossRef]  

45. L. Xu, Q. Yan, Y. Xia, and J. Jia, “Structure extraction from texture via relative total variation,” ACM Trans. Graph. 31(6), 1–10 (2012). [CrossRef]  

46. L. Yan, T. Wu, S. Zhong, and Q. Zhang, “A variation-based ring artifact correction method with sparse constraint for flat-detector CT,” Phys. Med. Biol. 61(3), 1278–1292 (2016). [CrossRef]  

47. X. Liang, Z. Zhang, T. Niu, S. Yu, S. Wu, Z. Li, H. Zhang, and Y. Xie, “Iterative image-domain ring artifact removal in cone-beam CT,” Phys. Med. Biol. 62(13), 5276–5292 (2017). [CrossRef]  

48. Z. Chao and H. Kim, “Removal of computed tomography ring artifacts via radial basis function artificial neural networks,” Phys. Med. Biol. 64(23), 235015 (2019). [CrossRef]  

49. W. Fang, L. Li, and Z. Chen, “Removing Ring Artefacts for Photon-Counting Detectors Using Neural Networks in Different Domains,” IEEE Access 8, 42447–42457 (2020). [CrossRef]  

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References

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  1. C. Han and J. Baek, “Multi-pass approach to reduce cone-beam artifacts in a circular orbit cone-beam CT system,” Opt. Express 27(7), 10108–10126 (2019).
    [Crossref]
  2. A. Cuadros, X. Ma, and G. R. Arce, “Compressive spectral X-ray tomography based on spatial and spectral coded illumination,” Opt. Express 27(8), 10745–10764 (2019).
    [Crossref]
  3. X. Zhao, P. Chen, J. Wei, and Z. Qu, “Spectral CT imaging method based on blind separation of polychromatic projections with Poisson prior,” Opt. Express 28(9), 12780–12794 (2020).
    [Crossref]
  4. Z. Wang, J. Li, and M. Enoh, “Removing ring artifacts in CBCT images via generative adversarial networks with unidirectional relative total variation loss,” Neural. Comput. Appl. 31(9), 5147–5158 (2019).
    [Crossref]
  5. H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
    [Crossref]
  6. X. Liang, S. Gong, Q. Zhou, Z. Zhang, Y. Xie, and T. Niu, “SU-F-J-211: Scatter Correction for Clinical Cone-Beam CT System Using An Optimized Stationary Beam Blocker with a Single Scan,” Med. Phys. 43(6), 3457 (2016).
    [Crossref]
  7. W. Yuan, “Extended Applications of Image Flat-Field Correction Method,” Acta. Photonica. Sin. 36(9), 1587–1590 (2007).
  8. X. Tang, R. Ning, R. Yu, and D. Conover, “Cone beam volume CT image artifacts caused by defective cells in x-ray flat panel imagers and the artifact removal using a wavelet-analysis-based algorithm,” Med. Phys. 28(5), 812–825 (2001).
    [Crossref]
  9. F. Sadi, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter,” Comput. Biol. Med. 40(1), 109–118 (2010).
    [Crossref]
  10. R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
    [Crossref]
  11. V. Titarenko, S. Titarenko, P. J. Withers, D. C. Francesco, and X. Xiao, “Improved tomographic reconstructions using adaptive time dependent intensity normalization,” J. Synchrotron Radiat. 17(5), 689–699 (2010).
    [Crossref]
  12. V. Astromskas, E. N. Gimenez, A. Lohstroh, and N. Tartoni, “Evaluation of Polarization Effects of, Collection Schottky CdTe Medipix3RX Hybrid Pixel Detector,” IEEE Trans. Nucl. Sci. 63(1), 252–258 (2016).
    [Crossref]
  13. P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
    [Crossref]
  14. J. Lifton and T. Liu, “Ring artefact reduction via multi-point piecewise linear flat field correction for X-ray computed tomography,” Opt. Express 27(3), 3217–3228 (2019).
    [Crossref]
  15. N. V. Van, B. J. De, C. F. De, L. Mancini, F. Marone, and J. Sijbers, “Dynamic intensity normalization using eigen flat fields in X-ray imaging,” Opt. Express 23(21), 27975–27989 (2015).
    [Crossref]
  16. G. R. Davis and J. C. Elliott, “X-ray microtomography scanner using time-delay integration for elimination of ring artefacts in the reconstructed image,” Nucl. Instrum. Methods Phys. Res., Sect. A 394(1-2), 157–162 (1997).
    [Crossref]
  17. W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
    [Crossref]
  18. C. Altunbas, C. J. Lai, Y. Zhong, and C. C Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
    [Crossref]
  19. L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
    [Crossref]
  20. X. Tang, R. Ning, R. Yu, and D. L. Conover, “2D wavelet-analysis-based calibration technique for flat panel imaging detectors: Application in cone beam volume CT,” Proc. SPIE 3659, 806–816 (1999).
    [Crossref]
  21. M. Boin and A. Haibel, “Compensation of ring artefacts in synchrotron tomographic images,” Opt. Express 14(25), 12071–12075 (2006).
    [Crossref]
  22. R. A. Ketcham, “New algorithms for ring artifact removal,” Proc. SPIE 6318(38), 63180O (2006).
    [Crossref]
  23. P. Wu, T. Mao, S. Xie, K. Sheng, and T. Niu, “WE-G-207-09: A Practical Bowtie Ring Artifact Correction Algorithm for Cone-Beam CT,” Med. Phys. 42(6Part41), 3698 (2015).
    [Crossref]
  24. E. M. A. Anas, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in CT imaging through detection and correction of stripes in the sinogram,” Phys. Med. Biol. 55(22), 6911–6930 (2010).
    [Crossref]
  25. A. N. M. Ashrafuzzaman, S. Y. Lee, and M. K. Hasan, “A self-adaptive approach for the detection and correction of stripes in the sinogram: suppression of ring artifacts in CT imaging,” EURASIP J. Adv. Signal. Process. 2011(1), 183547 (2011).
    [Crossref]
  26. G. Kowalski, “Suppression of Ring Artefacts in CT Fan-Beam Scanners,” IEEE Trans Nucl. Sci. 25(5), 1111–1116 (1978).
    [Crossref]
  27. C. Raven, “Numerical removal of ring artifacts in microtomography,” Rev. Sci. Instrum. 69(8), 2978–2980 (1998).
    [Crossref]
  28. B. Munch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet — Fourier filtering,” Opt. Express 17(10), 8567–8591 (2009).
    [Crossref]
  29. M. Boin and A. Haibel, “Compensation of ring artefacts in synchrotron tomographic images,” Opt. Express 14(25), 12071–12075 (2006).
    [Crossref]
  30. C. Altunbas, C. Lai, Y. Zhong, and C. C. Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
    [Crossref]
  31. S. Titarenko, P. J. Withers, and A. Yagola, “An analytical formula for ring artefact suppression in X-ray tomography,” Appl. Math. Lett. 23(12), 1489–1495 (2010).
    [Crossref]
  32. E. X. Miqueles, J. Rinkel, F. O’Dowd, and J. S. V. Bermúdez, “Generalized Titarenko’s algorithm for ring artefacts reduction,” J. Synchrotron Radiat. 21(6), 1333–1346 (2014).
    [Crossref]
  33. Y. Kim, J. Baek, and D. Hwang, “Ring artifact correction using detector line-ratios in computed tomography,” Opt. Express 22(11), 13380–13392 (2014).
    [Crossref]
  34. V. Titarenko, “1D Filter for Ring Artifact Suppression,” J. Synchrotron Radiat. 23(6), 800–804 (2016).
    [Crossref]
  35. W. Vågberg, J. C. Larsson, and H. M. Hertz, “Removal of ring artifacts in microtomography by characterization of scintillator variations,” Opt. Express 25(19), 23191–23198 (2017).
    [Crossref]
  36. N. T. Vo, R. C. Atwood, and M. Drakopoulos, “Superior techniques for eliminating ring artifacts in x-ray microtomography,” Opt. Express 26(22), 28396–28412 (2018).
    [Crossref]
  37. L. Croton, G. Ruben, K. S. Morgan, D. M. Pagnin, and M. Kitchen, “Ring artifact suppression in X-ray computed tomography using a simple, pixel-wise response correction,” Opt. Express 27(10), 14231–14245 (2019).
    [Crossref]
  38. W. Chen, D. Prell, Y. Kyriakou, and W. A. Kalender, “Accelerating Ring Artifact Correction for Flat-Detector CT using the CUDA Framework,” Proc. SPIE 7622, 76223A (2010).
    [Crossref]
  39. J. Sijbers and A. Postnov, “Reduction of ring artefacts in high resolution micro-CT reconstructions,” Phys. Med. Biol. 49(14), N247–N253 (2004).
    [Crossref]
  40. Z. Wei, S. Wiebe, and D. Chapman, “Ring artifacts removal from synchrotron CT image slices,” J. Instrum. 8(06), C06006 (2013).
    [Crossref]
  41. Y. Kyriakou, D. Prell, and W. A. Kalender, “Ring artifact correction for high-resolution micro CT,” Phys. Med. Biol. 54(17), N385–N391 (2009).
    [Crossref]
  42. D. Prell, Y. Kyriakou, and W. A. Kalender, “Comparison of ring artifact correction methods for flat-detector CT,” Phys. Med. Biol. 54(12), 3881–3895 (2009).
    [Crossref]
  43. M. Bouali and S. Ladjal, “Toward Optimal Destriping of MODIS Data Using a Unidirectional Variational Model,” IEEE Trans. Geosci. Remote. Sens. 49(8), 2924–2935 (2011).
    [Crossref]
  44. P. Wu, X. Sun, H. Hu, T. Mao, W. Zhao, K. Sheng, A. A. Cheung, and T. Niu, “Iterative CT shading correction with no prior information,” Phys. Med. Biol. 60(21), 8437–8455 (2015).
    [Crossref]
  45. L. Xu, Q. Yan, Y. Xia, and J. Jia, “Structure extraction from texture via relative total variation,” ACM Trans. Graph. 31(6), 1–10 (2012).
    [Crossref]
  46. L. Yan, T. Wu, S. Zhong, and Q. Zhang, “A variation-based ring artifact correction method with sparse constraint for flat-detector CT,” Phys. Med. Biol. 61(3), 1278–1292 (2016).
    [Crossref]
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2020 (2)

X. Zhao, P. Chen, J. Wei, and Z. Qu, “Spectral CT imaging method based on blind separation of polychromatic projections with Poisson prior,” Opt. Express 28(9), 12780–12794 (2020).
[Crossref]

W. Fang, L. Li, and Z. Chen, “Removing Ring Artefacts for Photon-Counting Detectors Using Neural Networks in Different Domains,” IEEE Access 8, 42447–42457 (2020).
[Crossref]

2019 (6)

2018 (2)

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

N. T. Vo, R. C. Atwood, and M. Drakopoulos, “Superior techniques for eliminating ring artifacts in x-ray microtomography,” Opt. Express 26(22), 28396–28412 (2018).
[Crossref]

2017 (3)

X. Liang, Z. Zhang, T. Niu, S. Yu, S. Wu, Z. Li, H. Zhang, and Y. Xie, “Iterative image-domain ring artifact removal in cone-beam CT,” Phys. Med. Biol. 62(13), 5276–5292 (2017).
[Crossref]

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

W. Vågberg, J. C. Larsson, and H. M. Hertz, “Removal of ring artifacts in microtomography by characterization of scintillator variations,” Opt. Express 25(19), 23191–23198 (2017).
[Crossref]

2016 (4)

V. Titarenko, “1D Filter for Ring Artifact Suppression,” J. Synchrotron Radiat. 23(6), 800–804 (2016).
[Crossref]

V. Astromskas, E. N. Gimenez, A. Lohstroh, and N. Tartoni, “Evaluation of Polarization Effects of, Collection Schottky CdTe Medipix3RX Hybrid Pixel Detector,” IEEE Trans. Nucl. Sci. 63(1), 252–258 (2016).
[Crossref]

X. Liang, S. Gong, Q. Zhou, Z. Zhang, Y. Xie, and T. Niu, “SU-F-J-211: Scatter Correction for Clinical Cone-Beam CT System Using An Optimized Stationary Beam Blocker with a Single Scan,” Med. Phys. 43(6), 3457 (2016).
[Crossref]

L. Yan, T. Wu, S. Zhong, and Q. Zhang, “A variation-based ring artifact correction method with sparse constraint for flat-detector CT,” Phys. Med. Biol. 61(3), 1278–1292 (2016).
[Crossref]

2015 (3)

P. Wu, X. Sun, H. Hu, T. Mao, W. Zhao, K. Sheng, A. A. Cheung, and T. Niu, “Iterative CT shading correction with no prior information,” Phys. Med. Biol. 60(21), 8437–8455 (2015).
[Crossref]

N. V. Van, B. J. De, C. F. De, L. Mancini, F. Marone, and J. Sijbers, “Dynamic intensity normalization using eigen flat fields in X-ray imaging,” Opt. Express 23(21), 27975–27989 (2015).
[Crossref]

P. Wu, T. Mao, S. Xie, K. Sheng, and T. Niu, “WE-G-207-09: A Practical Bowtie Ring Artifact Correction Algorithm for Cone-Beam CT,” Med. Phys. 42(6Part41), 3698 (2015).
[Crossref]

2014 (5)

C. Altunbas, C. J. Lai, Y. Zhong, and C. C Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
[Crossref]

E. X. Miqueles, J. Rinkel, F. O’Dowd, and J. S. V. Bermúdez, “Generalized Titarenko’s algorithm for ring artefacts reduction,” J. Synchrotron Radiat. 21(6), 1333–1346 (2014).
[Crossref]

Y. Kim, J. Baek, and D. Hwang, “Ring artifact correction using detector line-ratios in computed tomography,” Opt. Express 22(11), 13380–13392 (2014).
[Crossref]

C. Altunbas, C. Lai, Y. Zhong, and C. C. Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
[Crossref]

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

2013 (1)

Z. Wei, S. Wiebe, and D. Chapman, “Ring artifacts removal from synchrotron CT image slices,” J. Instrum. 8(06), C06006 (2013).
[Crossref]

2012 (1)

L. Xu, Q. Yan, Y. Xia, and J. Jia, “Structure extraction from texture via relative total variation,” ACM Trans. Graph. 31(6), 1–10 (2012).
[Crossref]

2011 (2)

M. Bouali and S. Ladjal, “Toward Optimal Destriping of MODIS Data Using a Unidirectional Variational Model,” IEEE Trans. Geosci. Remote. Sens. 49(8), 2924–2935 (2011).
[Crossref]

A. N. M. Ashrafuzzaman, S. Y. Lee, and M. K. Hasan, “A self-adaptive approach for the detection and correction of stripes in the sinogram: suppression of ring artifacts in CT imaging,” EURASIP J. Adv. Signal. Process. 2011(1), 183547 (2011).
[Crossref]

2010 (5)

E. M. A. Anas, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in CT imaging through detection and correction of stripes in the sinogram,” Phys. Med. Biol. 55(22), 6911–6930 (2010).
[Crossref]

S. Titarenko, P. J. Withers, and A. Yagola, “An analytical formula for ring artefact suppression in X-ray tomography,” Appl. Math. Lett. 23(12), 1489–1495 (2010).
[Crossref]

F. Sadi, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter,” Comput. Biol. Med. 40(1), 109–118 (2010).
[Crossref]

V. Titarenko, S. Titarenko, P. J. Withers, D. C. Francesco, and X. Xiao, “Improved tomographic reconstructions using adaptive time dependent intensity normalization,” J. Synchrotron Radiat. 17(5), 689–699 (2010).
[Crossref]

W. Chen, D. Prell, Y. Kyriakou, and W. A. Kalender, “Accelerating Ring Artifact Correction for Flat-Detector CT using the CUDA Framework,” Proc. SPIE 7622, 76223A (2010).
[Crossref]

2009 (3)

Y. Kyriakou, D. Prell, and W. A. Kalender, “Ring artifact correction for high-resolution micro CT,” Phys. Med. Biol. 54(17), N385–N391 (2009).
[Crossref]

D. Prell, Y. Kyriakou, and W. A. Kalender, “Comparison of ring artifact correction methods for flat-detector CT,” Phys. Med. Biol. 54(12), 3881–3895 (2009).
[Crossref]

B. Munch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet — Fourier filtering,” Opt. Express 17(10), 8567–8591 (2009).
[Crossref]

2008 (1)

R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
[Crossref]

2007 (1)

W. Yuan, “Extended Applications of Image Flat-Field Correction Method,” Acta. Photonica. Sin. 36(9), 1587–1590 (2007).

2006 (3)

2004 (1)

J. Sijbers and A. Postnov, “Reduction of ring artefacts in high resolution micro-CT reconstructions,” Phys. Med. Biol. 49(14), N247–N253 (2004).
[Crossref]

2001 (2)

X. Tang, R. Ning, R. Yu, and D. Conover, “Cone beam volume CT image artifacts caused by defective cells in x-ray flat panel imagers and the artifact removal using a wavelet-analysis-based algorithm,” Med. Phys. 28(5), 812–825 (2001).
[Crossref]

W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
[Crossref]

1999 (1)

X. Tang, R. Ning, R. Yu, and D. L. Conover, “2D wavelet-analysis-based calibration technique for flat panel imaging detectors: Application in cone beam volume CT,” Proc. SPIE 3659, 806–816 (1999).
[Crossref]

1998 (1)

C. Raven, “Numerical removal of ring artifacts in microtomography,” Rev. Sci. Instrum. 69(8), 2978–2980 (1998).
[Crossref]

1997 (1)

G. R. Davis and J. C. Elliott, “X-ray microtomography scanner using time-delay integration for elimination of ring artefacts in the reconstructed image,” Nucl. Instrum. Methods Phys. Res., Sect. A 394(1-2), 157–162 (1997).
[Crossref]

1978 (1)

G. Kowalski, “Suppression of Ring Artefacts in CT Fan-Beam Scanners,” IEEE Trans Nucl. Sci. 25(5), 1111–1116 (1978).
[Crossref]

1974 (1)

L. A. Shepp and B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. 21(3), 21–43 (1974).
[Crossref]

Altunbas, C.

C. Altunbas, C. J. Lai, Y. Zhong, and C. C Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
[Crossref]

C. Altunbas, C. Lai, Y. Zhong, and C. C. Shaw, “Reduction of ring artifacts in CBCT: Detection and correction of pixel gain variations in flat panel detectors,” Med. Phys. 41(9), 091913 (2014).
[Crossref]

Anas, E. M. A.

E. M. A. Anas, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in CT imaging through detection and correction of stripes in the sinogram,” Phys. Med. Biol. 55(22), 6911–6930 (2010).
[Crossref]

Arce, G. R.

Ashrafuzzaman, A. N. M.

A. N. M. Ashrafuzzaman, S. Y. Lee, and M. K. Hasan, “A self-adaptive approach for the detection and correction of stripes in the sinogram: suppression of ring artifacts in CT imaging,” EURASIP J. Adv. Signal. Process. 2011(1), 183547 (2011).
[Crossref]

Astromskas, V.

V. Astromskas, E. N. Gimenez, A. Lohstroh, and N. Tartoni, “Evaluation of Polarization Effects of, Collection Schottky CdTe Medipix3RX Hybrid Pixel Detector,” IEEE Trans. Nucl. Sci. 63(1), 252–258 (2016).
[Crossref]

Atwood, R. C.

Baek, J.

Bai, T.

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

Bermúdez, J. S. V.

E. X. Miqueles, J. Rinkel, F. O’Dowd, and J. S. V. Bermúdez, “Generalized Titarenko’s algorithm for ring artefacts reduction,” J. Synchrotron Radiat. 21(6), 1333–1346 (2014).
[Crossref]

Bleuet, P.

R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
[Crossref]

Boin, M.

Bottigli, U.

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Bouali, M.

M. Bouali and S. Ladjal, “Toward Optimal Destriping of MODIS Data Using a Unidirectional Variational Model,” IEEE Trans. Geosci. Remote. Sens. 49(8), 2924–2935 (2011).
[Crossref]

Brombal, L.

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Cervino, L.

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

Chao, Z.

Z. Chao and H. Kim, “Removal of computed tomography ring artifacts via radial basis function artificial neural networks,” Phys. Med. Biol. 64(23), 235015 (2019).
[Crossref]

Chapman, D.

Z. Wei, S. Wiebe, and D. Chapman, “Ring artifacts removal from synchrotron CT image slices,” J. Instrum. 8(06), C06006 (2013).
[Crossref]

Chen, P.

Chen, W.

W. Chen, D. Prell, Y. Kyriakou, and W. A. Kalender, “Accelerating Ring Artifact Correction for Flat-Detector CT using the CUDA Framework,” Proc. SPIE 7622, 76223A (2010).
[Crossref]

Chen, Z.

W. Fang, L. Li, and Z. Chen, “Removing Ring Artefacts for Photon-Counting Detectors Using Neural Networks in Different Domains,” IEEE Access 8, 42447–42457 (2020).
[Crossref]

Cheung, A. A.

P. Wu, X. Sun, H. Hu, T. Mao, W. Zhao, K. Sheng, A. A. Cheung, and T. Niu, “Iterative CT shading correction with no prior information,” Phys. Med. Biol. 60(21), 8437–8455 (2015).
[Crossref]

Cloetens, P.

R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
[Crossref]

Conover, D.

X. Tang, R. Ning, R. Yu, and D. Conover, “Cone beam volume CT image artifacts caused by defective cells in x-ray flat panel imagers and the artifact removal using a wavelet-analysis-based algorithm,” Med. Phys. 28(5), 812–825 (2001).
[Crossref]

Conover, D. L.

X. Tang, R. Ning, R. Yu, and D. L. Conover, “2D wavelet-analysis-based calibration technique for flat panel imaging detectors: Application in cone beam volume CT,” Proc. SPIE 3659, 806–816 (1999).
[Crossref]

Croton, L.

Cuadros, A.

Davis, G. R.

G. R. Davis and J. C. Elliott, “X-ray microtomography scanner using time-delay integration for elimination of ring artefacts in the reconstructed image,” Nucl. Instrum. Methods Phys. Res., Sect. A 394(1-2), 157–162 (1997).
[Crossref]

De, B. J.

De, C. F.

Delogu, P.

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Diete, W.

W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
[Crossref]

Donato, S.

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Drakopoulos, M.

Dreossi, D.

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Elliott, J. C.

G. R. Davis and J. C. Elliott, “X-ray microtomography scanner using time-delay integration for elimination of ring artefacts in the reconstructed image,” Nucl. Instrum. Methods Phys. Res., Sect. A 394(1-2), 157–162 (1997).
[Crossref]

Enoh, M.

Z. Wang, J. Li, and M. Enoh, “Removing ring artifacts in CBCT images via generative adversarial networks with unidirectional relative total variation loss,” Neural. Comput. Appl. 31(9), 5147–5158 (2019).
[Crossref]

Fang, W.

W. Fang, L. Li, and Z. Chen, “Removing Ring Artefacts for Photon-Counting Detectors Using Neural Networks in Different Domains,” IEEE Access 8, 42447–42457 (2020).
[Crossref]

Fanti, V.

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

Folkerts, M.

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

Frahm, R.

W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
[Crossref]

Francesco, B.

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

Francesco, D. C.

V. Titarenko, S. Titarenko, P. J. Withers, D. C. Francesco, and X. Xiao, “Improved tomographic reconstructions using adaptive time dependent intensity normalization,” J. Synchrotron Radiat. 17(5), 689–699 (2010).
[Crossref]

Gimenez, E. N.

V. Astromskas, E. N. Gimenez, A. Lohstroh, and N. Tartoni, “Evaluation of Polarization Effects of, Collection Schottky CdTe Medipix3RX Hybrid Pixel Detector,” IEEE Trans. Nucl. Sci. 63(1), 252–258 (2016).
[Crossref]

Golosio, B.

L. Brombal, D. Sandro, B. Francesco, P. Delogu, V. Fanti, P. Oliva, L. Rigon, V. D. Trapani, R. Longo, and B. Golosio, “Large-area single-photon-counting CdTe detector for synchrotron radiation computed tomography: a dedicated pre-processing procedure,” J. Synchrotron Radiat. 25(4), 1068–1077 (2018).
[Crossref]

P. Delogu, L. Brombal, V. D. Trapani, S. Donato, U. Bottigli, D. Dreossi, B. Golosio, P. Oliva, L. Rigon, and R. Longo, “Optimization of the equalization procedure for a single-photon counting CdTe detector used for CT,” J. Instrum. 12(11), C11014 (2017).
[Crossref]

Gong, S.

X. Liang, S. Gong, Q. Zhou, Z. Zhang, Y. Xie, and T. Niu, “SU-F-J-211: Scatter Correction for Clinical Cone-Beam CT System Using An Optimized Stationary Beam Blocker with a Single Scan,” Med. Phys. 43(6), 3457 (2016).
[Crossref]

Gorner, W.

W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
[Crossref]

Guilloud, C.

R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
[Crossref]

Haibel, A.

Han, C.

Hasan, M. K.

A. N. M. Ashrafuzzaman, S. Y. Lee, and M. K. Hasan, “A self-adaptive approach for the detection and correction of stripes in the sinogram: suppression of ring artifacts in CT imaging,” EURASIP J. Adv. Signal. Process. 2011(1), 183547 (2011).
[Crossref]

E. M. A. Anas, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in CT imaging through detection and correction of stripes in the sinogram,” Phys. Med. Biol. 55(22), 6911–6930 (2010).
[Crossref]

F. Sadi, S. Y. Lee, and M. K. Hasan, “Removal of ring artifacts in computed tomographic imaging using iterative center weighted median filter,” Comput. Biol. Med. 40(1), 109–118 (2010).
[Crossref]

Hentschel, M. P.

W. Gorner, M. P. Hentschel, B. R. Mullera, H. Riesemeier, M. Krumrey, G. Ulm, W. Diete, U. Klein, and R. Frahm, “BAMline: the first hard X-ray beamline at BESSY II,” Nucl. Instrum. Methods Phys. Res., Sect. A 467-468, 703–706 (2001).
[Crossref]

Hertz, H. M.

Hu, H.

P. Wu, X. Sun, H. Hu, T. Mao, W. Zhao, K. Sheng, A. A. Cheung, and T. Niu, “Iterative CT shading correction with no prior information,” Phys. Med. Biol. 60(21), 8437–8455 (2015).
[Crossref]

Hwang, D.

Jia, J.

L. Xu, Q. Yan, Y. Xia, and J. Jia, “Structure extraction from texture via relative total variation,” ACM Trans. Graph. 31(6), 1–10 (2012).
[Crossref]

Jia, X.

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

Jiang, S. B.

H. Yan, X. Wang, F. Shi, T. Bai, M. Folkerts, L. Cervino, S. B. Jiang, and X. Jia, “Towards the clinical implementation of iterative low-dose cone-beam CT reconstruction in image-guided radiation therapy: Cone/ring artifact correction and multiple GPU implementation,” Med. Phys. 41(11), 111912 (2014).
[Crossref]

Kalender, W. A.

W. Chen, D. Prell, Y. Kyriakou, and W. A. Kalender, “Accelerating Ring Artifact Correction for Flat-Detector CT using the CUDA Framework,” Proc. SPIE 7622, 76223A (2010).
[Crossref]

Y. Kyriakou, D. Prell, and W. A. Kalender, “Ring artifact correction for high-resolution micro CT,” Phys. Med. Biol. 54(17), N385–N391 (2009).
[Crossref]

D. Prell, Y. Kyriakou, and W. A. Kalender, “Comparison of ring artifact correction methods for flat-detector CT,” Phys. Med. Biol. 54(12), 3881–3895 (2009).
[Crossref]

Ketcham, R. A.

R. A. Ketcham, “New algorithms for ring artifact removal,” Proc. SPIE 6318(38), 63180O (2006).
[Crossref]

Kieffer, I.

R. Tucoulou, G. Martinezcriado, P. Bleuet, I. Kieffer, P. Cloetens, S. Laboure, T. Martin, C. Guilloud, and J. Susini, “High-resolution angular beam stability monitoring at a nanofocusing beamline,” J. Synchrotron Radiat. 15(4), 392–398 (2008).
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Figures (9)

Fig. 1.
Fig. 1. The framework for removing the ring artifacts.
Fig. 2.
Fig. 2. The process for extracting the intensity-dependent stripe artifacts.
Fig. 3.
Fig. 3. The 3D graph of the classification function $F({p,q} ){\kern 1pt} {\kern 1pt}$ changes with p and q for a certain stripe artifact.
Fig. 4.
Fig. 4. The process for extracting the time-dependent stripe artifacts.
Fig. 5.
Fig. 5. The comparisons of the corrected images of the Shepp-Logan phantom with the mixed artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al. (d) The corrected image obtained by the first part of the proposed method. (e) The corrected image obtained by the second part of the proposed method. (f) The reference image; the images in the second row are zoomed-in views of the images in the first row; all above display windows are [0.98, 1.05]. The images in third row are the ring artifacts of the images in the first row, and the display windows are [-0.001, 0.005].
Fig. 6.
Fig. 6. The comparisons of the corrected images of the aluminum part with the real mixed ring artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al. (d) The corrected image obtained by the first part of the proposed method. (e) The corrected image obtained by the second part of the proposed method. The images in the second row are zoomed-in views of the images in the first row; all above display windows are [-0.2,1.2]. The images in third row are the ring artifacts of the images in the first row (if we assume Fig. 7(e) as the reference image), and the display windows are [-0.05,0.25].
Fig. 7.
Fig. 7. The comparisons of the corrected images of the real synchrotron data sets with ring artifacts. (a) The uncorrected image. (b) The corrected image obtained by the method of Wei et al. (c) The corrected image obtained by the method of Liang et al.; (d) the corrected image obtained by the first part of the proposed method; (e) the corrected image obtained by the second part of the proposed method; the images in the second row are zoomed-in views of the images in the first row. The images in third row are the ring artifacts of the images in the first row (if we assume Fig. 7(e) as the reference image), and all above display windows are [-0.0015,0.0005].
Fig. 8.
Fig. 8. The variation of SSIM with the methods, the smoothing weights and the number of iterations.
Fig. 9.
Fig. 9. The variation of SSIM with the methods, the smoothing weights and the number of iterations.

Tables (2)

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Table 1. Reconstruction parameters.

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Table 2. Numerical evaluations of the simulation and experimental data.

Equations (3)

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S = t s m o o t h ( I , λ , σ , max T t e r , ε ) ,
arg min F ( p , q ) = i = p q ( R i E C 1 ) 2 + j = 1 ( p 1 ) ( R j E C 2 ) 2 + j = ( q + 1 ) M ( R j E C 2 ) 2 R i C 1 , R j C 2 ,
C s = | | R I N G K | | 1 / | | R I N G 1 | | 1 ,

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