Low-dose Computed Tomography Reconstruction Research Articles

Noise-Generating and Imaging Mechanism Inspired Implicit Regularization Learning Network for Low Dose CT Reconstrution.

Low-dose computed tomography (LDCT) helps to reduce radiation risks in CT scanning while maintaining image quality, which involves a consistent pursuit of lower incident rays and higher reconstruction performance. Although deep learning approaches have achieved encouraging success in LDCT reconstruction, most of them treat the task as a general inverse problem in either the image domain or the dual (sinogram and image) domains. Such frameworks have not considered the original noise generation of the projection data and suffer from limited performance improvement for the LDCT task. In this paper, we propose a novel reconstruction model based on noise-generating and imaging mechanism in full-domain, which fully considers the statistical properties of intrinsic noises in LDCT and prior information in sinogram and image domains. To solve the model, we propose an optimization algorithm based on the proximal gradient technique. Specifically, we derive the approximate solutions of the integer programming problem on the projection data theoretically. Instead of hand-crafting the sinogram and image regularizers, we propose to unroll the optimization algorithm to be a deep network. The network implicitly learns the proximal operators of sinogram and image regularizers with two deep neural networks, providing a more interpretable and effective reconstruction procedure. Numerical results demonstrate our proposed method improvements of > 2.9 dB in peak signal to noise ratio, > 1.4% promotion in structural similarity metric, and > 9 HU decrements in root mean square error over current state-of-the-art LDCT methods.

Generation model meets swin transformer for unsupervised low-dose CT reconstruction

Computed tomography (CT) has evolved into an indispensable tool for clinical diagnosis. Reducing radiation dose crucially minimizes adverse effects but may introduce noise and artifacts in reconstructed images, affecting diagnostic processes for physicians. Scholars have tackled deep learning training instability by exploring diffusion models. Given the scarcity of clinical data, we propose the unsupervised image domain score generation model (UISG) for low-dose CT reconstruction. During training, normal-dose CT images are utilized as network inputs to train a score-based generative model that captures the prior distribution of CT images. In the iterative reconstruction, the initial CT image is obtained using a filtered back-projection algorithm. Subsequently, diffusion-based prior, high-frequency convolutional sparse coding prior, and data-consistency steps are employed to obtain the high-quality reconstructed image. Given the global characteristics of noise, the score network of the diffusion model utilizes a swin transformer structure to enhance the model’s ability to capture long-range dependencies. Furthermore, convolutional sparse coding is applied exclusively to the high-frequency components of the image, to prevent over-smoothing or the loss of crucial anatomical details during the denoising process. Quantitative and qualitative results indicate that UISG outperforms competing methods in terms of denoising and generalization performance.

A priority-based self-guided serial–parallel genetic algorithm for low-dose computed tomography

Computed tomography (CT) is a non-destructive evaluation technique to know the internal structure of the objects under scan. It has numerous applications in engineering as well as in the medical field. The prime objective of this manuscript is to reduce the radiation dose to the object under scanning and to reconstruct the low-noise CT images, even in limited-view projections data scenarios. The present manuscript proposes a novel priority-based self-guided serial-parallel hybrid genetic algorithm for low-dose CT reconstruction. The current algorithm combines serial and parallel processing elements to reduce the loss of diversity in the population and increase the convergence rate. The proposed algorithm uses a novel priority-based self-guided competition operator. It consists of two approaches for maintaining exploration and exploitation tradeoffs. Here, the first approach uses serial processing, whereas the second approach uses parallel processing. The algorithm also uses priority-based hybrid crossover (PHCO) and adaptive priority-based mutation operator (PMO) to generate the most suitable offspring in every generation. Experimental results reveal that the presented algorithm produces satisfactory results with limited-view projections. The presented algorithm also outperforms other limited-view CT reconstruction algorithms.

Proj2Proj: self-supervised low-dose CT reconstruction.

In Computed Tomography (CT) imaging, one of the most serious concerns has always been ionizing radiation. Several approaches have been proposed to reduce the dose level without compromising the image quality. With the emergence of deep learning, thanks to the increasing availability of computational power and huge datasets, data-driven methods have recently received a lot of attention. Deep learning based methods have also been applied in various ways to address the low-dose CT reconstruction problem. However, the success of these methods largely depends on the availability of labeled data. On the other hand, recent studies showed that training can be done successfully without the need for labeled datasets. In this study, a training scheme was defined to use low-dose projections as their own training targets. The self-supervision principle was applied in the projection domain. The parameters of a denoiser neural network were optimized through self-supervised training. It was shown that our method outperformed both traditional and compressed sensing-based iterative methods, and deep learning based unsupervised methods, in the reconstruction of analytic CT phantoms and human CT images in low-dose CT imaging. Our method's reconstruction quality is also comparable to a well-known supervised method.

Unsupervised Domain Adaptation for Low-Dose CT Reconstruction via Bayesian Uncertainty Alignment.

Low-dose computed tomography (LDCT) image reconstruction techniques can reduce patient radiation exposure while maintaining acceptable imaging quality. Deep learning (DL) is widely used in this problem, but the performance of testing data (also known as target domain) is often degraded in clinical scenarios due to the variations that were not encountered in training data (also known as source domain). Unsupervised domain adaptation (UDA) of LDCT reconstruction has been proposed to solve this problem through distribution alignment. However, existing UDA methods fail to explore the usage of uncertainty quantification, which is crucial for reliable intelligent medical systems in clinical scenarios with unexpected variations. Moreover, existing direct alignment for different patients would lead to content mismatch issues. To address these issues, we propose to leverage a probabilistic reconstruction framework to conduct a joint discrepancy minimization between source and target domains in both the latent and image spaces. In the latent space, we devise a Bayesian uncertainty alignment to reduce the epistemic gap between the two domains. This approach reduces the uncertainty level of target domain data, making it more likely to render well-reconstructed results on target domains. In the image space, we propose a sharpness-aware distribution alignment (SDA) to achieve a match of second-order information, which can ensure that the reconstructed images from the target domain have similar sharpness to normal-dose CT (NDCT) images from the source domain. Experimental results on two simulated datasets and one clinical low-dose imaging dataset show that our proposed method outperforms other methods in quantitative and visualized performance.

Multi-layer clustering-based residual sparsifying transform for low-dose CT image reconstruction.

The recently proposed sparsifying transform (ST) models incur low computational cost and have been applied to medical imaging. Meanwhile, deep models with nested network structure reveal great potential for learning features in different layers. In this study, we propose a network-structured ST learning approach for X-ray computed tomography (CT), which we refer to as multi-layer clustering-based residual sparsifying transform (MCST) learning. The proposed MCST scheme learns multiple different unitary transforms in each layer by dividing each layer's input into several classes. We apply the MCST model to low-dose CT (LDCT) reconstruction by deploying the learned MCST model into the regularizer in penalized weighted least squares (PWLS) reconstruction. The proposed MCST model combines a multi-layer sparse representation structure with multiple clusters for the features in each layer that are modeled by a rich collection of transforms. We train the MCST model in an unsupervised manner via a block coordinate descent (BCD) algorithm. Since our method is patch-based, the training can be performed with a limited set of images. For CT image reconstruction, we devise a novel algorithm called PWLS-MCST by integrating the pre-learned MCST signal model with PWLSoptimization. We conducted LDCT reconstruction experiments on XCAT phantom data, Numerical Mayo Clinical CT dataset and "LDCT image and projection dataset" (Clinical LDCT dataset). We trained the MCST model with two (or three) layers and with five clusters in each layer. The learned transforms in the same layer showed rich features while additional information is extracted from representation residuals. Our simulation results and clinical results demonstrate that PWLS-MCST achieves better image reconstruction quality than the conventional filtered back-projection (FBP) method and PWLS with edge-preserving (EP) regularizer. It also outperformed recent advanced methods like PWLS with a learned multi-layer residual sparsifying transform (MARS) prior and PWLS with a union of learned transforms (ULTRA), especially for displaying clear edges and preserving subtledetails. In this work, a multi-layer sparse signal model with a nested network structure is proposed. We refer this novel model as the MCST model that exploits multi-layer residual maps to sparsify the underlying image and clusters the inputs in each layer for accurate sparsification. We presented a new PWLS framework with a learned MCST regularizer for LDCT reconstruction. Experimental results show that the proposed PWLS-MCST provides clearer reconstructions than several baseline methods. The code for PWLS-MCST is released at https://github.com/Xikai97/PWLS-MCST.

An efficient reversible data hiding using SVD over a novel weighted iterative anisotropic total variation based denoised medical images

Computed tomography (CT) advancement and extensive usage have raised the public’s worry regarding the patient’s associated radiation dose. Reducing the radiation dose may lead to more noise and artifacts, which may harm the reputation of radiologists. The instability of low-dose CT reconstruction necessitates better image reconstruction, which increases the diagnostic performance. More modern low-dose CT tests have demonstrated outstanding results. Many times these low-dose denoised medical images with medical related information are also required to transmit over a network. Hence in this article, firstly is a novel denoising method is proposed to improve the quality of low-dose CT images that is based on the total variation method by utilizing whale optimization algorithm (WHA). WHA method is important for getting the best possible weighted function. Reduction of noise occurs by the comparison of a given output to the ground truth, although total variation tends to statistically migrate the data noise distribution from strong to weak. Following denoising, a reversible watermarking approach based on SVD and multi-local extrema (MLE) approaches is provided. Individual results of denoising and watermarking are excellent in terms of visual and performance metrics, according to comparative experimental investigation. Also it was also analyzed that if the watermark is embedded over the denoised CT images then the results of watermarking methods are impressive. So, resultant image offers us the chance to use our visual perception abilities to allow us to cut noise and keep vital and secure information.

Noise Characteristics Modeled Unsupervised Network for Robust CT Image Reconstruction.

Deep learning (DL)-based methods show great potential in computed tomography (CT) imaging field. The DL-based reconstruction methods are usually evaluated on the training and testing datasets which are obtained from the same distribution, i.e., the same CT scan protocol (i.e., the region setting, kVp, mAs, etc.). In this work, we focus on analyzing the robustness of the DL-based methods against protocol-specific distribution shifts (i.e., the training and testing datasets are from different region settings, different kVp settings, or different mAs settings, respectively). The results show that the DL-based reconstruction methods are sensitive to the protocol-specific perturbations which can be attributed to the noise distribution shift between the training and testing datasets. Based on these findings, we presented a low-dose CT reconstruction method using an unsupervised strategy with the consideration of noise distribution to address the issue of protocol-specific perturbations. Specifically, unpaired sinogram data is enrolled into the network training, which represents unique information for specific imaging protocol, and a Gaussian mixture model (GMM) is introduced to characterize the noise distribution in CT images. It can be termed as GMM based unsupervised CT reconstruction network (GMM-unNet) method. Moreover, an expectation-maximization algorithm is designed to optimize the presented GMM-unNet method. Extensive experiments are performed on three datasets from different scan protocols, which demonstrate that the presented GMM-unNet method outperforms the competing methods both qualitatively and quantitatively.

Bilateral Weighted Relative Total Variation for Low-Dose CT Reconstruction.

Low-dose computed tomography (LDCT) has been widely used for various clinic applications to reduce the X-ray dose absorbed by patients. However, LDCT is usually degraded by severe noise over the image space. The image quality of LDCT has attracted aroused attentions of scholars. In this study, we propose the bilateral weighted relative total variation (BRTV) used for image restoration to simultaneously maintain edges and further reduce noise, then propose the BRTV-regularized projections onto convex sets (POCS-BRTV) model for LDCT reconstruction. Referring to the spacial closeness and the similarity of gray value between two pixels in a local rectangle, POCS-BRTV can adaptively extract sharp edges and minor details during the iterative reconstruction process. Evaluation indexes and visual effects are used to measure the performances among different algorithms. Experimental results indicate that the proposed POCS-BRTV model can achieve superior image quality than the compared algorithms in terms of the structure and texture preservation.

MAGIC: Manifold and Graph Integrative Convolutional Network for Low-Dose CT Reconstruction.

Low-dose computed tomography (LDCT) scans, which can effectively alleviate the radiation problem, will degrade the imaging quality. In this paper, we propose a novel LDCT reconstruction network that unrolls the iterative scheme and performs in both image and manifold spaces. Because patch manifolds of medical images have low-dimensional structures, we can build graphs from the manifolds. Then, we simultaneously leverage the spatial convolution to extract the local pixel-level features from the images and incorporate the graph convolution to analyze the nonlocal topological features in manifold space. The experiments show that our proposed method outperforms both the quantitative and qualitative aspects of state-of-the-art methods. In addition, aided by a projection loss component, our proposed method also demonstrates superior performance for semi-supervised learning. The network can remove most noise while maintaining the details of only 10% (40 slices) of the training data labeled.

Prior-image-based CT reconstruction using attenuation-mismatched priors

Prior-image-based reconstruction (PIBR) methods are powerful tools for reducing radiation doses and improving the image quality of low-dose computed tomography (CT). Apart from anatomical changes, prior and current images can also have different attenuations because they originated from different scanners or from the same scanner but with different x-ray beam qualities (e.g., kVp settings, beam filters) during data acquisition. In such scenarios, with attenuation-mismatched priors, PIBR is challenging. In this work, we investigate a specific PIBR method, called statistical image reconstruction, using normal-dose image-induced nonlocal means regularization (SIR-ndiNLM), to address PIBR with such attenuation-mismatched priors and achieve quantitative low-dose CT imaging. We propose two corrective schemes for the original SIR-ndiNLM method, (1) a global histogram-matching approach and (2) a local attenuation correction approach, to account for the attenuation differences between the prior and current images in PIBR. We validate the efficacy of the proposed schemes using images acquired from dual-energy CT scanners to simulate attenuation mismatches. Meanwhile, we utilize different CT slices to simulate anatomical mismatches or changes between the prior and the current low-dose image. We observe that the original SIR-ndiNLM introduces artifacts to the reconstruction when an attenuation-mismatched prior is used. Furthermore, we find that a larger attenuation mismatch between the prior and current images results in more severe artifacts in the SIR-ndiNLM reconstruction. Our two proposed corrective schemes enable SIR-ndiNLM to effectively handle the attenuation mismatch and anatomical changes between the two images and successfully eliminate the artifacts. We demonstrate that the proposed techniques permit SIR-ndiNLM to leverage the attenuation-mismatched prior and achieve quantitative low-dose CT reconstruction from both low-flux and sparse-view data acquisitions. This work permits robust and reliable PIBR for CT data acquired using different beam settings.

Deep learning based low-dose synchrotron radiation CT reconstruction

Synchrotron radiation sources are widely used in various fields, among which computed tomography (CT) is one of the most important. The amount of effort expended by the operator varies depending on the subject. If the number of angles needed to be used can be greatly reduced under the condition of similar imaging effects, the working time and workload of the experimentalists will be greatly reduced. However, decreasing the sampling angle can produce serious artifacts and blur the details. We try to use a deep learning model which can build high quality reconstruction sparse data sampling from the angle of the image and ResAttUnet are put forward. ResAttUnet is roughly a symmetrical U-shaped network that incorporates similar mechanisms to ResNet and attention. In addition, the mixed precision is adopted to reduce the demand for video memory of the model and training time.

Low-dose spectral CT reconstruction based on image-gradient L0-norm and adaptive spectral PICCS

The photon-counting detector based spectral computed tomography (CT) is promising for lesion detection, tissue characterization, and material decomposition. However, the lower signal-to-noise ratio within multi-energy projection dataset can result in poorly reconstructed image quality. Recently, as prior information, a high-quality spectral mean image was introduced into the prior image constrained compressed sensing (PICCS) framework to suppress noise, leading to spectral PICCS (SPICCS). In the original SPICCS model, the image gradient L1-norm is employed, and it can cause blurred edge structures in the reconstructed images. Encouraged by the advantages in edge preservation and finer structure recovering, the image gradient L0-norm was incorporated into the PICCS model. Furthermore, due to the difference of energy spectrum in different channels, a weighting factor is introduced and adaptively adjusted for different channel-wise images, leading to an L0-norm based adaptive SPICCS (L0-ASPICCS) algorithm for low-dose spectral CT reconstruction. The split-Bregman method is employed to minimize the objective function. Extensive numerical simulations and physical phantom experiments are performed to evaluate the proposed method. By comparing with the state-of-the-art algorithms, such as the simultaneous algebraic reconstruction technique, total variation minimization, and SPICCS, the advantages of our proposed method are demonstrated in terms of both qualitative and quantitative evaluation results.

Weighted adaptive non-local dictionary for low-dose CT reconstruction

Low-dose computed tomography (LDCT) image reconstruction has been attracting much attention in medical applications because it can reduce the radiation risk. Sometimes, traditional methods are difficult to reconstruct satisfying image quality from low-dose projections. It is a challenging task for LDCT reconstruction with image quality improvement. Recently, various patch-based methods including dictionary learning were developed for LDCT and have achieved promising performance. Most of these patch-based methods assume that the noise follows the uniform Gaussian distribution, while noise is much more complex than Gaussian distribution in practice. In this study, considering the varying statistics of noise in different patches, we develop a weighted adaptive non-local dictionary (WAND) method for LDCT. Concretely, instead of establishing a complex model for the noise distribution in the whole image, we iteratively characterize the noise property in each local patch during the iteration processing. Besides, we also adaptively describe the different distributions of sparse coefficients of each patch to better characterize the sparsity priors of the image. The simulated and realistic experiments have shown that WAND can achieve better image quality in terms of small details preservation and noise suppression.

Low-dose CT reconstruction method based on prior information of normal-dose image.

Radiation risk from computed tomography (CT) is always an issue for patients, especially those in clinical conditions in which repeated CT scanning is required. For patients undergoing repeated CT scanning, a low-dose protocol, such as sparse scanning, is often used, and consequently, an advanced reconstruction algorithm is also needed. To develop a novel algorithm used for sparse-view CT reconstruction associated with the prior image. A low-dose CT reconstruction method based on prior information of normal-dose image (PI-NDI) involving a transformed model for attenuation coefficients of the object to be reconstructed and prior information application in the forward-projection process was used to reconstruct CT images from sparse-view projection data. A digital extended cardiac-torso (XCAT) ventral phantom and a diagnostic head phantom were employed to evaluate the performance of the proposed PI-NDI method. The root-mean-square error (RMSE), peak signal-to-noise ratio (PSNR) and mean percent absolute error (MPAE) of the reconstructed images were measured for quantitative evaluation of the proposed PI-NDI method. The reconstructed images with sparse-view projection data via the proposed PI-NDI method have higher quality by visual inspection than that via the compared methods. In terms of quantitative evaluations, the RMSE measured on the images reconstructed by the PI-NDI method with sparse projection data is comparable to that by MLEM-TV, PWLS-TV and PWLS-PICCS with fully sampled projection data. When the projection data are very sparse, images reconstructed by the PI-NDI method have higher PSNR values and lower MPAE values than those from the compared algorithms. This study presents a new low-dose CT reconstruction method based on prior information of normal-dose image (PI-NDI) for sparse-view CT image reconstruction. The experimental results validate that the new method has superior performance over other state-of-art methods.

VVBP-Tensor in the FBP Algorithm: Its Properties and Application in Low-Dose CT Reconstruction

For decades, commercial X-ray computed tomography (CT) scanners have been using the filtered backprojection (FBP) algorithm for image reconstruction. However, the desire for lower radiation doses has pushed the FBP algorithm to its limit. Previous studies have made significant efforts to improve the results of FBP through preprocessing the sinogram, modifying the ramp filter, or postprocessing the reconstructed images. In this paper, we focus on analyzing and processing the stacked view-by-view backprojections (named VVBP-Tensor) in the FBP algorithm. A key challenge for our analysis lies in the radial structures in each backprojection slice. To overcome this difficulty, a sorting operation was introduced to the VVBP-Tensor in its z direction (the direction of the projection views). The results show that, after sorting, the tensor contains structures that are similar to those of the object, and structures in different slices of the tensor are correlated. We then analyzed the properties of the VVBP-Tensor, including structural self-similarity, tensor sparsity, and noise statistics. Considering these properties, we have developed an algorithm using the tensor singular value decomposition (named VVBP-tSVD) to denoise the VVBP-Tensor for low-mAs CT imaging. Experiments were conducted using a physical phantom and clinical patient data with different mAs levels. The results demonstrate that the VVBP-tSVD is superior to all competing methods under different reconstruction schemes, including sinogram preprocessing, image postprocessing, and iterative reconstruction. We conclude that the VVBP-Tensor is a suitable processing target for improving the quality of FBP reconstruction, and the proposed VVBP-tSVD is an effective algorithm for noise reduction in low-mAs CT imaging. This preliminary work might provide a heuristic perspective for reviewing and rethinking the FBP algorithm.

An efficient low-dose CT reconstruction technique using partial derivatives based guided image filter

Low-dose Computed Tomography (CT) reconstruction techniques have been implemented to minimize the X-ray radiation in a human body. Many researchers have designed different low-dose CT reconstruction techniques to reduce the effect of radiation in a human body. However, the majority of these techniques suffer from over-smoothing, edge distortion, halo artifacts, gradient reversal artifacts etc. problems. Therefore, in this paper, novel partial differential equations and dictionary learning based reconstruction technique have been designed to reconstruct the low-dose CT images. Extensive experiments have been carried out to evaluate the effectiveness of the proposed technique that existing image reconstruction techniques. It has been observed that the proposed technique significantly preserves the radiometric information of low-dose CT images with a lesser number of edge distortion, halo and gradient reversal artifacts. Also, the proposed technique is computationally faster than existing techniques, therefore most suitable for real-time low-dose CT reconstruction systems.

Low dose CT reconstruction via L1 norm dictionary learning using alternating minimization algorithm and balancing principle

Excessive radiation exposure in computed tomography (CT) scans increases the chance of developing cancer and has become a major clinical concern. Recently, statistical iterative reconstruction (SIR) with l0-norm dictionary learning regularization has been developed to reconstruct CT images from the low dose and few-view dataset in order to reduce radiation dose. Nonetheless, the sparse regularization term adopted in this approach is l0-norm, which cannot guarantee the global convergence of the proposed algorithm. To address this problem, in this study we introduced the l1-norm dictionary learning penalty into SIR framework for low dose CT image reconstruction, and developed an alternating minimization algorithm to minimize the associated objective function, which transforms CT image reconstruction problem into a sparse coding subproblem and an image updating subproblem. During the image updating process, an efficient model function approach based on balancing principle is applied to choose the regularization parameters. The proposed alternating minimization algorithm was evaluated first using real projection data of a sheep lung CT perfusion and then using numerical simulation based on sheep lung CT image and chest image. Both visual assessment and quantitative comparison using terms of root mean square error (RMSE) and structural similarity (SSIM) index demonstrated that the new image reconstruction algorithm yielded similar performance with l0-norm dictionary learning penalty and outperformed the conventional filtered backprojection (FBP) and total variation (TV) minimization algorithms.

Compressed sensing based CT reconstruction algorithm combined with modified Canny edge detection

Given that the computed tomography (CT) reconstruction algorithm based on compressed sensing (CS) results in blurred edges, we propose a modified Canny operator that assists the CS algorithm to accurately capture an object’s edge, to preserve and further enhance the contrasts in the reconstructed image, thereby improving image quality. We modified two procedures of the traditional Canny operator, namely non-maximum suppression and edge tracking by hysteresis according to the characteristics of low-dose CT reconstruction, and proposed two major modifications: double-response edge detection and directional edge tracking. The newly modified Canny operator was combined with the CS reconstruction algorithm to become an edge-enhanced CS (EECS). Both a 2D Shepp–Logan phantom and a 3D dental phantom were used to conduct reconstruction testing. Root-mean-square error, peak signal-to-noise ratio, and universal quality index were employed to verify the reconstruction results. Qualitative and quantitative results of EECS reconstruction showed its superiority over conventional CS or CS combined with different edge detection techniques, such as Laplacian, Prewitt, Sobel operators, etc. The experiments verified that the proposed modified Canny operator is able to effectively detect the edge location of an object during low-dose reconstruction, enabling EECS to reconstruct images with better quality than those produced by other algorithms.

Low-dose spectral CT reconstruction using image gradient ℓ 0-norm and tensor dictionary.

Spectral computed tomography (CT) has a great superiority in lesion detection, tissue characterization and material decomposition. To further extend its potential clinical applications, in this work, we propose an improved tensor dictionary learning method for low-dose spectral CT reconstruction with a constraint of image gradient ℓ 0-norm, which is named as ℓ 0TDL. The ℓ 0TDL method inherits the advantages of tensor dictionary learning (TDL) by employing the similarity of spectral CT images. On the other hand, by introducing the ℓ 0-norm constraint in gradient image domain, the proposed method emphasizes the spatial sparsity to overcome the weakness of TDL on preserving edge information. The split-bregman method is employed to solve the proposed method. Both numerical simulations and real mouse studies are perform to evaluate the proposed method. The results show that the proposed ℓ 0TDL method outperforms other competing methods, such as total variation (TV) minimization, TV with low rank (TV+LR), and TDL methods.