Group Sparsity

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Zhiyuan Zha - One of the best experts on this subject based on the ideXlab platform.

  • Group Sparsity Residual Constraint with Non-Local Priors for Image Restoration.
    IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 2020
    Co-Authors: Zhiyuan Zha, Xin Yuan, Bihan Wen, Jiantao Zhou, Ce Zhu
    Abstract:

    Group sparse representation (GSR) has made great strides in image restoration producing superior performance, realized through employing a powerful mechanism to integrate the local Sparsity and nonlocal self-similarity of images. However, due to some form of degradation (e.g., noise, down-sampling or pixels missing), traditional GSR models may fail to faithfully estimate Sparsity of each Group in an image, thus resulting in a distorted reconstruction of the original image. This motivates us to design a simple yet effective model that aims to address the above mentioned problem. Specifically, we propose Group Sparsity residual constraint with nonlocal priors (GSRC-NLP) for image restoration. Through introducing the Group Sparsity residual constraint, the problem of image restoration is further defined and simplified through attempts at reducing the Group Sparsity residual. Towards this end, we first obtain a good estimation of the Group sparse coefficient of each original image Group by exploiting the image nonlocal self-similarity (NSS) prior along with self-supervised learning scheme, and then the Group sparse coefficient of the corresponding degraded image Group is enforced to approximate the estimation. To make the proposed scheme tractable and robust, two algorithms, i.e., iterative shrinkage/thresholding (IST) and alternating direction method of multipliers (ADMM), are employed to solve the proposed optimization problems for different image restoration tasks. Experimental results on image denoising, image inpainting and image compressive sensing (CS) recovery, demonstrate that the proposed GSRC-NLP based image restoration algorithm is comparable to state-of-the-art denoising methods and outperforms several state-of-the-art image inpainting and image CS recovery methods in terms of both objective and perceptual quality metrics.

  • Group Sparsity Residual Constraint With Non-Local Priors for Image Restoration
    IEEE Transactions on Image Processing, 2020
    Co-Authors: Zhiyuan Zha, Xin Yuan, Bihan Wen, Jiantao Zhou, Ce Zhu
    Abstract:

    Group sparse representation (GSR) has made great strides in image restoration producing superior performance, realized through employing a powerful mechanism to integrate the local Sparsity and nonlocal self-similarity of images. However, due to some form of degradation ( e.g. , noise, down-sampling or pixels missing), traditional GSR models may fail to faithfully estimate Sparsity of each Group in an image, thus resulting in a distorted reconstruction of the original image. This motivates us to design a simple yet effective model that aims to address the above mentioned problem. Specifically, we propose Group Sparsity residual constraint with nonlocal priors (GSRC-NLP) for image restoration. Through introducing the Group Sparsity residual constraint, the problem of image restoration is further defined and simplified through attempts at reducing the Group Sparsity residual. Towards this end, we first obtain a good estimation of the Group sparse coefficient of each original image Group by exploiting the image nonlocal self-similarity (NSS) prior along with self-supervised learning scheme, and then the Group sparse coefficient of the corresponding degraded image Group is enforced to approximate the estimation. To make the proposed scheme tractable and robust, two algorithms, i.e. , iterative shrinkage/thresholding (IST) and alternating direction method of multipliers (ADMM), are employed to solve the proposed optimization problems for different image restoration tasks. Experimental results on image denoising, image inpainting and image compressive sensing (CS) recovery, demonstrate that the proposed GSRC-NLP based image restoration algorithm is comparable to state-of-the-art denoising methods and outperforms several testing image inpainting and image CS recovery methods in terms of both objective and perceptual quality metrics.

  • Group Sparsity residual with non local samples for image denoising
    arXiv: Computer Vision and Pattern Recognition, 2018
    Co-Authors: Zhiyuan Zha, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang, Xin Yuan
    Abstract:

    Inspired by Group-based sparse coding, recently proposed Group Sparsity residual (GSR) scheme has demonstrated superior performance in image processing. However, one challenge in GSR is to estimate the residual by using a proper reference of the Group-based sparse coding (GSC), which is desired to be as close to the truth as possible. Previous researches utilized the estimations from other algorithms (i.e., GMM or BM3D), which are either not accurate or too slow. In this paper, we propose to use the Non-Local Samples (NLS) as reference in the GSR regime for image denoising, thus termed GSR-NLS. More specifically, we first obtain a good estimation of the Group sparse coefficients by the image nonlocal self-similarity, and then solve the GSR model by an effective iterative shrinkage algorithm. Experimental results demonstrate that the proposed GSR-NLS not only outperforms many state-of-the-art methods, but also delivers the competitive advantage of speed.

  • Group Sparsity Residual Constraint for Image Denoising with External Nonlocal Self-Similarity Prior
    Neurocomputing, 2018
    Co-Authors: Zhiyuan Zha, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang, Yang Chen, Xin Liu
    Abstract:

    Abstract Nonlocal image representation has been successfully used in many image-related inverse problems including denoising, deblurring and deblocking. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of degraded observation image, and few methods use the NSS prior from natural images. In this paper we propose a novel method for image denoising via Group Sparsity residual constraint with external NSS prior (GSRC-ENSS). Different from the previous NSS prior-based denoising methods, two kinds of NSS prior (e.g., NSS priors of noisy image and natural images) are used for image denoising. In particular, to enhance the performance of image denoising, the Group Sparsity residual is proposed, and thus the problem of image denoising is translated into reducing the Group Sparsity residual. Because the Groups contain a large amount of NSS information of natural images, to reduce the Group Sparsity residual, we obtain a good estimation of the Group sparse coefficients of the original image by the external NSS prior based on Gaussian Mixture Model (GMM) learning, and the Group sparse coefficients of noisy image are used to approximate the estimation. To combine these two NSS priors better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC-ENSS model. Experimental results demonstrate that the proposed GSRC-ENSS not only outperforms several state-of-the-art methods, but also delivers the best qualitative denoising results with finer details and less ringing artifacts.

  • Group Sparsity Residual Constraint for Image Denoising.
    arXiv: Computer Vision and Pattern Recognition, 2017
    Co-Authors: Zhiyuan Zha, Yechao Bai, Qiong Wang, Xinggan Zhang, Lan Tang
    Abstract:

    Group-based sparse representation has shown great potential in image denoising. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of noisy input image. That is, the similar patches are collected only from degraded input, which makes the quality of image denoising largely depend on the input itself. However, such methods often suffer from a common drawback that the denoising performance may degrade quickly with increasing noise levels. In this paper we propose a new prior model, called Group Sparsity residual constraint (GSRC). Unlike the conventional Group-based sparse representation denoising methods, two kinds of prior, namely, the NSS priors of noisy and pre-filtered images, are used in GSRC. In particular, we integrate these two NSS priors through the mechanism of Sparsity residual, and thus, the task of image denoising is converted to the problem of reducing the Group Sparsity residual. To this end, we first obtain a good estimation of the Group sparse coefficients of the original image by pre-filtering, and then the Group sparse coefficients of the noisy image are used to approximate this estimation. To improve the accuracy of the nonlocal similar patch selection, an adaptive patch search scheme is designed. Furthermore, to fuse these two NSS prior better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC model. Experimental results demonstrate that the proposed GSRC modeling outperforms many state-of-the-art denoising methods in terms of the objective and the perceptual metrics.

Xin Liu - One of the best experts on this subject based on the ideXlab platform.

  • background subtraction using spatio temporal Group Sparsity recovery
    IEEE Transactions on Circuits and Systems for Video Technology, 2018
    Co-Authors: Xin Liu, Ziheng Zhou, Xiaohua Huang, Jiawen Yao, Xiaopeng Hong, Guoying Zhao
    Abstract:

    Background subtraction is a key step in a wide spectrum of video applications, such as object tracking and human behavior analysis. Compressive sensing-based methods, which make little specific assumptions about the background, have recently attracted wide attention in background subtraction. Within the framework of compressive sensing, background subtraction is solved as a decomposition and optimization problem, where the foreground is typically modeled as pixel-wised sparse outliers. However, in real videos, foreground pixels are often not randomly distributed, but instead, Group clustered. Moreover, due to costly computational expenses, most compressive sensing-based methods are unable to process frames online. In this paper, we take into account the Group properties of foreground signals in both spatial and temporal domains, and propose a greedy pursuit-based method called spatio-temporal Group Sparsity recovery, which prunes data residues in an iterative process, according to both Sparsity and Group clustering priors, rather than merely Sparsity. Furthermore, a random strategy for background dictionary learning is used to handle complex background variations, while foreground-free training is not required. Finally, we propose a two-pass framework to achieve online processing. The proposed method is validated on multiple challenging video sequences. Experiments demonstrate that our approach effectively works on a wide range of complex scenarios and achieves a state-of-the-art performance with far fewer computations.

  • Group Sparsity Residual Constraint for Image Denoising with External Nonlocal Self-Similarity Prior
    Neurocomputing, 2018
    Co-Authors: Zhiyuan Zha, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang, Yang Chen, Xin Liu
    Abstract:

    Abstract Nonlocal image representation has been successfully used in many image-related inverse problems including denoising, deblurring and deblocking. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of degraded observation image, and few methods use the NSS prior from natural images. In this paper we propose a novel method for image denoising via Group Sparsity residual constraint with external NSS prior (GSRC-ENSS). Different from the previous NSS prior-based denoising methods, two kinds of NSS prior (e.g., NSS priors of noisy image and natural images) are used for image denoising. In particular, to enhance the performance of image denoising, the Group Sparsity residual is proposed, and thus the problem of image denoising is translated into reducing the Group Sparsity residual. Because the Groups contain a large amount of NSS information of natural images, to reduce the Group Sparsity residual, we obtain a good estimation of the Group sparse coefficients of the original image by the external NSS prior based on Gaussian Mixture Model (GMM) learning, and the Group sparse coefficients of noisy image are used to approximate the estimation. To combine these two NSS priors better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC-ENSS model. Experimental results demonstrate that the proposed GSRC-ENSS not only outperforms several state-of-the-art methods, but also delivers the best qualitative denoising results with finer details and less ringing artifacts.

  • ICASSP - Image denoising via Group Sparsity residual constraint
    2017 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2017
    Co-Authors: Zhiyuan Zha, Xin Liu, Ziheng Zhou, Xiaohua Huang, Jingang Shi, Zhenhong Shang, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang
    Abstract:

    Group Sparsity has shown great potential in various low-level vision tasks (e.g, image denoising, deblurring and inpainting). In this paper, we propose a new prior model for image denoising via Group Sparsity residual constraint (GSRC). To enhance the performance of Group sparse-based image denoising, the concept of Group Sparsity residual is proposed, and thus, the problem of image denoising is translated into one that reduces the Group Sparsity residual. To reduce the residual, we first obtain some good estimation of the Group sparse coefficients of the original image by the first-pass estimation of noisy image, and then centralize the Group sparse coefficients of noisy image to the estimation. Experimental results have demonstrated that the proposed method not only outperforms many state-of-the-art denoising methods such as BM3D and WNNM, but results in a faster speed.

  • Image denoising via Group Sparsity residual constraint
    arXiv: Computer Vision and Pattern Recognition, 2016
    Co-Authors: Zhiyuan Zha, Xin Liu, Ziheng Zhou, Xiaohua Huang, Jingang Shi, Zhenhong Shang, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang
    Abstract:

    Group Sparsity has shown great potential in various low-level vision tasks (e.g, image denoising, deblurring and inpainting). In this paper, we propose a new prior model for image denoising via Group Sparsity residual constraint (GSRC). To enhance the performance of Group sparse-based image denoising, the concept of Group Sparsity residual is proposed, and thus, the problem of image denoising is translated into one that reduces the Group Sparsity residual. To reduce the residual, we first obtain some good estimation of the Group sparse coefficients of the original image by the first-pass estimation of noisy image, and then centralize the Group sparse coefficients of noisy image to the estimation. Experimental results have demonstrated that the proposed method not only outperforms many state-of-the-art denoising methods such as BM3D and WNNM, but results in a faster speed.

Xinggan Zhang - One of the best experts on this subject based on the ideXlab platform.

  • Group Sparsity residual with non local samples for image denoising
    arXiv: Computer Vision and Pattern Recognition, 2018
    Co-Authors: Zhiyuan Zha, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang, Xin Yuan
    Abstract:

    Inspired by Group-based sparse coding, recently proposed Group Sparsity residual (GSR) scheme has demonstrated superior performance in image processing. However, one challenge in GSR is to estimate the residual by using a proper reference of the Group-based sparse coding (GSC), which is desired to be as close to the truth as possible. Previous researches utilized the estimations from other algorithms (i.e., GMM or BM3D), which are either not accurate or too slow. In this paper, we propose to use the Non-Local Samples (NLS) as reference in the GSR regime for image denoising, thus termed GSR-NLS. More specifically, we first obtain a good estimation of the Group sparse coefficients by the image nonlocal self-similarity, and then solve the GSR model by an effective iterative shrinkage algorithm. Experimental results demonstrate that the proposed GSR-NLS not only outperforms many state-of-the-art methods, but also delivers the competitive advantage of speed.

  • Group Sparsity Residual Constraint for Image Denoising with External Nonlocal Self-Similarity Prior
    Neurocomputing, 2018
    Co-Authors: Zhiyuan Zha, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang, Yang Chen, Xin Liu
    Abstract:

    Abstract Nonlocal image representation has been successfully used in many image-related inverse problems including denoising, deblurring and deblocking. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of degraded observation image, and few methods use the NSS prior from natural images. In this paper we propose a novel method for image denoising via Group Sparsity residual constraint with external NSS prior (GSRC-ENSS). Different from the previous NSS prior-based denoising methods, two kinds of NSS prior (e.g., NSS priors of noisy image and natural images) are used for image denoising. In particular, to enhance the performance of image denoising, the Group Sparsity residual is proposed, and thus the problem of image denoising is translated into reducing the Group Sparsity residual. Because the Groups contain a large amount of NSS information of natural images, to reduce the Group Sparsity residual, we obtain a good estimation of the Group sparse coefficients of the original image by the external NSS prior based on Gaussian Mixture Model (GMM) learning, and the Group sparse coefficients of noisy image are used to approximate the estimation. To combine these two NSS priors better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC-ENSS model. Experimental results demonstrate that the proposed GSRC-ENSS not only outperforms several state-of-the-art methods, but also delivers the best qualitative denoising results with finer details and less ringing artifacts.

  • Group Sparsity Residual Constraint for Image Denoising.
    arXiv: Computer Vision and Pattern Recognition, 2017
    Co-Authors: Zhiyuan Zha, Yechao Bai, Qiong Wang, Xinggan Zhang, Lan Tang
    Abstract:

    Group-based sparse representation has shown great potential in image denoising. However, most existing methods only consider the nonlocal self-similarity (NSS) prior of noisy input image. That is, the similar patches are collected only from degraded input, which makes the quality of image denoising largely depend on the input itself. However, such methods often suffer from a common drawback that the denoising performance may degrade quickly with increasing noise levels. In this paper we propose a new prior model, called Group Sparsity residual constraint (GSRC). Unlike the conventional Group-based sparse representation denoising methods, two kinds of prior, namely, the NSS priors of noisy and pre-filtered images, are used in GSRC. In particular, we integrate these two NSS priors through the mechanism of Sparsity residual, and thus, the task of image denoising is converted to the problem of reducing the Group Sparsity residual. To this end, we first obtain a good estimation of the Group sparse coefficients of the original image by pre-filtering, and then the Group sparse coefficients of the noisy image are used to approximate this estimation. To improve the accuracy of the nonlocal similar patch selection, an adaptive patch search scheme is designed. Furthermore, to fuse these two NSS prior better, an effective iterative shrinkage algorithm is developed to solve the proposed GSRC model. Experimental results demonstrate that the proposed GSRC modeling outperforms many state-of-the-art denoising methods in terms of the objective and the perceptual metrics.

  • Image denoising using Group Sparsity residual and external nonlocal self-similarity prior
    arXiv: Computer Vision and Pattern Recognition, 2017
    Co-Authors: Zhiyuan Zha, Yechao Bai, Qiong Wang, Xinggan Zhang, Lan Tang
    Abstract:

    Nonlocal image representation has been successfully used in many image-related inverse problems including denoising, deblurring and deblocking. However, a majority of reconstruction methods only exploit the nonlocal self-similarity (NSS) prior of the degraded observation image, it is very challenging to reconstruct the latent clean image. In this paper we propose a novel model for image denoising via Group Sparsity residual and external NSS prior. To boost the performance of image denoising, the concept of Group Sparsity residual is proposed, and thus the problem of image denoising is transformed into one that reduces the Group Sparsity residual. Due to the fact that the Groups contain a large amount of NSS information of natural images, we obtain a good estimation of the Group sparse coefficients of the original image by the external NSS prior based on Gaussian Mixture model (GMM) learning and the Group sparse coefficients of noisy image is used to approximate the estimation. Experimental results have demonstrated that the proposed method not only outperforms many state-of-the-art methods, but also delivers the best qualitative denoising results with finer details and less ringing artifacts.

  • ICASSP - Image denoising via Group Sparsity residual constraint
    2017 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2017
    Co-Authors: Zhiyuan Zha, Xin Liu, Ziheng Zhou, Xiaohua Huang, Jingang Shi, Zhenhong Shang, Lan Tang, Yechao Bai, Qiong Wang, Xinggan Zhang
    Abstract:

    Group Sparsity has shown great potential in various low-level vision tasks (e.g, image denoising, deblurring and inpainting). In this paper, we propose a new prior model for image denoising via Group Sparsity residual constraint (GSRC). To enhance the performance of Group sparse-based image denoising, the concept of Group Sparsity residual is proposed, and thus, the problem of image denoising is translated into one that reduces the Group Sparsity residual. To reduce the residual, we first obtain some good estimation of the Group sparse coefficients of the original image by the first-pass estimation of noisy image, and then centralize the Group sparse coefficients of noisy image to the estimation. Experimental results have demonstrated that the proposed method not only outperforms many state-of-the-art denoising methods such as BM3D and WNNM, but results in a faster speed.

Jun Liu - One of the best experts on this subject based on the ideXlab platform.

  • total variation with overlapping Group Sparsity for speckle noise reduction
    Neurocomputing, 2016
    Co-Authors: Jun Liu, Tingzhu Huang, Gang Liu, Si Wang
    Abstract:

    Staircase effect usually happens on the total variation (TV) regularized solutions, while the overlapping Group Sparsity total variation (OGSTV) as a regularization has been proved to be effective for alleviating this drawback. For coherent imaging systems, such as the synthetic aperture radar, the acquired images are corrupted by speckles. In this paper, we propose a speckle noise reduction model based on the regularization of OGSTV. Under the framework of efficient alternating direction method of multipliers, we develop the corresponding algorithm for solving the proposed model. Numerical experiments are presented to illustrate the superiority of the proposed model and efficiency of the corresponding algorithm.

  • total variation with overlapping Group Sparsity for image deblurring under impulse noise
    PLOS ONE, 2015
    Co-Authors: Gang Liu, Tingzhu Huang, Jun Liu
    Abstract:

    The total variation (TV) regularization method is an effective method for image deblurring in preserving edges. However, the TV based solutions usually have some staircase effects. In order to alleviate the staircase effects, we propose a new model for restoring blurred images under impulse noise. The model consists of an l1-fidelity term and a TV with overlapping Group Sparsity (OGS) regularization term. Moreover, we impose a box constraint to the proposed model for getting more accurate solutions. The solving algorithm for our model is under the framework of the alternating direction method of multipliers (ADMM). We use an inner loop which is nested inside the majorization minimization (MM) iteration for the subproblem of the proposed method. Compared with other TV-based methods, numerical results illustrate that the proposed method can significantly improve the restoration quality, both in terms of peak signal-to-noise ratio (PSNR) and relative error (ReE).

  • image restoration using total variation with overlapping Group Sparsity
    Information Sciences, 2015
    Co-Authors: Jun Liu, Tingzhu Huang, Ivan W Selesnick, Poyu Chen
    Abstract:

    Image restoration is one of the most fundamental issues in imaging science. Total variation regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase artifacts. In this work we extend the total variation with overlapping Group Sparsity, which we previously developed for one dimension signal processing, to image restoration. A convex cost function is given and an efficient algorithm is proposed for solving the corresponding minimization problem. In the experiments, we compare our method with several state-of-the-art methods. The results illustrate the efficiency and effectiveness of the proposed method in terms of PSNR and computing time.

  • image restoration using total variation with overlapping Group Sparsity
    arXiv: Computer Vision and Pattern Recognition, 2013
    Co-Authors: Jun Liu, Tingzhu Huang, Ivan W Selesnick, Poyu Chen
    Abstract:

    Image restoration is one of the most fundamental issues in imaging science. Total variation (TV) regularization is widely used in image restoration problems for its capability to preserve edges. In the literature, however, it is also well known for producing staircase-like artifacts. Usually, the high-order total variation (HTV) regularizer is an good option except its over-smoothing property. In this work, we study a minimization problem where the objective includes an usual $l_2$ data-fidelity term and an overlapping Group Sparsity total variation regularizer which can avoid staircase effect and allow edges preserving in the restored image. We also proposed a fast algorithm for solving the corresponding minimization problem and compare our method with the state-of-the-art TV based methods and HTV based method. The numerical experiments illustrate the efficiency and effectiveness of the proposed method in terms of PSNR, relative error and computing time.

Yi Zhang - One of the best experts on this subject based on the ideXlab platform.

  • few view ct reconstruction with Group Sparsity regularization
    International Journal for Numerical Methods in Biomedical Engineering, 2018
    Co-Authors: Peng Bao, Jiliu Zhou, Yi Zhang
    Abstract:

    Classical total variation-based iterative reconstruction algorithm is effective for the reconstruction of piecewise smooth image, but it causes oversmoothing effect for textured regions in the reconstructed image. To address this problem, this work presents a novel computed tomography reconstruction method for the few-view problem called the Group-Sparsity regularization-based simultaneous algebraic reconstruction technique (SART). Group-based sparse representation, which uses the concept of a Group as the basic unit of sparse representation instead of a patch, is introduced as the image domain prior regularization term to eliminate the oversmoothing effect. By Grouping the nonlocal patches into different clusters with similarity measured by Euclidean distance, the Sparsity and nonlocal similarity in a single image are simultaneously explored. The split Bregman iteration algorithm is applied to obtain the numerical scheme. Experimental results demonstrate that our method both qualitatively and quantitatively outperforms several existing reconstruction methods, including filtered back projection, SART, total variation-based projections onto convex sets, and SART-based dictionary learning.

  • few view ct reconstruction with Group Sparsity regularization
    arXiv: Medical Physics, 2018
    Co-Authors: Peng Bao, Jiliu Zhou, Yi Zhang
    Abstract:

    Classical total variation (TV) based iterative reconstruction algorithms assume that the signal is piecewise smooth, which causes reconstruction results to suffer from the over-smoothing effect. To address this problem, this work presents a novel computed tomography (CT) reconstruction method for the few-view problem called the Group-Sparsity regularization-based simultaneous algebraic reconstruction technique (GSR-SART). Group-based sparse representation, which utilizes the concept of a Group as the basic unit of sparse representation instead of a patch, is introduced as the image domain prior regularization term to eliminate the over-smoothing effect. By Grouping the nonlocal patches into different clusters with similarity measured by Euclidean distance, the Sparsity and nonlocal similarity in a single image are simultaneously explored. The split Bregman iteration algorithm is applied to obtain the numerical scheme. Experimental results demonstrate that our method both qualitatively and quantitatively outperforms several existing reconstruction methods, including filtered back projection, expectation maximization, SART, and TV-based projections onto convex sets.