基于低秩正則聯(lián)合稀疏建模的圖像去噪算法

查志遠(yuǎn); 袁鑫; 張嘉超; 朱策

doi:10.11999/JEIT240324

基于低秩正則聯(lián)合稀疏建模的圖像去噪算法

doi: 10.11999/JEIT240324 cstr: 32379.14.JEIT240324

1.
吉林大學(xué)通信工程學(xué)院長(zhǎng)春 130012
2.
西湖大學(xué)工學(xué)院杭州 310024
3.
南京工程學(xué)院人工智能工業(yè)技術(shù)研究所南京 211167
4.
電子科技大學(xué)信息與通信工程學(xué)院成都 611731

基金項(xiàng)目: 國(guó)家自然科學(xué)基金(62471199, 62020106011, 62271414, 61971476, 62002160和62072238)，吉林大學(xué)唐敖慶英才教授啟動(dòng)基金，浙江省杰出青年基金(LR23F010001)和西湖基金會(huì)(2023GD007)

詳細(xì)信息

作者簡(jiǎn)介:
查志遠(yuǎn)：男，教授，研究方向?yàn)閳D像復(fù)原、計(jì)算成像、機(jī)器學(xué)習(xí)

袁鑫：男，研究員，研究方向?yàn)橛?jì)算成像

張嘉超：女，助理教授，研究方向?yàn)閳D像復(fù)原

朱策：男，教授，研究方向?yàn)橐曨l編碼

通訊作者:
查志遠(yuǎn)　zhiyuan_zha@jlu.edu.cn

中圖分類號(hào): TN911.73
計(jì)量
- 文章訪問數(shù): 389
- HTML全文瀏覽量: 140
- PDF下載量: 72
- 被引次數(shù): 0
出版歷程
- 收稿日期: 2024-04-23
- 修回日期: 2025-01-24
- 網(wǎng)絡(luò)出版日期: 2025-02-09
- 刊出日期: 2025-02-28

Low-Rank Regularized Joint Sparsity Modeling for Image Denoising

1.
College of Communication Engineering, Jilin University, Changchun 130012, China
2.
School of Engineering, Westlake University, Hangzhou 310024, China
3.
Artificial Intelligence Institute of Industrial Technology, Nanjing Institute of Technology, Nanjing 211167, China
4.
School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China

Funds: The National Natural Science Foundation of China (62471199, 62020106011, 62271414, 61971476, 62002160 and 62072238), The Start-up Grant from the Tang Aoqing talent professor of Jilin University, The Science Fund for Distinguished Young Scholars of Zhejiang Province (LR23F010001), The Westlake Foundation (2023GD007)

摘要

摘要: 非局部稀疏表示模型，如聯(lián)合稀疏(JS)模型、低秩(LR)模型和組稀疏表示(GSR)模型，通過有效利用圖像的非局部自相似(NSS)屬性，在圖像去噪研究中展現(xiàn)出巨大的潛力。流行的基于字典的JS算法在其目標(biāo)函數(shù)中利用松馳的凸懲罰，避免了NP-hard稀疏編碼，但只能得到近似的稀疏表示。這種近似的JS模型未能對(duì)潛在的圖像數(shù)據(jù)施加低秩性，從而導(dǎo)致圖像去噪質(zhì)量降低。該文提出一種新穎的低秩正則聯(lián)合稀疏(LRJS)模型，用于求解圖像去噪問題。提出的LRJS模型同時(shí)利用非局部相似塊的LR和JS先驗(yàn)信息，可以增強(qiáng)非局部相似塊之間的相關(guān)性(即低秩性)，從而可以更好地抑制噪聲，提升去噪圖像的質(zhì)量。為了提高優(yōu)化過程的可處理性和魯棒性，該文設(shè)計(jì)了一種具有自適應(yīng)參數(shù)調(diào)整策略的交替最小化算法來求解目標(biāo)函數(shù)。在兩個(gè)圖像去噪問題(包括高斯噪聲去除和泊松噪聲去除)上的實(shí)驗(yàn)結(jié)果表明，提出的LRJS方法在客觀度量和視覺感知上均優(yōu)于許多現(xiàn)有的流行或先進(jìn)的圖像去噪算法，特別是在處理具有高度自相似性的圖像數(shù)據(jù)時(shí)表現(xiàn)更為出色。提出的LRJS圖像去噪算法的源代碼通過以下鏈接下載：https://pan.baidu.com/s/14bt6u94NBTZXxhWjBHxn6A?pwd=1234，提取碼：1234。
- 圖像去噪 /
- 泊松去噪 /
- 非局部稀疏表示 /
- 低秩正則聯(lián)合稀疏 /
- 交替最小化算法 /
- 自適應(yīng)參數(shù)
Abstract: Objective Image denoising aims to reduce unwanted noise in images, which has been a long-standing issue in imaging science. Noise significantly degrades image quality, affecting their use in applications such as medical imaging, remote sensing, and image reconstruction. Over recent decades, various image prior models have been developed to address this problem, focusing on different image characteristics. These models, utilizing priors like sparsity, Low-Rankness (LR), and Nonlocal Self-Similarity (NSS), have proven highly effective. Nonlocal sparse representation models, including Joint Sparsity (JS), LR, and Group Sparse Representation (GSR), effectively leverage the NSS property of images. They capture the structural similarity of image patches, even when spatially distant. Popular dictionary-based JS algorithms use a relaxed convex penalty to avoid NP-hard sparse coding, leading to an approximately sparse representation. However, these approximations fail to enforce LR on the image data, reducing denoising quality, especially in cases of complex noise patterns or high self-similarity. This paper proposes a novel Low-Rank Regularized Joint Sparsity (LRJS) model for image denoising, integrating the benefits of LR and JS priors. The LRJS model enhances denoising performance, particularly where traditional methods underperform. By exploiting the NSS in images, the LRJS model better preserves fine details and structures, offering a robust solution for real-world applications. Methods The proposed LRJS model integrates low-rank and JS priors to enhance image denoising performance. By exploiting the NSS property of images, the LRJS model strengthens the dependency between nonlocal similar patches, improving image structure representation and noise suppression. The low-rank prior reflects the smoothness and regularity inherent in the image, whereas the JS prior captures the sparsity of the image patches. Incorporating these priors ensures a more accurate representation of the underlying clean image, enhancing denoising performance. An alternating minimization algorithm is proposed to solve this optimization problem, alternating between the low-rank and JS terms to simplify the optimization process. Additionally, an adaptive parameter adjustment strategy dynamically tunes the regularization parameters, balancing LR and sparsity throughout the optimization. The LRJS model offers an effective approach for image denoising by combining low-rank and JS priors, solved using an alternating minimization framework with adaptive parameter tuning. Results and Discussions Experimental results on two image denoising tasks, Gaussian noise removal (Fig. 4, Fig. 5, Table 1, Table 2) and Poisson denoising (Fig. 6, Table 3), demonstrate that the proposed LRJS method outperforms several popular and state-of-the-art denoising algorithms in both objective metrics and visual perceptual quality, particularly for images with high self-similarity. In Gaussian noise removal, the LRJS method achieves significant improvements, especially with highly self-similar images. This improvement results from LRJS effectively leveraging the NSS prior, which strengthens the dependencies among similar patches, leading to better noise suppression while preserving image details. Compared with other methods, LRJS demonstrates greater robustness, particularly in retaining fine details and structures often lost with traditional denoising techniques. For Poisson denoising, the LRJS method also yields notable performance gains. It better manages the complexity of Poisson noise compared with other approaches, highlighting its versatility and robustness across different noise types. The visual quality of the denoised images shows fewer artifacts and more accurate recovery of details. Quantitative results in terms of PSNR and SSIM further validate the effectiveness of LRJS, positioning it as a competitive solution in image denoising. Overall, these experimental findings confirm that LRJS offers a reliable and effective approach, particularly for images with high self-similarity and complex noise models. Conclusions The LRJS model proposed in this paper improves image denoising performance by combining LR and JS priors. This dual-prior framework better captures the underlying image structure while suppressing noise, particularly benefiting images with high self-similarity. Experimental results demonstrate that the LRJS method not only outperforms traditional denoising techniques but also exceeds many state-of-the-art algorithms in both objective metrics and visual quality. By leveraging the NSS property of image patches, the LRJS model enhances the dependencies among similar patches, making it particularly effective for tasks requiring the preservation of fine details and structures. The LRJS method significantly enhances the quality of denoised images, especially in complex noise scenarios such as Gaussian and Poisson noise. Its robust alternating minimization algorithm with adaptive parameter adjustment ensures effective optimization, contributing to superior performance. The results further highlight the LRJS model’s ability to preserve image edges, textures, and other fine details often degraded in other denoising algorithms. Compared with existing techniques, the LRJS method demonstrates superior performance in handling high noise levels while maintaining image clarity and detail, making it a promising tool for applications such as medical imaging, remote sensing, and image restoration. Future research could focus on optimizing the model for more complex noise environments, such as mixed noise or real-world noise that is challenging to model. Additionally, exploring more efficient algorithms and integrating advanced techniques, such as deep learning, may further improve the LRJS model’s capability and applicability to diverse denoising tasks.
- Image denoising /
- Poisson denoising /
- Nonlocal sparse representation /
- Low-Rank Regularized Joint Sparsity (LRJS) /
- Alternating minimization /
- Self-adaptive parameter

HTML全文

圖 1 稀疏系數(shù)對(duì)比示意圖

下載: 全尺寸圖片幻燈片

圖 2 比較提出的LRJS方法和幾種先進(jìn)的方法的圖像去噪結(jié)果

下載: 全尺寸圖片幻燈片

圖 3 實(shí)驗(yàn)中的一些測(cè)試圖像

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圖 4 圖像House在噪聲標(biāo)準(zhǔn)差為100時(shí)，不同方法的去噪視覺比對(duì)結(jié)果

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圖 5 圖像Urban25在噪聲標(biāo)準(zhǔn)差為50時(shí)，不同方法的去噪視覺比對(duì)結(jié)果

下載: 全尺寸圖片幻燈片

圖 6 圖像Barbara伴隨著像素強(qiáng)度峰值$ P = 5 $的泊松噪聲的不同方法的去噪視覺比對(duì)結(jié)果

下載: 全尺寸圖片幻燈片

圖 7 真實(shí)圖像去噪場(chǎng)景1

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圖 8 真實(shí)圖像去噪場(chǎng)景2

下載: 全尺寸圖片幻燈片

1 基于LRJS的高斯噪聲去除算法

輸入：噪聲圖像$ {\boldsymbol{y}} $。
初始化：$ {\sigma _n} $, $ {\hat {\boldsymbol{x}}^0} = {\boldsymbol{y}} $, $ {{\boldsymbol{y}}^0} = {\boldsymbol{y }}$。
For $ k = 1 $ do
迭代正則調(diào)整： $ {\boldsymbol{{y}}^k} = {\hat {\boldsymbol{x}}^{(k - 1)}} + \gamma ({\boldsymbol{y}} - {\hat {\boldsymbol{x}}^{(k - 1)}}) $。
更新噪聲標(biāo)準(zhǔn)差$ {\sigma _e} $通過式(26)。
For 噪聲圖像$ {\boldsymbol{y}} $中每個(gè)塊$ {y_i} $ do
收集相似塊生成一個(gè)組$ {{\boldsymbol{Y}}_i} $。
使用PCA從組$ {{\boldsymbol{Y}}_i} $中學(xué)習(xí)一個(gè)字典$ {{\boldsymbol{D}}_i} $。
獲得組稀疏$ {{\boldsymbol{A}}_i} $通過計(jì)算$ {{\boldsymbol{A}}_i} = {\boldsymbol{D}}_i^{\mathrm{T}}{{\boldsymbol{Y}}_i} $。
對(duì)組稀疏$ {{\boldsymbol{A}}_i} $執(zhí)行SVD：$ [{{\boldsymbol{U}}_i},{\Delta _i},{{\boldsymbol{V}}_i}] = {\text{SVD}}({{\boldsymbol{A}}_i}) $。
更新參數(shù)$ \mu $通過計(jì)算式(25)。
更新參數(shù)$ \tau $通過計(jì)算式(28)。
估計(jì)LR矩陣$ {\hat {\boldsymbol{L}}_i} $通過計(jì)算式(9)。
更新參數(shù)$ \eta $通過計(jì)算式(25)。
更新參數(shù)$ \lambda $通過計(jì)算式(28)。
估計(jì)組稀疏系數(shù)$ {\hat {\boldsymbol{A}}_i} $通過計(jì)算式(5)。
End for
估計(jì)噪聲圖像$ \hat {\boldsymbol{x}} $通過計(jì)算式(13)。
End for
輸出：最終的去噪圖像$ \hat {\boldsymbol{x}} $。

下載: 導(dǎo)出CSV

2 基于LRJS的泊松噪聲去除算法

輸入：噪聲圖像${\boldsymbol{ y}} $。
初始化：估計(jì)$ {\sigma _n} $通過計(jì)算式(27)，$ {\hat {\boldsymbol{x}}^0} = {\boldsymbol{y}} $，$ {{\boldsymbol{y}}^0} = {\boldsymbol{y}} $。
For $ k = 1 $ do
迭代正則調(diào)整： $ {{\boldsymbol{y}}^k} = {\hat {\boldsymbol{x}}^{(k - 1)}} + \gamma ({\boldsymbol{y}} - {\hat {\boldsymbol{x}}^{(k - 1)}}) $。
更新噪聲標(biāo)準(zhǔn)差$ {\sigma _e} $通過式(26)。
For 噪聲圖像$ {\boldsymbol{y}} $中每個(gè)塊$ {y_i} $ do
收集相似塊生成一個(gè)組$ {{\boldsymbol{Y}}_i} $。
使用PCA從組$ {{\boldsymbol{Y}}_i} $中學(xué)習(xí)一個(gè)字典$ {{\boldsymbol{D}}_i} $。
獲得組稀疏$ {{\boldsymbol{A}}_i} $通過計(jì)算$ {{\boldsymbol{A}}_i} = {\boldsymbol{D}}_i^{\mathrm{T}}{{\boldsymbol{Y}}_i} $。
對(duì)組稀疏$ {{\boldsymbol{A}}_i} $執(zhí)行SVD：$ [{{\boldsymbol{U}}_i},{\Delta _i},{{\boldsymbol{V}}_i}] = {\text{SVD}}({{\boldsymbol{A}}_i}) $。
更新參數(shù)$ \mu $通過計(jì)算式(25)。
更新參數(shù)$ \tau $通過計(jì)算式(28)。
估計(jì)LR矩陣$ {\hat {\boldsymbol{L}}_i} $通過計(jì)算式(9)。
更新參數(shù)$ \eta $通過計(jì)算式(25)。
更新參數(shù)$ \lambda $通過計(jì)算式(28)。
估計(jì)組稀疏系數(shù)$ {\hat {\boldsymbol{A}}_i} $通過計(jì)算式(5)。
End for
調(diào)用ADMM算法：
初始化：$ {\boldsymbol{g }}= {{\textit{0}}} $，${\boldsymbol{ z}} = {\hat {\boldsymbol{x}}^{(k)}} $。
更新$ \hat {\boldsymbol{z}} $通過計(jì)算式(21)。
更新$ \hat {\boldsymbol{x}} $通過計(jì)算式(24)。
更新$ \hat {\boldsymbol{g}} $通過計(jì)算式(20)。
End for
輸出：最終的去噪圖像$ \hat {\boldsymbol{x}} $。

下載: 導(dǎo)出CSV

表 1 不同方法用于高斯噪聲去除的平均PSNR比較結(jié)果(dB)

$ {\sigma _{\boldsymbol{n}}} $	BM3D	LSSC	EPLL	NCSR	GID	PGPD	aGMM	OGLR	NLNCDR	LRJS
20	31.20	31.36	30.72	31.26	30.25	31.30	31.04	31.05	30.44	31.56
40	27.53	27.77	27.16	27.66	26.65	27.79	27.37	27.69	27.04	28.02
75	24.66	24.56	24.01	24.47	23.20	24.71	24.17	24.41	24.02	24.90
100	23.30	23.09	22.66	23.00	21.56	23.36	22.81	22.69	22.72	23.62

下載: 導(dǎo)出CSV

表 2 不同方法測(cè)試Urban100數(shù)據(jù)集用于高斯噪聲去除的平均PSNR比較結(jié)果(dB)

$ {\sigma _{\boldsymbol{n}}} $	BM3D	NCSR	PGPD	OGLR	Dn-CNN	IRCNN	FFDNet	LRJS
10	33.39	33.66	33.40	32.94	33.83	33.65	33.42	34.25
20	29.50	29.68	29.47	29.27	29.75	29.64	29.61	30.16
30	27.33	27.39	27.19	27.18	27.44	27.40	27.49	27.85
40	25.44	25.77	25.70	25.67	25.86	25.90	26.03	26.28
50	24.55	24.59	24.59	24.51	24.77	24.75	24.93	25.07
平均	28.04	28.22	28.07	27.91	28.33	28.27	28.30	28.72

下載: 導(dǎo)出CSV

表 3 不同方法用于泊松噪聲去除的平均PSNR比較結(jié)果(dB)

$ P $	TNRD	Dn-CNN	IRCNN	LRPD	LRS	LRJS
5	19.50	22.31	22.78	21.66	22.23	23.56
10	23.58	23.22	24.67	23.63	24.61	25.44
15	24.11	25.47	25.81	24.29	25.93	26.54
20	24.40	25.95	26.54	25.39	26.83	27.44

下載: 導(dǎo)出CSV

表 4 消融學(xué)習(xí)：JS和提出的LRJS模型在Set12數(shù)據(jù)集上用于圖像去噪的平均PSNR結(jié)果(dB)

高斯噪聲去除									泊松噪聲去除
$ {\sigma _{\boldsymbol{n}}} $	10	20	30	40	50	75	100	平均	$ P $	1	5	10	15	20	25	30	平均
JS	34.30	31.02	29.06	27.78	26.74	24.96	23.68	28.22	JS	19.37	23.65	25.10	26.29	27.01	27.58	27.93	25.28
LRJS	34.54	31.14	29.21	27.87	26.80	24.98	23.73	28.32	LRJS	20.07	23.86	25.40	26.41	27.14	27.69	28.09	25.52

下載: 導(dǎo)出CSV

參考文獻(xiàn)(36)

[1]	OSHER S, BURGER M, GOLDFARB D, et al. An iterative regularization method for total variation-based image restoration[J]. Multiscale Modeling & Simulation, 2005, 4(2): 460–489. doi: 10.1137/040605412.
[2]	AHARON M, ELAD M, and BRUCKSTEIN A. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[J]. IEEE Transactions on Signal Processing, 2006, 54(11): 4311–4322. doi: 10.1109/TSP.2006.881199.
[3]	BUADES A, COLL B, and MOREL J M. A non-local algorithm for image denoising[C]. IEEE Conference on Computer Vision and Pattern Recognition, San Diego, USA, 2005: 60–65. doi: 10.1109/CVPR.2005.38.
[4]	ELAD M, KAWAR B, and VAKSMAN G. Image denoising: The deep learning revolution and beyond—a survey paper[J]. SIAM Journal on Imaging Sciences, 2023, 16(3): 1594–1654. doi: 10.1137/23M1545859.
[5]	趙冬冬, 葉逸飛, 陳朋, 等. 基于殘差和注意力網(wǎng)絡(luò)的聲吶圖像去噪方法[J]. 光電工程, 2023, 50(6): 230017. doi: 10.12086/oee.2023.230017. ZHAO Dongdong, YE Yifei, CHEN Peng, et al. Sonar image denoising method based on residual and attention network[J]. Opto-Electronic Engineering, 2023, 50(6): 230017. doi: 10.12086/oee.2023.230017.
[6]	周建新, 周鳳祺. 基于改進(jìn)協(xié)同量子粒子群的小波去噪分析研究[J]. 電光與控制, 2022, 29(1): 47–50. doi: 10.3969/j.issn.1671-637x.2022.01.010. ZHOU Jianxin and ZHOU Fengqi. Wavelet denoising analysis based on cooperative quantum-behaved particle swarm optimization[J]. Electronics Optics & Control, 2022, 29(1): 47–50. doi: 10.3969/j.issn.1671-637x.2022.01.010.
[7]	DABOV K, FOI A, KATKOVNIK V, et al. Image denoising by sparse 3-D transform-domain collaborative filtering[J]. IEEE Transactions on Image Processing, 2007, 16(8): 2080–2095. doi: 10.1109/TIP.2007.901238.
[8]	羅亮, 馮象初, 張選德, 等. 基于非局部雙邊隨機(jī)投影低秩逼近圖像去噪算法[J]. 電子與信息學(xué)報(bào), 2013, 35(1): 99–105. doi: 10.3724/SP.J.1146.2012.00819. LUO Liang, FENG Xiangchu, ZHANG Xuande, et al. An image denoising method based on non-local two-side random projection and low rank approximation[J]. Journal of Electronics & Information Technology, 2013, 35(1): 99–105. doi: 10.3724/SP.J.1146.2012.00819.
[9]	ZHANG Jian, ZHAO Debin, and GAO Wen. Group-based sparse representation for image restoration[J]. IEEE Transactions on Image Processing, 2014, 23(8): 3336–3351. doi: 10.1109/TIP.2014.2323127.
[10]	XU Jun, ZHANG Lei, ZUO Wangmeng, et al. Patch group based nonlocal self-similarity prior learning for image denoising[C]. IEEE International Conference on Computer Vision, Santiago, Chile, 2015: 244–252. doi: 10.1109/ICCV.2015.36.
[11]	ZHA Zhiyuan, YUAN Xin, WEN Bihan, et al. A benchmark for sparse coding: When group sparsity meets rank minimization[J]. IEEE Transactions on Image Processing, 2020, 29: 5094–5109. doi: 10.1109/TIP.2020.2972109.
[12]	MAIRAL J, BACH F, PONCE J, et al. Non-local sparse models for image restoration[C]. 12th IEEE International Conference on Computer Vision, Kyoto, Japan, 2009: 2272–2279. doi: 10.1109/ICCV.2009.5459452.
[13]	ZHANG Chengfang, ZHANG Ziyou, FENG Ziliang, et al. Joint sparse model with coupled dictionary for medical image fusion[J]. Biomedical Signal Processing and Control, 2023, 79: 104030. doi: 10.1016/j.bspc.2022.104030.
[14]	LI Chunzhi and CHEN Xiaohua. A staged approach with structural sparsity for hyperspectral unmixing[J]. IEEE Sensors Journal, 2023, 23(12): 13248–13260. doi: 10.1109/JSEN.2023.3270885.
[15]	ZHA Zhiyuan, WEN Bihan, YUAN Xin, et al. Low-rank regularized joint sparsity for image denoising[C]. IEEE International Conference on Image Processing, Anchorage, USA, 2021: 1644–1648. doi: 10.1109/ICIP42928.2021.9506726.
[16]	ZHANG Kai, ZUO Wangmeng, CHEN Yunjin, et al. Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising[J]. IEEE Transactions on Image Processing, 2017, 26(7): 3142–3155. doi: 10.1109/TIP.2017.2662206.
[17]	ZHANG Kai, ZUO Wangmeng, GU Shuhang, et al. Learning deep CNN denoiser prior for image restoration[C]. IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, USA, 2017: 2808–2817. doi: 10.1109/CVPR.2017.300.
[18]	ZHANG Kai, ZUO Wangmeng, and ZHANG Lei. FFDNet: Toward a fast and flexible solution for CNN-based image denoising[J]. IEEE Transactions on Image Processing, 2018, 27(9): 4608–4622. doi: 10.1109/TIP.2018.2839891.
[19]	ZHAO Mingchao, WEN Youwei, NG M, et al. A nonlocal low rank model for Poisson noise removal[J]. Inverse Problems and Imaging, 2021, 15(3): 519–537. doi: 10.3934/ipi.2021003.
[20]	CHEN Yunjin and PORK T. Trainable nonlinear reaction diffusion: A flexible framework for fast and effective image restoration[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017, 39(6): 1256–1272. doi: 10.1109/TPAMI.2016.2596743.
[21]	ELAD M and YAVNEH I. A plurality of sparse representations is better than the sparsest one alone[J]. IEEE Transactions on Information Theory, 2009, 55(10): 4701–4714. doi: 10.1109/TIT.2009.2027565.
[22]	DONG Weisheng, ZHANG Lei, SHI Guangming, et al. Nonlocally centralized sparse representation for image restoration[J]. IEEE Transactions on Image Processing, 2013, 22(4): 1620–1630. doi: 10.1109/TIP.2012.2235847.
[23]	CHEN Yong, HE Wei, YOKOYA N, et al. Hyperspectral image restoration using weighted group sparsity-regularized low-rank tensor decomposition[J]. IEEE Transactions on Cybernetics, 2020, 50(8): 3556–3570. doi: 10.1109/TCYB.2019.2936042.
[24]	CAI Jianfeng, CANDèS E J, and SHEN Zuowei. A singular value thresholding algorithm for matrix completion[J]. SIAM Journal on Optimization, 2010, 20(4): 1956–1982. doi: 10.1137/080738970.
[25]	KUMAR G P and SAHAY R R. Low rank Poisson denoising (LRPD): A low rank approach using split Bregman algorithm for Poisson noise removal from images[C]. IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, Long Beach, USA, 2019: 1907–1916. doi: 10.1109/CVPRW.2019.00242.
[26]	BOYD S, PARIKH N, CHU E, et al. Distributed optimization and statistical learning via the alternating direction method of multipliers[J]. Foundations and Trends? in Machine Learning, 2011, 3(1): 1–122. doi: 10.1561/2200000016.
[27]	WEN Youwei and CHAN R H. Parameter selection for total-variation-based image restoration using discrepancy principle[J]. IEEE Transactions on Image Processing, 2012, 21(4): 1770–1781. doi: 10.1109/TIP.2011.2181401.
[28]	ZORAN D and WEISS Y. From learning models of natural image patches to whole image restoration[C]. IEEE International Conference on Computer Vision, Barcelona, Spain, 2011: 479–486. doi: 10.1109/ICCV.2011.6126278.
[29]	TALEBI H and MILANFAR P. Global image denoising[J]. IEEE Transactions on Image Processing, 2014, 23(2): 755–768. doi: 10.1109/TIP.2013.2293425.
[30]	LUO Enming, CHAN S H, and NGUYEN T Q. Adaptive image denoising by mixture adaptation[J]. IEEE Transactions on Image Processing, 2016, 25(10): 4489–4503. doi: 10.1109/TIP.2016.2590318.
[31]	PANG Jiahao and CHEUNG G. Graph Laplacian regularization for image denoising: Analysis in the continuous domain[J]. IEEE Transactions on Image Processing, 2017, 26(4): 1770–1785. doi: 10.1109/TIP.2017.2651400.
[32]	LIU Haosen and TAN Shan. Image regularizations based on the sparsity of corner points[J]. IEEE Transactions on Image Processing, 2019, 28(1): 72–87. doi: 10.1109/TIP.2018.2862357.
[33]	HUANG Jiabin, SINGH A, and AHUJA N. Single image super-resolution from transformed self-exemplars[C]. IEEE Conference on Computer Vision and Pattern Recognition, Boston, USA, 2015: 5197–5206. doi: 10.1109/CVPR.2015.7299156.
[34]	AZZARI L and FOI A. Variance stabilization for noisy+estimate combination in iterative Poisson denoising[J]. IEEE Signal Processing Letters, 2016, 23(8): 1086–1090. doi: 10.1109/LSP.2016.2580600.
[35]	LEHTINEN J, MUNKBERG J, HASSELGREN J, et al. Noise2Noise: Learning image restoration without clean data[C]. Proceedings of the 35th International Conference on Machine Learning, Stockholm, Sweden, 2018: 2965–2974.
[36]	MITTAL A, MOORTHY A K, and BOVIK A C. No-reference image quality assessment in the spatial domain[J]. IEEE Transactions on Image Processing, 2012, 21(12): 4695–4708. doi: 10.1109/TIP.2012.2214050.