{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:01:14Z","timestamp":1760238074023,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T00:00:00Z","timestamp":1657152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["41971396","61971290","X21024","KM202110016001"],"award-info":[{"award-number":["41971396","61971290","X21024","KM202110016001"]}]},{"name":"Research Ability Enhancement Program for Young Teachers of Beijing University of Civil Engineering and Architecture","award":["41971396","61971290","X21024","KM202110016001"],"award-info":[{"award-number":["41971396","61971290","X21024","KM202110016001"]}]},{"name":"Outstanding Youth Program of Beijing University of Civil Engineering and Architecture, and the Beijing Municipal Education Commission Science and Technology General Project","award":["41971396","61971290","X21024","KM202110016001"],"award-info":[{"award-number":["41971396","61971290","X21024","KM202110016001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Group sparse coding (GSC) uses the non-local similarity of images as constraints, which can fully exploit the structure and group sparse features of images. However, it only imposes the sparsity on the group coefficients, which limits the effectiveness of reconstructing real images. Low-rank regularized group sparse coding (LR-GSC) reduces this gap by imposing low-rankness on the group sparse coefficients. However, due to the use of non-local similarity, the edges and details of the images are over-smoothed, resulting in the blocking artifact of the images. In this paper, we propose a low-rank matrix restoration model based on sparse coding and dual weighting. In addition, total variation (TV) regularization is integrated into the proposed model to maintain local structure smoothness and edge features. Finally, to solve the problem of the proposed optimization, an optimization method is developed based on the alternating direction method. Extensive experimental results show that the proposed SDWLR-GSC algorithm outperforms state-of-the-art algorithms for image restoration when the images have large and sparse noise, such as salt and pepper noise.<\/jats:p>","DOI":"10.3390\/e24070946","type":"journal-article","created":{"date-parts":[[2022,7,7]],"date-time":"2022-07-07T07:51:56Z","timestamp":1657180316000},"page":"946","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Structural Smoothing Low-Rank Matrix Restoration Based on Sparse Coding and Dual-Weighted Model"],"prefix":"10.3390","volume":"24","author":[{"given":"Jiawei","family":"Wu","sequence":"first","affiliation":[{"name":"School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6693-0161","authenticated-orcid":false,"given":"Hengyou","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Elad, M., and Elad, M. (2010). Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer.","DOI":"10.1007\/978-1-4419-7011-4"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1109\/JPROC.2010.2044470","article-title":"Sparse representation for computer vision and pattern recognition","volume":"98","author":"Wright","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"972","DOI":"10.1109\/JPROC.2009.2037655","article-title":"On the role of sparse and redundant representations in image processing","volume":"98","author":"Elad","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Xu, J., Lei, Z., Zuo, W., and Zhang, D. (2015, January 7\u201313). Patch group based nonlocal self-similarity prior learning for image denoising. Proceedings of the 2015 IEEE International Conference on Computer Vision (ICCV), Santiago, Chile.","DOI":"10.1109\/ICCV.2015.36"},{"key":"ref_5","first-page":"513","article-title":"Principal component analysis","volume":"87","author":"Jolliffe","year":"2002","journal-title":"J. Mark. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4311","DOI":"10.1109\/TSP.2006.881199","article-title":"rmK-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation","volume":"54","author":"Aharon","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1553","DOI":"10.1109\/TSP.2009.2036477","article-title":"Double sparsity: Learning sparse dictionaries for sparse signal approximation","volume":"58","author":"Rubinstein","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Mairal, J., Bach, F., Ponce, J., Sapiro, G., and Zisserman, A. (October, January 27). Non-local sparse models for image restoration. Proceedings of the 2009 IEEE 12th International Conference on Computer Vision (ICCV), Kyoto, Japan.","DOI":"10.1109\/ICCV.2009.5459452"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3336","DOI":"10.1109\/TIP.2014.2323127","article-title":"Group-based sparse representation for image restoration","volume":"23","author":"Zhang","year":"2014","journal-title":"IEEE Trans. Image Process. Publ. IEEE Signal Process. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/s11263-015-0808-y","article-title":"Image restoration via simultaneous sparse coding: Where structured sparsity meets gaussian scale mixture","volume":"114","author":"Dong","year":"2015","journal-title":"Int. J. Comput. Vis."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1686","DOI":"10.1109\/LSP.2017.2731791","article-title":"Non-convex weighted lp minimization based group sparse representation framework for image denoising","volume":"24","author":"Wang","year":"2017","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"302","DOI":"10.1109\/TMM.2016.2614427","article-title":"Retrieval compensated group structured sparsity for image super-resolution","volume":"19","author":"Liu","year":"2016","journal-title":"IEEE Trans. Multimed."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3043","DOI":"10.1109\/TGRS.2019.2947032","article-title":"Global spatial and local spectral similarity-based manifold learning group sparse representation for hyperspectral imagery classification","volume":"58","author":"Yu","year":"2020","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1007\/s00034-018-0859-8","article-title":"Image block compressive sensing reconstruction via group-based sparse representation and nonlocal total variation","volume":"38","author":"Xu","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zha, Z., Wen, B., Yuan, X., Zhou, J., and Zhu, C. (2020, January 6\u201310). Reconciliation Of Group Sparsity And Low-Rank Models For Image Restoration. Proceedings of the 2020 IEEE International Conference on Multimedia and Expo (ICME), London, UK.","DOI":"10.1109\/ICME46284.2020.9102930"},{"key":"ref_16","unstructured":"Wright, J., Ganesh, A., Rao, S., and Ma, Y. (2009). Robust principal component analysis: Exact recovery of corrupted low-rank matrices. NIPS\u201909: Proceedings of the 22nd International Conference on Neural Information Processing Systems Red Hook, NY, USA, 7\u20139 December 2009, Curran Associates Inc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1109\/JPROC.2009.2035722","article-title":"Matrix completion with noise","volume":"98","author":"Plan","year":"2010","journal-title":"Proc. IEEE"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1007\/s00041-008-9045-x","article-title":"Enhancing sparsity by reweighted l1 minimization","volume":"14","author":"Wakin","year":"2008","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gu, S., Lei, Z., Zuo, W., and Feng, X. (2014, January 23\u201328). Weighted Nuclear Norm Minimization with Application to Image Denoising. Proceedings of the 2014 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Columbus, OH, USA.","DOI":"10.1109\/CVPR.2014.366"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11263-016-0930-5","article-title":"Weighted nuclear norm minimization and its applications to low level vision","volume":"121","author":"Gu","year":"2017","journal-title":"Int. J. Comput. Vis."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2418","DOI":"10.1109\/TCYB.2014.2307854","article-title":"Reweighted low-rank matrix recovery and its application in image restoration","volume":"44","author":"Peng","year":"2014","journal-title":"IEEE Trans. Cybern."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1970392.1970395","article-title":"Robust principal component analysis?","volume":"58","author":"Candes","year":"2011","journal-title":"J. ACM"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2419","DOI":"10.1109\/TIP.2009.2028250","article-title":"Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems","volume":"18","author":"Beck","year":"2009","journal-title":"IEEE Trans. Image Process."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/0167-2789(92)90242-F","article-title":"Nonlinear total variation based noise removal algorithms","volume":"60","author":"Rudin","year":"1992","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/18.382009","article-title":"De-noising by soft-thresholding","volume":"41","author":"Donoho","year":"2002","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1620","DOI":"10.1109\/TIP.2012.2235847","article-title":"Nonlocally centralized sparse representation for image restoration","volume":"22","author":"Dong","year":"2013","journal-title":"IEEE Trans. Image Process. Publ. IEEE Signal Process. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1109\/TCSVT.2006.869964","article-title":"Reversible data hiding","volume":"16","author":"Ni","year":"2006","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1109\/TIP.2017.2781425","article-title":"Reweighted low-rank matrix analysis with structural smoothness for image denoising","volume":"27","author":"Wang","year":"2018","journal-title":"IEEE Trans. Image Process."},{"key":"ref_30","unstructured":"Lin, Z., Chen, M., and Ma, Y. (2010). The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1109\/TNNLS.2012.2235082","article-title":"Low-rank structure learning via nonconvex heuristic recovery","volume":"24","author":"Yue","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4642","DOI":"10.1109\/TGRS.2016.2547879","article-title":"Hyperspectral image restoration via iteratively regularized weighted schatten p-norm minimization","volume":"54","author":"Xie","year":"2016","journal-title":"IEEE Trans. Geosci. Remote. Sens."},{"key":"ref_33","first-page":"1","article-title":"Image denoising in impulsive noise via weighted schatten p-norm regularization","volume":"28","author":"Chen","year":"2019","journal-title":"IEEE Trans. Image Process."},{"key":"ref_34","unstructured":"Dong, H., Yu, J., and Xiao, C. (2018). Dual reweighted lp-norm minimization for salt-and-pepper noise removal. arXiv."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/946\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:43:57Z","timestamp":1760139837000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/946"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,7]]},"references-count":34,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["e24070946"],"URL":"https:\/\/doi.org\/10.3390\/e24070946","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,7,7]]}}}