{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T15:21:14Z","timestamp":1771514474124,"version":"3.50.1"},"reference-count":47,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2020,10,3]],"date-time":"2020-10-03T00:00:00Z","timestamp":1601683200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No.61372069"],"award-info":[{"award-number":["No.61372069"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Defencse Pre-research Foundation","award":["none"],"award-info":[{"award-number":["none"]}]},{"name":"the SRF for ROCS, SEM","award":["JY0600090102"],"award-info":[{"award-number":["JY0600090102"]}]},{"DOI":"10.13039\/501100012176","name":"Project 211","doi-asserted-by":"publisher","award":["No.B08038"],"award-info":[{"award-number":["No.B08038"]}],"id":[{"id":"10.13039\/501100012176","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["none"],"award-info":[{"award-number":["none"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper focuses on image compressive sensing (CS). As the intrinsic properties of natural images, nonlocal self-similarity and sparse representation have been widely used in various image processing tasks. Most existing image CS methods apply either self-adaptive dictionary (e.g., principle component analysis (PCA) dictionary and singular value decomposition (SVD) dictionary) or fixed dictionary (e.g., discrete cosine transform (DCT), discrete wavelet transform (DWT), and Curvelet) as the sparse basis, while single dictionary could not fully explore the sparsity of images. In this paper, a Hybrid NonLocal Sparsity Regularization (HNLSR) is developed and applied to image compressive sensing. The proposed HNLSR measures nonlocal sparsity in 2D and 3D transform domain simultaneously, and both self-adaptive singular value decomposition (SVD) dictionary and fixed 3D transform are utilized. We use an efficient alternating minimization method to solve the optimization problem. Experimental results demonstrate that the proposed method outperforms existing methods in both objective evaluation and visual quality.<\/jats:p>","DOI":"10.3390\/s20195666","type":"journal-article","created":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T08:35:57Z","timestamp":1601886957000},"page":"5666","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Image Compressive Sensing via Hybrid Nonlocal Sparsity Regularization"],"prefix":"10.3390","volume":"20","author":[{"given":"Lizhao","family":"Li","sequence":"first","affiliation":[{"name":"State Key Lab of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Song","family":"Xiao","sequence":"additional","affiliation":[{"name":"State Key Lab of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7050-8318","authenticated-orcid":false,"given":"Yimin","family":"Zhao","sequence":"additional","affiliation":[{"name":"State Key Lab of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2006.871582","article-title":"Compressed sensing","volume":"52","author":"Donoho","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1109\/TIT.2005.862083","article-title":"Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information","volume":"52","author":"Romberg","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1109\/MSP.2007.914730","article-title":"Single-pixel imaging via compressive sampling","volume":"25","author":"Duarte","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Cevher, V., Sankaranarayanan, A., Duarte, M.F., Reddy, D., Baraniuk, R.G., and Chellappa, R. (2008, January 12\u201318). Compressive sensing for background subtraction. Proceedings of the European Conference on Computer Vision (ECCV), Marseille, France.","DOI":"10.1007\/978-3-540-88688-4_12"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1109\/MSP.2007.914728","article-title":"Compressed sensing MRI","volume":"25","author":"Lustig","year":"2008","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4285","DOI":"10.1109\/TGRS.2010.2051231","article-title":"A novel strategy for radar imaging based on compressive sensing","volume":"48","author":"Alonso","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1002\/cpa.20042","article-title":"An iterative thresholding algorithm for linear inverse problems with a sparsity constraint","volume":"57","author":"Daubechies","year":"2004","journal-title":"Common. Pure Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4655","DOI":"10.1109\/TIT.2007.909108","article-title":"Signal recovery from random measurements via orthogonal matching pursuit","volume":"53","author":"Tropp","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1137\/080725891","article-title":"The split Bregman method for L1-regularized problems","volume":"2","author":"Goldstein","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1002\/cpa.3160420503","article-title":"Optimal approximations by piecewise smooth functions and associated variational problems","volume":"42","author":"Mumford","year":"1989","journal-title":"Common. Pure Appl. Math."},{"key":"ref_11","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":"Candes","year":"2008","journal-title":"J. Fourier Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1137\/080724265","article-title":"A new alternating minimization algorithm for total variation image reconstruction","volume":"1","author":"Wang","year":"2008","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1007\/s10589-013-9576-1","article-title":"An efficient augmented Lagrangian method with applications to total variation minimization","volume":"56","author":"Li","year":"2013","journal-title":"Comput. Optim. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3488","DOI":"10.1109\/TSP.2009.2022003","article-title":"Exploiting structure in wavelet-based Bayesian compressive sensing","volume":"57","author":"He","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kim, Y., Nadar, M.S., and Bilgin, A. (2010, January 12\u201315). Compressed sensing using a Gaussian scale mixtures model in wavelet domain. Proceedings of the IEEE International Conference on Image Processing (ICIP), Hong Kong, China.","DOI":"10.1109\/ICIP.2010.5652744"},{"key":"ref_16","first-page":"233","article-title":"Tree-structured compressive sensing with variational Bayesian analysis","volume":"17","author":"He","year":"2009","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4311","DOI":"10.1109\/TSP.2006.881199","article-title":"K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation","volume":"54","author":"Aharon","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3736","DOI":"10.1109\/TIP.2006.881969","article-title":"Image denoising via sparse and redundant representations over learned dictionaries","volume":"15","author":"Elad","year":"2006","journal-title":"IEEE Trans. Image Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1109\/TPAMI.2008.79","article-title":"Robust face recognition via sparse representation","volume":"31","author":"Wright","year":"2008","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2861","DOI":"10.1109\/TIP.2010.2050625","article-title":"Image super-resolution via sparse representation","volume":"19","author":"Yang","year":"2010","journal-title":"IEEE Trans. Image Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1838","DOI":"10.1109\/TIP.2011.2108306","article-title":"Image deblurring and super-resolution by adaptive sparse domain selection and adaptive regularization","volume":"20","author":"Dong","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_22","unstructured":"Buades, A., Coll, B., and Morel, J.-M. (2005, January 20\u201325). A non-local algorithm for image denoising. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), San Diego, CA, USA."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2080","DOI":"10.1109\/TIP.2007.901238","article-title":"Image denoising by sparse 3-D transform-domain collaborative filtering","volume":"16","author":"Dabov","year":"2007","journal-title":"IEEE Trans. Image Process."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Egiazarian, K., Foi, A., and Katkovnik, V. (2007, January 16\u201319). Compressed sensing image reconstruction via recursive spatially adaptive filtering. Proceedings of the IEEE International Conference on Image Processing (ICIP), San Antonio, TX, USA.","DOI":"10.1109\/ICIP.2007.4379013"},{"key":"ref_25","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":"2012","journal-title":"IEEE Trans. Image Process."},{"key":"ref_26","unstructured":"Zhang, J., Zhao, D., Jiang, F., and Gao, W. (2013, January 20\u201322). Structural group sparse representation for image compressive sensing recovery. Proceedings of the IEEE Data Compression Conference (DCC), Snowbird, UT, USA."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.sigpro.2013.09.025","article-title":"Image compressive sensing recovery using adaptively learned sparsifying basis via L0 minimization","volume":"103","author":"Zhang","year":"2014","journal-title":"Signal Process."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3618","DOI":"10.1109\/TIP.2014.2329449","article-title":"Compressive sensing via nonlocal low-rank regularization","volume":"23","author":"Dong","year":"2014","journal-title":"IEEE Trans. Image Process."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.neucom.2018.03.027","article-title":"Group-based sparse representation for image compressive sensing reconstruction with non-convex regularization","volume":"296","author":"Zha","year":"2018","journal-title":"Neurocomputing"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"90640","DOI":"10.1109\/ACCESS.2019.2927009","article-title":"Image compressive sensing reconstruction based on z-score standardized group sparse representation","volume":"7","author":"Gao","year":"2019","journal-title":"IEEE Access"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/j.image.2019.07.021","article-title":"LLp norm regularization based group sparse representation for image compressed sensing recovery","volume":"78","author":"Keshavarzian","year":"2019","journal-title":"Signal Process. Image Commun."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1016\/j.neucom.2020.04.065","article-title":"Joint group and residual sparse coding for image compressives sensing","volume":"405","author":"Li","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1109\/JETCAS.2012.2220391","article-title":"Image compressive sensing recovery via collaborative sparsity","volume":"2","author":"Zhang","year":"2012","journal-title":"IEEE J. Emerg. Sel. Top. Circuits Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1016\/j.image.2012.09.003","article-title":"Image reconstruction with locally adaptive sparsity and nonlocal robust regularization","volume":"27","author":"Dong","year":"2012","journal-title":"Signal Process. Image Commun."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1049\/el.2012.2536","article-title":"Compressive sensing via reweighted TV and nonlocal sparsity regularisation","volume":"49","author":"Dong","year":"2013","journal-title":"Electron. Lett."},{"key":"ref_36","unstructured":"Zhang, J., Liu, S., Xiong, R., Ma, S., and Zhao, D. (2013, January 19\u201323). Improved total variation based image compressive sensing recovery by nonlocal regularization. Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS), Beijing, China."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3126","DOI":"10.1109\/TIP.2016.2562563","article-title":"Compressive sensing image restoration using adaptive curvelet thresholding and nonlocal sparse regularization","volume":"25","author":"Eslahi","year":"2016","journal-title":"IEEE Trans. Image Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.ins.2018.08.064","article-title":"Collaborative block compressed sensing reconstruction with dual-domain sparse representation","volume":"472","author":"Zhou","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Kulkarni, K., Lohit, S., Turaga, P., Kerviche, R., and Ashok, A. (2016, January 27\u201330). Reconnet: Non-iterative reconstruction of images from compressively sensed measurements. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.55"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Zhang, J., and Ghanem, B. (2018, January 18\u201322). ISTA-Net: Interpretable optimization-inspired deep network for image compressive sensing. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Salt Lake City, UT, USA.","DOI":"10.1109\/CVPR.2018.00196"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Zhang, X., Yuan, X., and Carin, L. (2018, January 18\u201322). Nonlocal low-rank tensor factor analysis for image restoration. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Salt Lake City, UT, USA.","DOI":"10.1109\/CVPR.2018.00859"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.neucom.2019.05.006","article-title":"DR2-Net: Deep Residual Reconstruction Network for image compressive sensing","volume":"359","author":"Yao","year":"2019","journal-title":"Neurocomputing"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1109\/TPAMI.2018.2883941","article-title":"ADMM-CSNet: A deep learning approach for image compressive sensing","volume":"42","author":"Yang","year":"2020","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1109\/JSTSP.2020.2977507","article-title":"Optimization-inspired compact deep compressive sensing","volume":"14","author":"Zhang","year":"2020","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_45","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."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2378","DOI":"10.1109\/TIP.2011.2109730","article-title":"FSIM: A feature similarity index for image quality assessment","volume":"20","author":"Zhang","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Chen, C., Tramel, E.W., and Fowler, J.E. (2011, January 6-9). Compressed-sensing recovery of images and video using multihypothesis predictions. Proceedings of the Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Pacific Grove, CA, USA.","DOI":"10.1109\/ACSSC.2011.6190204"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/19\/5666\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:16:21Z","timestamp":1760177781000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/19\/5666"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,3]]},"references-count":47,"journal-issue":{"issue":"19","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["s20195666"],"URL":"https:\/\/doi.org\/10.3390\/s20195666","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,10,3]]}}}