{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T16:20:05Z","timestamp":1764433205229,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T00:00:00Z","timestamp":1539043200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In the reconstruction of sparse signals in compressed sensing, the reconstruction algorithm is required to reconstruct the sparsest form of signal. In order to minimize the objective function, minimal norm algorithm and greedy pursuit algorithm are most commonly used. The minimum L1 norm algorithm has very high reconstruction accuracy, but this convex optimization algorithm cannot get the sparsest signal like the minimum L0 norm algorithm. However, because the L0 norm method is a non-convex problem, it is difficult to get the global optimal solution and the amount of calculation required is huge. In this paper, a new algorithm is proposed to approximate the smooth L0 norm from the approximate L2 norm. First we set up an approximation function model of the sparse term, then the minimum value of the objective function is solved by the gradient projection, and the weight of the function model of the sparse term in the objective function is adjusted adaptively by the reconstruction error value to reconstruct the sparse signal more accurately. Compared with the pseudo inverse of L2 norm and the L1 norm algorithm, this new algorithm has a lower reconstruction error in one-dimensional sparse signal reconstruction. In simulation experiments of two-dimensional image signal reconstruction, the new algorithm has shorter image reconstruction time and higher image reconstruction accuracy compared with the usually used greedy algorithm and the minimum norm algorithm.<\/jats:p>","DOI":"10.3390\/s18103373","type":"journal-article","created":{"date-parts":[[2018,10,9]],"date-time":"2018-10-09T11:10:44Z","timestamp":1539083444000},"page":"3373","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Gradient Projection with Approximate L0 Norm Minimization for Sparse Reconstruction in Compressed Sensing"],"prefix":"10.3390","volume":"18","author":[{"given":"Ziran","family":"Wei","sequence":"first","affiliation":[{"name":"Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China"},{"name":"School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100039, China"}]},{"given":"Jianlin","family":"Zhang","sequence":"additional","affiliation":[{"name":"Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China"}]},{"given":"Zhiyong","family":"Xu","sequence":"additional","affiliation":[{"name":"Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China"}]},{"given":"Yongmei","family":"Huang","sequence":"additional","affiliation":[{"name":"Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China"}]},{"given":"Yong","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China"}]},{"given":"Xiangsuo","family":"Fan","sequence":"additional","affiliation":[{"name":"Institute of Optics and Electronics, Chinese Academy of Science, Chengdu 610209, China"},{"name":"School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100039, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,9]]},"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","unstructured":"Boche, H., Calderbank, R., Kutyniok, G., and Vybiral, J. (2015). Compressed Sensing and Its Applications, Springer.","DOI":"10.1007\/978-3-319-16042-9"},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"5406","DOI":"10.1109\/TIT.2006.885507","article-title":"Near-optimal signal recovery from random projections: Universal encoding strategies","volume":"52","author":"Tao","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Cand\u00e8s, E.J., and Romberg, J.K. (2004). Practical signal recovery from random projections. Computational Imaging III, SPIE.","DOI":"10.1117\/12.600722"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1109\/MSP.2007.4286571","article-title":"Compressive sensing","volume":"24","author":"Baraniuk","year":"2007","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Chen, C., Tramel, E.W., and Fowler, J.E. (2011, January 6\u20139). Compressed-sensing recovery of images and video using multihypothesis predictions. Proceedings of the 2011 Conference Record of the Forty Fifth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), Pacific Grove, CA, USA.","DOI":"10.1109\/ACSSC.2011.6190204"},{"key":"ref_8","first-page":"1","article-title":"Adaptive compressed sensing for wireless image sensor networks","volume":"76","author":"Zhang","year":"2017","journal-title":"Multimed. Tools Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1207","DOI":"10.1002\/cpa.20124","article-title":"Stable signal recovery from incomplete and inaccurate measurements","volume":"59","author":"Romberg","year":"2006","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_10","first-page":"533","article-title":"Extensions of compressed sensing","volume":"86","author":"Donoho","year":"2006","journal-title":"Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1109\/JSTSP.2007.910971","article-title":"A Interior-Point Method for Large-Scale l1-Regularized Least Square","volume":"1","author":"Kim","year":"2007","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1137\/S003614450037906X","article-title":"Atomic decomposition by basis pursuit","volume":"43","author":"Chen","year":"2001","journal-title":"SIAM Rev."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Statist. Soc. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1109\/JSTSP.2007.910281","article-title":"Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems","volume":"1","author":"Figueiredo","year":"2008","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"4347","DOI":"10.1109\/TSP.2009.2025979","article-title":"Relaxed conditions for sparse signal recovery with general concave priors","volume":"57","author":"TRzasko","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.neucom.2016.10.051","article-title":"Intelligent nonconvex compressive sensing using prior information for image reconstruction by sparse representation","volume":"224","author":"Wang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Eldar, Y.C., and Kutyniok, G. (2012). Compressed Sensing: Theory and Applications, Cambridge University Press. [1st ed.]. ISBN-10 1107005582.","DOI":"10.1017\/CBO9780511794308"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1088\/0266-5611\/23\/3\/008","article-title":"Sparsity and incoherence in compressive sampling","volume":"23","author":"Romberg","year":"2007","journal-title":"Inverse Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2203","DOI":"10.1109\/TIT.2009.2016030","article-title":"Restricted isometry constants where Lp sparse recovery can fail for 0 < p <= 1","volume":"55","author":"Davies","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1145\/1859204.1859229","article-title":"Cosamp: Iterative signal recovery from incomplete and inaccurate samples","volume":"53","author":"Needell","year":"2010","journal-title":"Commun. ACM"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1002\/cpa.20303","article-title":"Iteratively reweighted least squares minimization for sparse recovery","volume":"63","author":"Daubechies","year":"2010","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Liu, Z., and Zhen, X. (2011). Variable-p iteratively weighted algorithm for image reconstruction. Artificial Intelligence and Computational Intelligence, Springer.","DOI":"10.1007\/978-3-642-23887-1_43"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1109\/TNNLS.2012.2197412","article-title":"L1\/2 regularization: A thresholding representation theory and a fast solver","volume":"23","author":"Xu","year":"2012","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1109\/78.738263","article-title":"Multipath time-delay detection and estimation","volume":"47","author":"Fuchs","year":"1999","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","unstructured":"Bertsekas, D.P. (1999). Nonlinear Programming, Athena Scientific. [2nd ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1341","DOI":"10.1109\/TIT.2004.828141","article-title":"More on sparse representations in arbitrary bases","volume":"50","author":"Fuchs","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2230","DOI":"10.1109\/TIT.2009.2016006","article-title":"Subspace pursuit for compressive sensing signal reconstruction","volume":"55","author":"Dai","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chartrand, R., and Yin, W. (2008). Iteratively Reweighted Algorithms for Compressive Sensing, CAAM Technical Reports [719]; Rice University.","DOI":"10.1109\/ICASSP.2008.4518498"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1109\/JSTSP.2010.2042412","article-title":"Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit","volume":"4","author":"Needell","year":"2010","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"6202","DOI":"10.1109\/TSP.2012.2218810","article-title":"Generalized orthogonal matching pursuit","volume":"60","author":"Wang","year":"2012","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/10\/3373\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:24:36Z","timestamp":1760196276000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/10\/3373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,9]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["s18103373"],"URL":"https:\/\/doi.org\/10.3390\/s18103373","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2018,10,9]]}}}