{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T07:08:39Z","timestamp":1775372919879,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2013,2,5]],"date-time":"2013-02-05T00:00:00Z","timestamp":1360022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Electrical capacitance tomography (ECT) attempts to reconstruct the permittivity distribution of the cross-section of measurement objects from the capacitance measurement data, in which reconstruction algorithms play a crucial role in real applications. Based on the robust principal component analysis (RPCA) method, a dynamic reconstruction model that utilizes the multiple measurement vectors is presented in this paper, in which the evolution process of a dynamic object is considered as a sequence of images with different temporal sparse deviations from a common background. An objective functional that simultaneously considers the temporal constraint and the spatial constraint is proposed, where the images are reconstructed by a batching pattern. An iteration scheme that integrates the advantages of the alternating direction iteration optimization (ADIO) method and the forward-backward splitting (FBS) technique is developed for solving the proposed objective functional. Numerical simulations are implemented to validate the feasibility of the proposed algorithm.<\/jats:p>","DOI":"10.3390\/s130202076","type":"journal-article","created":{"date-parts":[[2013,2,5]],"date-time":"2013-02-05T11:11:11Z","timestamp":1360062671000},"page":"2076-2092","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["An Image Reconstruction Algorithm for Electrical Capacitance Tomography Based on Robust Principle Component Analysis"],"prefix":"10.3390","volume":"13","author":[{"given":"Jing","family":"Lei","sequence":"first","affiliation":[{"name":"Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Beijing 102206, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shi","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, Ministry of Education, North China Electric Power University, Beijing 102206, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xueyao","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Engineering  Thermophysics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qibin","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Engineering  Thermophysics, Chinese Academy of Sciences, Beijing 100190, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,2,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.ces.2012.03.028","article-title":"Moisture distribution and hydrodynamics of wet granules during fluidized-bed drying characterized with volumetric electrical capacitance tomography","volume":"75","author":"Rimpilainen","year":"2012","journal-title":"Chem. Eng. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4090","DOI":"10.1016\/j.ces.2011.05.044","article-title":"Electrical capacitance tomography as a monitoring tool for high-shear mixing and granulation","volume":"66","author":"Rimpilainen","year":"2011","journal-title":"Chem. Eng. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.powtec.2009.04.019","article-title":"Measurement of fluidised bed dryer by different frequency and different normalisation methods with electrical capacitance tomography","volume":"199","author":"Wang","year":"2010","journal-title":"Powder Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.cej.2009.12.031","article-title":"Wavelet enhanced visualization of solids distribution in the top of a CFB","volume":"158","author":"Xiong","year":"2010","journal-title":"Chem. Eng. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.flowmeasinst.2009.12.008","article-title":"ECT measurement and CFD-DEM simulation of particle distribution in a down-flow fluidized bed","volume":"21","author":"Zhao","year":"2010","journal-title":"Flow Meas. Instrum."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1016\/j.ces.2009.06.040","article-title":"Dynamics of filtration in monolith reactors using electrical capacitance tomography","volume":"65","author":"Hamidipour","year":"2010","journal-title":"Chem. Eng. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.ces.2008.08.035","article-title":"Determination of bulk solid concentration changes during granular flow in a model silo with ECT sensors","volume":"64","author":"Niedostatkiewicz","year":"2009","journal-title":"Chem. Eng. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1088\/0957-0233\/19\/9\/094017","article-title":"Preliminary study on ECT imaging of flames in porous media","volume":"19","author":"Liu","year":"2008","journal-title":"Meas. Sci. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4304","DOI":"10.1016\/j.ces.2006.11.048","article-title":"Integration of ECT measurement with hydrodynamic modelling of conventional gas-solid bubbling bed","volume":"62","author":"Makkawi","year":"2007","journal-title":"Chem. Eng. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5384","DOI":"10.1021\/ie051401w","article-title":"Imaging the choking transition in gas-solid risers using electrical capacitance tomography","volume":"45","author":"Du","year":"2006","journal-title":"Ind. Eng. Chem. Res."},{"key":"ref_11","first-page":"89","article-title":"Electrical capacitance for flow imaging: system model for development of image reconstruction algorithms and design of primary sensors","volume":"139","author":"Xie","year":"1992","journal-title":"IEE Proc. G"},{"key":"ref_12","unstructured":"Tikhonov, A.N., and Arsenin, V.Y. (1977). Solution of Ill-posed Problems, V.H. Winston & Sons."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"615","DOI":"10.2307\/2372313","article-title":"An iteration formula for fredholm integral equations of the first kind","volume":"73","author":"Landweber","year":"1951","journal-title":"Am. J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1088\/0957-0233\/10\/11\/315","article-title":"An image reconstruction algorithm based on Landweber's iteration method for electrical capacitance tomography","volume":"10","author":"Yang","year":"1999","journal-title":"Meas. Sci. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.1088\/0957-0233\/17\/7\/032","article-title":"Modified iterative Landweber method in electrical capacitance tomography","volume":"17","author":"Jang","year":"2006","journal-title":"Meas. Sci. Technol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1049\/ip-smt:20040246","article-title":"Prior-online iteration for image reconstruction with electrical capacitance tomography","volume":"151","author":"Liu","year":"2004","journal-title":"IEE Proc. Sci. Meas. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"L1","DOI":"10.1088\/0957-0233\/14\/1\/201","article-title":"Image reconstruction algorithms for electrical capacitance tomography","volume":"14","author":"Yang","year":"2003","journal-title":"Meas. Sci. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/S1007-0214(05)70123-1","article-title":"Image reconstruction using a genetic algorithm for electrical capacitance tomography","volume":"10","author":"Mou","year":"2005","journal-title":"Tsinghua Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2104","DOI":"10.1088\/0957-0233\/17\/8\/008","article-title":"GVSPM image reconstruction for capacitance CT images of particles in a vertical pipe and comparison with the conventional method","volume":"17","author":"Takei","year":"2006","journal-title":"Meas. Sci. Technol."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1987","DOI":"10.1088\/0957-0233\/16\/10\/014","article-title":"Nonlinear image reconstruction for electrical capacitance tomography using experimental data","volume":"16","author":"Soleimani","year":"2005","journal-title":"Meas. Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1016\/j.flowmeasinst.2007.07.004","article-title":"An image reconstruction algorithm based on total variation with adaptive mesh refinement for ECT","volume":"18","author":"Wang","year":"2007","journal-title":"Flow Meas. Instrum."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2124","DOI":"10.1088\/0957-0233\/15\/10\/023","article-title":"A nonlinear image reconstruction algorithm for electrical capacitance tomography","volume":"15","author":"Fang","year":"2004","journal-title":"Meas. Sci. Technol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2843","DOI":"10.1016\/j.camwa.2011.03.060","article-title":"An image reconstruction algorithm based on the semiparametric model for electrical capacitance tomography","volume":"61","author":"Lei","year":"2011","journal-title":"Comp. Math. Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1382","DOI":"10.1088\/0957-0233\/15\/7\/022","article-title":"Reconstruction of permittivity images from capacitance tomography data by using very fast simulated annealing","volume":"15","author":"Martin","year":"2004","journal-title":"Meas. Sci. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2198","DOI":"10.1088\/0957-0233\/12\/12\/323","article-title":"Neural network based multi-criterion optimization image reconstruction technique for imaging two-and three-phase flow systems using electrical capacitance tomography","volume":"12","author":"Warsito","year":"2001","journal-title":"Meas. Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.ndteint.2009.12.001","article-title":"Shape based reconstruction of experimental data in 3D electrical capacitance tomography","volume":"43","author":"Banasiak","year":"2010","journal-title":"NDT & E Int."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1088\/0957-0233\/18\/1\/009","article-title":"Formulation of cost functionals for different measurement principles in nonlinear capacitance tomography","volume":"18","author":"Kortschak","year":"2007","journal-title":"Meas. Sci. Technol."},{"key":"ref_28","unstructured":"Lei, J. (2008). Research on Image Reconstruction Algorithms of Electrical Capacitance Tomography for the Multiphase Flow. [PhD thesis, Institute of Engineering Thermophysics, Chinese Academy of Sciences]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1088\/0957-0233\/18\/1\/004","article-title":"A particle filter approach for tomographic imaging based on different state-space representations","volume":"18","author":"Watzenig","year":"2007","journal-title":"Meas. Sci. Technol."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3287","DOI":"10.1088\/0957-0233\/18\/11\/004","article-title":"Dynamic imaging in electrical capacitance tomography and electromagnetic induction tomography using a Kalman filter","volume":"18","author":"Soleimani","year":"2007","journal-title":"Meas. Sci. Technol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"171","DOI":"10.2528\/PIER09010202","article-title":"Four-dimensional electrical capacitance tomography imaging using experimental data","volume":"90","author":"Soleimani","year":"2009","journal-title":"Prog. Electromagn. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1088\/0957-0233\/7\/3\/018","article-title":"Combustion imaging from electrical impedance measurements","volume":"7","author":"Waterfall","year":"1996","journal-title":"Meas. Sci. Technol."},{"key":"ref_33","first-page":"1554","article-title":"Evaluation of effect of number of electrodes in ECT sensors on image quality","volume":"12","author":"Peng","year":"2012","journal-title":"IEEE Sens. J."},{"key":"ref_34","unstructured":"Zhang, Z.L., Huang, Q.Y., Wen, H.Y., and Deng, Y.J. (2007). Deformation Monitoring Analysis and Prediction for Engineering Constructions, Surveying and Mapping Press."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/1970392.1970395","article-title":"Robust principle component analysis?","volume":"58","author":"Candes","year":"2011","journal-title":"J. ACM"},{"key":"ref_36","unstructured":"Wright, J., Peng, Y., Ma, Y., Ganesh, A., and Rao, S. (2009, January 7\u201310). Robust principal component analysis: exact recovery of corrupted low-rank matrices via convex optimization. Vancouver, BC, Canada."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3181","DOI":"10.1088\/0031-9155\/56\/11\/002","article-title":"Robust principle component analysis based four-dimensional computed tomography","volume":"56","author":"Gao","year":"2011","journal-title":"Phys. Med. Biol."},{"key":"ref_38","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. Imag. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1137\/070703983","article-title":"Bregman iterative algorithms for L1 minimization with applications to compressed sensing","volume":"1","author":"Yin","year":"2008","journal-title":"SIAM J. Imag. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1137\/090746379","article-title":"Bregmanized nonlocal regularization for deconvolution and sparse reconstruction","volume":"3","author":"Zhang","year":"2010","journal-title":"SIAM J. Imag. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A fast iteration shrinkage-thresholding algorithm for linear inverse problems","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imag. Sci."},{"key":"ref_42","first-page":"2899","article-title":"Efficient online and batch learning using forward backward splitting","volume":"10","author":"Duchi","year":"2009","journal-title":"J. Mach. Learn. Res."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1137\/050626090","article-title":"Signal recovery by proximal forward-backward splitting","volume":"4","author":"Combettes","year":"2005","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1064","DOI":"10.1109\/TMI.2010.2068306","article-title":"A fast compressed sensing approach to 3D MR image reconstruction","volume":"30","author":"Montefusco","year":"2011","journal-title":"IEEE Trans. Med. Imag."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1956","DOI":"10.1137\/080738970","article-title":"A singular value thresholding algorithm for matrix completion","volume":"20","author":"Cai","year":"2008","journal-title":"SIAM J. Optim."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/2\/2076\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:44:47Z","timestamp":1760219087000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/13\/2\/2076"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,2,5]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,2]]}},"alternative-id":["s130202076"],"URL":"https:\/\/doi.org\/10.3390\/s130202076","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,2,5]]}}}