{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:05Z","timestamp":1760060105622,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T00:00:00Z","timestamp":1754352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["61973232"],"award-info":[{"award-number":["61973232"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The regularization parameter plays an important role in regularization-based electrical tomography (ET) algorithms, but the existing methods generally cannot determine the parameter. Moreover, these methods are not real-time since a thorough search must be performed for the best parameter. To address the issue, a reproducing kernel-based interpolation approximation method is proposed to efficiently estimate the best regularization parameter from a group of representative samples. The optimization and generation of the new method have been verified by theoretical analysis and experimental demonstration. The theoretical evaluation is conducted in a Hilbert space with a known reproducing kernel, and its symmetry ensures the uniqueness of the interpolation. And experimental validation is carried out using both simulated and actual models, each with a range of distinct features. Results indicate that the new method can approximately find the best regularization parameter. Consequently, when using the regularization parameter, the new method can effectively improve both the spatial resolution and steadiness of ET imaging process.<\/jats:p>","DOI":"10.3390\/sym17081242","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T07:49:58Z","timestamp":1754380198000},"page":"1242","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two-Dimensional Reproducing Kernel-Based Interpolation Approximation for Best Regularization Parameter in Electrical Tomography Algorithm"],"prefix":"10.3390","volume":"17","author":[{"given":"Fanpeng","family":"Dong","sequence":"first","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3473-2704","authenticated-orcid":false,"given":"Shihong","family":"Yue","sequence":"additional","affiliation":[{"name":"School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,5]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"062003","DOI":"10.1088\/1361-6501\/add8ad","article-title":"Deep learning-based image reconstruction for electrical capacitance tomography","volume":"36","author":"Peng","year":"2025","journal-title":"Meas. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1117\/1.1377308","article-title":"Status of electrical tomography in industrial applications","volume":"10","author":"York","year":"2001","journal-title":"J. Electron. Imaging"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"20714","DOI":"10.1109\/JSEN.2021.3100265","article-title":"Measurement of Flow Velocity Using Electrical Resistance Tomography and Cross-Correlation Technique","volume":"21","author":"Tan","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107419","DOI":"10.1016\/j.neunet.2025.107419","article-title":"Deep prior embedding method for electrical impedance tomography","volume":"188","author":"Wang","year":"2025","journal-title":"Neural Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"13618","DOI":"10.1109\/JSEN.2025.3546972","article-title":"DELTA: Delving into high-quality reconstruction for electrical impedance tomography","volume":"25","author":"Wang","year":"2025","journal-title":"IEEE Sens. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3049","DOI":"10.1109\/JSEN.2019.2892179","article-title":"An Improved Tikhonov Regularization Method for Lung Cancer Monitoring Using Electrical Impedance Tomography","volume":"19","author":"Sun","year":"2019","journal-title":"IEEE Sens. J."},{"key":"ref_7","unstructured":"Vauhkonen, M. (1997). Electrical Impedance Tomography and Prior Information. [Ph.D. Thesis, Department of Physics, University of Kuopio]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"065016","DOI":"10.1088\/1361-6420\/accdfc","article-title":"A Bayesian interpretation of the L-curve","volume":"39","author":"Antoni","year":"2023","journal-title":"Inverse Probl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1093\/jrsssb\/qkae092","article-title":"Corrected generalized cross-validation for finite ensembles of penalized estimators","volume":"87","author":"Bellec","year":"2025","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1088\/0266-5611\/12\/4\/013","article-title":"Non-convergence of the l-curve regularization parameter selection method","volume":"12","author":"Vogel","year":"1996","journal-title":"Inverse Probl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1007\/s11075-021-01087-9","article-title":"Generalized cross validation for lp \u2212 lq minimization","volume":"88","author":"Buccini","year":"2021","journal-title":"Numer. Algorithms"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/BF01731984","article-title":"Limitations of the L-curve method in ill-posed problems","volume":"36","author":"Hanke","year":"1996","journal-title":"BIT Numer. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1023\/B:NUMA.0000021764.16526.47","article-title":"L-curve and curvature bounds for Tikhonov regularization","volume":"35","author":"Calvetti","year":"2004","journal-title":"Numer. Algorithms"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1088\/0266-5611\/19\/3\/311","article-title":"Direct analytic model of the L-curve for Tikhonov regularization parameter selection","volume":"19","year":"2003","journal-title":"Inverse Prob."},{"key":"ref_15","first-page":"116887","article-title":"Least squares B-spline approximation with applications to geospatial point clouds","volume":"221","author":"Esmaeili","year":"2025","journal-title":"Measurement"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1023\/A:1018981505752","article-title":"Polynomial interpolation in several variables","volume":"12","author":"Gasca","year":"2000","journal-title":"Adv. Comput. Math."},{"key":"ref_17","unstructured":"Berlinet, A., and Thomas-Agnan, C. (2011). Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer Science & Business Media."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1353","DOI":"10.1002\/num.20428","article-title":"A best approximation for the solution of one-dimensional variable-coefficient burgers\u2019 equation","volume":"25","author":"Li","year":"2009","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_19","unstructured":"Polydorides, N. (2002). Image Reconstruction Algorithm for Soft-Field Tomography. [Ph.D. Thesis, Department of Electrical Engineering and Electronics]."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"15477","DOI":"10.1109\/JSEN.2025.3547254","article-title":"Detection of magnetic samples by electromagnetic tomography with PID controlled iterative L1 regularization method","volume":"25","author":"Huang","year":"2025","journal-title":"IEEE Sens. J."},{"key":"ref_21","first-page":"0603005","article-title":"Choquet Integral-Based Fusion of Multiple Patterns for Improving EIT Spatial Resolution","volume":"29","author":"Li","year":"2019","journal-title":"IEEE Trans. Appl. Supercond."},{"key":"ref_22","first-page":"5506413","article-title":"Tuning-free plug-and-play hyperspectral image deconvolution with deep priors","volume":"61","author":"Wang","year":"2023","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1016\/j.cam.2009.05.016","article-title":"A new variant L-curve for Tikhonov regularization","volume":"231","author":"Rezghi","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Barbour, A.D., Holst, L., and Janson, S. (1992). Poisson Approximation, Clarendon Press. Oxford Studies in Probability.","DOI":"10.1093\/oso\/9780198522355.001.0001"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Chen, L.H.Y., Goldstein, L., and Shao, Q.-M. (2011). Normal Approximation by Stein\u2019s Method: Probability and Its Applications, Springer.","DOI":"10.1007\/978-3-642-15007-4"},{"key":"ref_26","first-page":"209","article-title":"On the best operator of interpolation in W21","volume":"8","author":"Cui","year":"1986","journal-title":"Math. Numer. Sin."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.acha.2014.03.007","article-title":"Solving support vector machines in reproducing kernel Banach spaces with positive definite functions","volume":"38","author":"Fasshauer","year":"2015","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_28","unstructured":"Borg, I., and Groenen, P. (2005). Modern Multidimensional Scaling: Theory and Applications, Springer. Springer Series in Statistics."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Paulsen, V.I., and Raghupathi, M. (2016). An Introduction to the Theory of Reproducing Kernel Hilbert Spaces, Cambridge University Press. Cambridge Studies in Advanced Mathematics.","DOI":"10.1017\/CBO9781316219232"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"112277","DOI":"10.1016\/j.disc.2020.112277","article-title":"Near MDS codes from oval polynomials","volume":"344","author":"Wang","year":"2021","journal-title":"Discrete Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1162\/neco.1995.7.2.219","article-title":"Regularization Theory and Neural Networks Architectures","volume":"7","author":"Girosi","year":"1995","journal-title":"Neural Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1002\/fld.1650200824","article-title":"Reproducing kernel particle methods","volume":"20","author":"Liu","year":"1995","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_33","first-page":"4506312","article-title":"Unsupervised Evaluation and Optimization for Electrical Impedance Tomography","volume":"70","author":"Wang","year":"2021","journal-title":"IEEE Trans. Instr. Meas."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/S0020-0255(02)00208-6","article-title":"An evolutionary technique based on k-means algorithm for optimal clustering in RN","volume":"146","author":"Bandyopadhyay","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2849","DOI":"10.1007\/s11432-012-4748-7","article-title":"Clustering mechanism for electric tomography imaging","volume":"55","author":"Yue","year":"2012","journal-title":"Sci. China Inf. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1242\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:23:16Z","timestamp":1760034196000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1242"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,5]]},"references-count":35,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081242"],"URL":"https:\/\/doi.org\/10.3390\/sym17081242","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,8,5]]}}}