{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:26:10Z","timestamp":1760711170663,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,30]],"date-time":"2023-04-30T00:00:00Z","timestamp":1682812800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The inverse conductivity problem in electrical impedance tomography involves the solving of a nonlinear and under-determined system of equations. This paper presents a new approach, which leads to a quadratic and overdetermined system of equations. The aim of the paper is to establish new research directions in handling of the inverse conductivity problem. The basis of the proposed method is that the material, which can be considered as an isotropic continuum, is modeled as a linear network with concentrated parameters. The weights of the obtained graph represent the properties of the discretized continuum. Further, the application of the developed procedure allows for the dielectric constant to be used in the multi-frequency approach, as a result of which the optimized system of equations always remains overdetermined. Through case studies, the efficacy of the reconstruction method by changing the mesh resolution applied for discretizing is presented and evaluated. The presented results show, that, due to the application of discrete, symmetric mathematical structures, the new approach even at coarse mesh resolution is capable of localizing the inhomogeneities of the material.<\/jats:p>","DOI":"10.3390\/sym15051008","type":"journal-article","created":{"date-parts":[[2023,5,1]],"date-time":"2023-05-01T12:02:31Z","timestamp":1682942551000},"page":"1008","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes"],"prefix":"10.3390","volume":"15","author":[{"given":"Zoltan","family":"Sari","sequence":"first","affiliation":[{"name":"Department of Technical Informatics, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag Str. 20, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mihaly","family":"Klincsik","sequence":"additional","affiliation":[{"name":"Department of Technical Informatics, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag Str. 20, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Odry","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Institute of Information Technology, University of Dunaujvaros, Tancsics M. Str. 1\/A, H-2401 Dunaujvaros, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8796-6822","authenticated-orcid":false,"given":"Vladimir","family":"Tadic","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Institute of Information Technology, University of Dunaujvaros, Tancsics M. Str. 1\/A, H-2401 Dunaujvaros, Hungary"},{"name":"John von Neumann Faculty of Informatics, University of Obuda, Becsi Str. 96\/B, H-1034 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Attila","family":"Toth","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag Str. 20, H-7624 Pecs, Hungary"},{"name":"Institute of Physiology, Medical School, University of Pecs, Szigeti Str. 12, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0313-9959","authenticated-orcid":false,"given":"Zoltan","family":"Vizvari","sequence":"additional","affiliation":[{"name":"Symbolic Methods in Material Analysis and Tomography Research Group, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 6, H-7624 Pecs, Hungary"},{"name":"Cellular Bioimpedance Research Group, Szentagothai Research Centre, University of Pecs, Ifjusag Str. 20, H-7624 Pecs, Hungary"},{"name":"Department of Environmental Engineering, Faculty of Engineering and Information Technology, University of Pecs, Boszorkany Str. 2, H-7624 Pecs, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Holder, D. (2005). Electrical Impedance Tomography: Methods, History, and Applications, Institute of Physics Pub.","DOI":"10.1201\/9781420034462.ch4"},{"key":"ref_2","unstructured":"Amalia, I. (2003). Continuous and Discrete Simulation in Electrodynamics, Akademiai Kiado. (In Hungarian)."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cardoso, J. (2017). Electromagnetics through the Finite Element Method: A Simplified Approach Using Maxwell\u2019s Equations, Taylor & Francis Group.","DOI":"10.1201\/9781315366777"},{"key":"ref_4","unstructured":"Jin, J.M. (2014). The Finite Element Method in Electromagnetics, John Wiley and Sons. [3rd ed.]."},{"key":"ref_5","first-page":"2","article-title":"On an inverse boundary value problem","volume":"25","year":"2006","journal-title":"Comput. Appl. Math. Braz. Soc. Comput. Appl. Math. (SBMAC)"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"201","DOI":"10.12700\/APH.18.7.2021.7.11","article-title":"Lumped Element Method\u2014A Discrete Calculus Approach for Solving Elliptic and Parabolic PDEs","volume":"18","author":"Vizvari","year":"2021","journal-title":"Acta Polytech. Hung."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Uhlmann, G. (2023, April 27). 30 Years of Calder\u00f3n\u2019s Problem, Seminaire Laurent Schwartz\u2014EDP et Applications, Cellule MathDoc\/CEDRAM, 2012, S\u00e9minaire Laurent Schwartz\u2014EDP et Applications, 1\u201325. Available online: https:\/\/slsedp.centre-mersenne.org\/item\/SLSEDP_2012-2013____A13_0\/.","DOI":"10.5802\/slsedp.40"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"123011","DOI":"10.1088\/0266-5611\/25\/12\/123011","article-title":"Electrical Impedance Tomography and Calder\u00f3n\u2019s Problem","volume":"25","author":"Uhlmann","year":"2009","journal-title":"Inverse Probl."},{"key":"ref_9","first-page":"241","article-title":"Reconstruction of high contrast 2-D conductivities by the algorithm of A. Nachman","volume":"278","author":"Siltanen","year":"2001","journal-title":"Am. Math. Soc."},{"key":"ref_10","first-page":"555","article-title":"A direct reconstruction algorithm for electrical impedance tomography","volume":"21","author":"Mueller","year":"2002","journal-title":"IEEE Trans. Med Imaging Inst. Electr. Electron. Eng."},{"key":"ref_11","first-page":"012075","article-title":"Studies of an Adaptive Kaczmarz Method for Electrical Impedance Imaging","volume":"434","author":"Li","year":"2013","journal-title":"J. Physics: Conf. Ser."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1088\/0967-3334\/34\/6\/595","article-title":"Adaptive Kaczmarz Method for Image Reconstruction in Electrical Impedance Tomography","volume":"34","author":"Li","year":"2013","journal-title":"Physiol. Meas."},{"key":"ref_13","unstructured":"Chen, Z.Q. (1990). Reconstruction Algorithms for Electrical Impedance Tomography. [Ph.D. Thesis, Department of Electrical and Computer Engineering, University of Wollongong]. Available online: http:\/\/ro.uow.edu.au\/theses\/1348."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1663","DOI":"10.1088\/0967-3334\/32\/10\/013","article-title":"Validation of Weighted Frequency-Difference EIT Using a Three-Dimensional Hemisphere Model and Phantom","volume":"32","author":"Ahn","year":"2011","journal-title":"Physiol. Meas."},{"key":"ref_15","first-page":"795","article-title":"Efficient Simultaneous Reconstruction of Time-Varying Images and Electrode Contact Impedances in Electrical Impedance Tomography","volume":"64","author":"Boverman","year":"2017","journal-title":"IEEE Trans. Biomed. Eng. Inst. Electr. Electron. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1088\/0967-3334\/21\/1\/308","article-title":"Sensitivity matrix and reconstruction algorithm for EIT assuming axial uniformity","volume":"21","author":"Jerbi","year":"2000","journal-title":"Physiol. Meas."},{"key":"ref_17","unstructured":"Lionheart, W.R. (2001, January 29\u201331). Reconstruction Algorithms for Permittivity and Conductivity Imaging. Proceedings of the 2nd World Congress on Industrial Process Tomography, Hannover, Germany."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"S29","DOI":"10.1088\/0967-3334\/28\/7\/S03","article-title":"Electrode placement configurations for 3D EIT","volume":"28","author":"Graham","year":"2007","journal-title":"Physiol. Meas."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"012049","DOI":"10.1088\/1742-6596\/224\/1\/012049","article-title":"Shape corrections for 3D EIT","volume":"224","author":"Paridis","year":"2010","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_20","unstructured":"Polydorides, N. (2002). Image Reconstruction Algorithms for Soft-Field Tomography. [Ph.D. Thesis, University of Manchester Institute of Science and Technology]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5595","DOI":"10.7314\/APJCP.2015.16.14.5595","article-title":"Electrical Impedance Tomography as a Primary Screening Technique for Breast Cancer Detection, Asian Pacific Journal of Cancer Prevention","volume":"16","author":"Latiff","year":"2015","journal-title":"Asian Pac. Organ. Cancer Prev."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Karpov, A., Korotkova, M., Shiferson, G., and Kotomina, E. (2020). Electrical Impedance Mammography: Screening and Basic Principles, Breast Cancer and Breast Reconstruction, IntechOpen. Available online: https:\/\/www.intechopen.com\/chapters\/69046.","DOI":"10.5772\/intechopen.89140"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Spatenkova, V., Teschner, E., and Jedlicka, J. (2020). Evaluation of regional ventilation by electric impedance tomography during percutaneous dilatational tracheostomy in neurocritical care: A pilot study. BMC Neurol., 20.","DOI":"10.1186\/s12883-020-01948-1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"044002","DOI":"10.1088\/1361-6579\/ab7edb","article-title":"The influence of an electrical impedance tomography belt on lung function determined by spirometry in sitting position","volume":"41","author":"Zhang","year":"2020","journal-title":"Physiol. Meas."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Luo, Y., Abiri, P., Chang, C.C., Tai, Y.C., and Hsiai, T.K. (2020). Epidermal EIT Electrode Arrays for Cardiopulmonary Application and Fatty Liver Infiltration. Interfacing Bioelectron. Biomed. Sens., 163\u2013184.","DOI":"10.1007\/978-3-030-34467-2_7"},{"key":"ref_26","first-page":"1341","article-title":"Electrical impedance tomography for decay diagnostics of Norway spruce (Picea abies): Possibilities and opportunities","volume":"50","author":"Humplik","year":"2016","journal-title":"Silva Fenn. Finn. Soc. For. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1137","DOI":"10.1007\/s00107-020-01591-0","article-title":"A tomographic approach to assessing the possibility of ring shake presence in standing chestnut trees","volume":"78","author":"Proto","year":"2020","journal-title":"Eur. J. Wood Wood Prod."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"790","DOI":"10.4236\/eng.2020.1210056","article-title":"Application of FPGA in Process Tomography Systems","volume":"12","author":"Hong","year":"2020","journal-title":"Engineering"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1002\/cite.201900172","article-title":"Controlled Inline Fluid Separation Based on Smart Process Tomography Sensors","volume":"92","author":"Sahovic","year":"2020","journal-title":"Chem. Ing. Tech."},{"key":"ref_30","unstructured":"Boyle, A. (2016). Geophysical Applications of Electrical Impedance Tomography. [Ph.D. Thesis, Carleton University]."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1088\/0266-5611\/10\/3\/010","article-title":"Impedance tomography: Imaging algorithms for geophysical applications","volume":"10","author":"Molyneux","year":"1994","journal-title":"Inverse Probl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"045012","DOI":"10.1088\/0266-5611\/26\/4\/045012","article-title":"A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem","volume":"26","author":"Beilina","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"125009","DOI":"10.1088\/0266-5611\/26\/12\/125009","article-title":"Reconstruction of dielectrics from experimental data via a hybrid globally convergent\/adaptive inverse algorithm","volume":"26","author":"Beilina","year":"2010","journal-title":"Inverse Probl."},{"key":"ref_34","unstructured":"Beilina, L., and Klibanov, M.V. (2010). Springer."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mueller, J.L., and Siltanen, S. (2012). Linear and Nonlinear Inverse Problems with Practical Applications, SIAM.","DOI":"10.1137\/1.9781611972344"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Isakov, V. (2017). Inverse Problems for Partial Differential Equations, Springer International Publishing.","DOI":"10.1007\/978-3-319-51658-5"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1186\/1687-2770-2013-272","article-title":"Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation","volume":"2013","author":"Koyunbakan","year":"2013","journal-title":"Bound. Value Probl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s12346-021-00548-9","article-title":"Reconstruction of Potential in Discrete Sturm\u2013Liouville Problem","volume":"21","author":"Koyunbakan","year":"2022","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Khaled, D.E., Novas, N., Gazquez, J.A., and Manzano-Agugliaro, F. (2018). Dielectric and Bioimpedance Research Studies: A Scientometric Approach Using the Scopus Database. Publications, 6.","DOI":"10.3390\/publications6010006"},{"key":"ref_40","first-page":"9210258","article-title":"Fundamentals, Recent Advances, and Future Challenges in Bioimpedance Devices for Healthcare Applications","volume":"2019","author":"Min","year":"2019","journal-title":"J. Sens."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Liu, D., Wang, J., Shan, Q., Smyl, D., Deng, J., and Du, J. (2023). DeepEIT: Deep Image Prior Enabled Electrical Impedance Tomography. IEEE Trans. Pattern Anal. Mach. Intell., 1\u201312.","DOI":"10.1109\/TPAMI.2023.3240565"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Wang, W., Yousaf, M., Liu, D., and Sohail, A. (2022). A Comparative Study of the Genetic Deep Learning Image Segmentation Algorithms. Symmetry, 14.","DOI":"10.3390\/sym14101977"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/s10915-021-01716-4","article-title":"Learning Nonlinear Electrical Impedance Tomography","volume":"90","author":"Colibazzi","year":"2022","journal-title":"J. Sci. Comput."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Wang, G., Feng, D., and Tang, W. (2022). Electrical Impedance Tomography Based on Grey Wolf Optimized Radial Basis Function Neural Network. Micromachines, 13.","DOI":"10.3390\/mi13071120"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Bibi, K. (2020). Particular Solutions of Ordinary Differential Equations Using Discrete Symmetry Groups. Symmetry, 12.","DOI":"10.3390\/sym12010180"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"065004","DOI":"10.1088\/1361-6420\/ac5f3a","article-title":"Mumford\u2013Shah regularization in electrical impedance tomography with complete electrode model","volume":"38","author":"Jauhiainen","year":"2022","journal-title":"Inverse Probl."},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Chen, Z., Xiang, J., PBagnaninchi, P.-O., and Yang, Y. (2022). MMV-Net: A Multiple Measurement Vector Network for Multifrequency Electrical Impedance Tomography. IEEE Trans. Neural Netw. Learn. Syst., 1\u201312.","DOI":"10.1109\/TNNLS.2022.3154108"},{"key":"ref_48","first-page":"2258","article-title":"Robust imaging using electrical impedance tomography: Review of current tools","volume":"478","author":"Benoit","year":"2022","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.jcp.2006.12.022","article-title":"Discrete calculus methods for diffusion","volume":"224","author":"Perot","year":"2007","journal-title":"J. Comput. Phys."},{"key":"ref_50","unstructured":"Leo, J., and Grady, J.P. (2010). Discrete Calculus, Springer."},{"key":"ref_51","unstructured":"Subramanian, V. (2007). Discrete Calculus Methods and Their Implementation. [Ph.D. Thesis, Mechanical Engineering, University of Massachusetts]."},{"key":"ref_52","first-page":"329","article-title":"A Simple Mesh Generator in MATLAB","volume":"46","author":"Persson","year":"2004","journal-title":"Siam Rev. Soc. Ind. Appl. Math. (SIAM)"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Alhevaz, A., Baghipur, M., Ganie, H.A., and Shang, Y. (2019). Bounds for the Generalized Distance Eigenvalues of a Graph. Symmetry, 11.","DOI":"10.3390\/sym11121529"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s00211-020-01170-8","article-title":"Mimicking relative continuum measurements by electrode data in two-dimensional electrical impedance tomography","volume":"147","author":"Garde","year":"2021","journal-title":"Numer. Math."},{"key":"ref_55","unstructured":"Aster, R., Borchers, B., and Thurber, C.H. (2005). Parameter Estimation and Inverse Problems, Elsevier Academic Press."},{"key":"ref_56","first-page":"47","article-title":"An efficient quadratic programming optimization method for deconvolution of gamma-ray spectra","volume":"9","author":"Hanka","year":"2010","journal-title":"AARMS Technol."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s11075-007-9136-9","article-title":"Regularization Tools Version 4.0 for Matlab 7.3","volume":"46","author":"Hansen","year":"2007","journal-title":"Numer. Algorithms"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Gramacki, A. (2018). Nonparametric Kernel Density Estimation and Its Computational Aspects, Springer.","DOI":"10.1007\/978-3-319-71688-6"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1008\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:27:06Z","timestamp":1760124426000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/5\/1008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,30]]},"references-count":58,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["sym15051008"],"URL":"https:\/\/doi.org\/10.3390\/sym15051008","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,4,30]]}}}