Microwave Tomography Using Neural Networks for Its Application in an Industrial Microwave Drying System
Abstract
:1. Introduction
2. Problem Formulation
2.1. Forward Model
2.2. Parametric Model for Moisture Distribution
Algorithm 1 Pseudocode for generating the moisture distribution. Note that a small diagonal component is added in matrix C to ensure the positive definiteness. 

3. Inverse Problem: Convolutional Neural Network
3.1. Training, Validation, and Test Datasets
3.2. Reconstruction Results
3.2.1. Sample with Low, and Moderate Moisture Content
3.2.2. Sample with High Moisture Distribution
3.2.3. Error Statistics
4. Experimental Setup and Result
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Osepchuk, J. A History of Microwave Heating Applications. IEEE Trans. Microw. Theory Tech. 1984, 32, 1200–1224. [Google Scholar] [CrossRef]
 Metaxas, A.C.; Meredith, R.J. Industrial Microwave Heating; P. Peregrinus on behalf of the Institution of Electrical Engineers: London, UK, 1988. [Google Scholar]
 Roussy, G.; Bennani, A.; Thiebaut, J. Temperature runaway of microwave irradiated materials. J. Appl. Phys. 1987, 62, 1167–1170. [Google Scholar] [CrossRef]
 Bykov, Y.V.; Rybakov, K.I.; Semenov, V.E. Hightemperature microwave processing of materials. J. Phys. D Appl. Phys. 2001, 34, R55–R75. [Google Scholar] [CrossRef]
 Sun, Y. Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources. Ph.D. Thesis, KIT Scientific Publishing, Karlsruhe, Germany, 2016. [Google Scholar]
 Link, G.; Ramopoulos, V. Simple analytical approach for industrial microwave applicator design. Chem. Eng. Process.Process Intensif. 2018, 125, 334–342. [Google Scholar] [CrossRef]
 Omrani, A.; Link, G.; Jelonnek, J. A Multistatic Uniform Diffraction Tomographic Algorithm for RealTime Moisture Detection. In Proceedings of the 2020 IEEE AsiaPacific Microwave Conference (APMC), Hong Kong, China, 10–13 November 2020; pp. 437–439. [Google Scholar] [CrossRef]
 Hosseini, M.; Kaasinen, A.; Link, G.; Lähivaara, T.; Vauhkonen, M. LQR Control of Moisture Distribution in Microwave Drying Process Based on a Finite Element Model of Parabolic PDEs. IFACPapersOnLine 2020, 53, 11470–11476. [Google Scholar] [CrossRef]
 Wu, Z. Developing a microwave tomographic system for multiphase flow imaging: Advances and challenges. Trans. Inst. Meas. Control 2015, 37, 760–768. [Google Scholar] [CrossRef]
 Wu, Z.; Wang, H. Microwave Tomography for Industrial Process Imaging: Example Applications and Experimental Results. IEEE Antennas Propag. Mag. 2017, 59, 61–71. [Google Scholar] [CrossRef]
 Trabelsi, S.; Kraszewski, A.W.; Nelson, S.O. A microwave method for online determination of bulk density and moisture content of particulate materials. IEEE Trans. Instrum. Meas. 1998, 47, 127–132. [Google Scholar] [CrossRef] [Green Version]
 Okamura, S. Microwave Technology for Moisture Measurement. Subsurf. Sens. Technol. Appl. 2000, 1, 205–227. [Google Scholar] [CrossRef]
 Trabelsi, S.; Nelson, S.O. Nondestructive sensing of physical properties of granular materials by microwave permittivity measurement. IEEE Trans. Instrum. Meas. 2006, 55, 953–963. [Google Scholar] [CrossRef]
 Trabelsi, S.; Nelson, S.O.; Lewis, M.A. Microwave nondestructive sensing of moisture content in shelled peanuts independent of bulk density and with temperature compensation. Sens. Instrum. Food Qual. Saf. 2009, 3, 114–121. [Google Scholar] [CrossRef]
 Trabelsi, S. New Calibration Algorithms for DielectricBased Microwave Moisture Sensors. IEEE Sens. Lett. 2017, 1, 1–4. [Google Scholar] [CrossRef]
 You, K.Y.; Lee, C.Y.; Then, Y.L.; Chong, S.H.C.; You, L.L.; Abbas, Z.; Cheng, E.M. Precise Moisture Monitoring for Various Soil Types Using Handheld MicrowaveSensor Meter. IEEE Sens. J. 2013, 13, 2563–2570. [Google Scholar] [CrossRef]
 Javed, N.; Habib, A.; Amin, Y.; Loo, J.; Akram, A.; Tenhunen, H. Directly Printable Moisture Sensor Tag for Intelligent Packaging. IEEE Sens. J. 2016, 16, 6147–6148. [Google Scholar] [CrossRef]
 Franchois, A.; Pichot, C. Microwave imagingcomplex permittivity reconstruction with a LevenbergMarquardt method. IEEE Trans. Antennas Propag. 1997, 45, 203–215. [Google Scholar] [CrossRef]
 Zhong, Y.; Lambert, M.; Lesselier, D.; Chen, X. A New Integral Equation Method to Solve Highly Nonlinear Inverse Scattering Problems. IEEE Trans. Antennas Propag. 2016, 64, 1788–1799. [Google Scholar] [CrossRef]
 Arridge, S.; Maass, P.; Öktem, O.; Schönlieb, C.B. Solving inverse problems using datadriven models. Acta Numer. 2019, 28, 1–174. [Google Scholar] [CrossRef] [Green Version]
 Higham, C.F.; Higham, D.J. Deep Learning: An Introduction for Applied Mathematicians. SIAM Rev. 2019, 61, 860–891. [Google Scholar] [CrossRef] [Green Version]
 Elshafiey, I.; Udpa, L.; Udpa, S.S. Application of neural networks to inverse problems in electromagnetics. IEEE Trans. Magn. 1994, 30, 3629–3632. [Google Scholar] [CrossRef]
 Bartley, P.G.; Nelson, S.O.; McClendon, R.W.; Trabelsi, S. Determining moisture content of wheat with an artificial neural network from microwave transmission measurements. IEEE Trans. Instrum. Meas. 1998, 47, 123–126. [Google Scholar] [CrossRef]
 Shrestha, B.L.; Wood, H.C.; Tabil, L.; Baik, O.; Sokhansanj, S. Microwave permittivityassisted artificial neural networks for determining moisture content of chopped alfalfa forage. IEEE Instrum. Meas. Mag. 2017, 20, 37–42. [Google Scholar] [CrossRef]
 Zhou, Y.; Zhong, Y.; Wei, Z.; Yin, T.; Chen, X. An Improved Deep Learning Scheme for Solving 2D and 3D Inverse Scattering Problems. IEEE Trans. Antennas Propag. 2021, 69, 2853–2863. [Google Scholar] [CrossRef]
 Ma, Z.; Xu, K.; Song, R.; Wang, C.F.; Chen, X. LearningBased Fast Electromagnetic Scattering Solver Through Generative Adversarial Network. IEEE Trans. Antennas Propag. 2021, 69, 2194–2208. [Google Scholar] [CrossRef]
 Guo, R.; Shan, T.; Song, X.; Li, M.; Yang, F.; Xu, S.; Abubakar, A. Physics Embedded Deep Neural Network for Solving Volume Integral Equation: 2D Case. IEEE Trans. Antennas Propag. 2021. [Google Scholar] [CrossRef]
 Guo, L.; Song, G.; Wu, H. ComplexValued Pix2pix—Deep Neural Network for Nonlinear Electromagnetic Inverse Scattering. Electronics 2021, 10, 752. [Google Scholar] [CrossRef]
 Sanghvi, Y.; Kalepu, Y.N.G.B.; Khankhoje, U. Embedding Deep Learning in Inverse Scattering Problems. IEEE Trans. Comput. Imaging 2019, 6, 46–56. [Google Scholar] [CrossRef]
 Lin, Z.; Guo, R.; Li, M.; Abubakar, A.; Zhao, T.; Yang, F.; Xu, S. LowFrequency Data Prediction With Iterative Learning for Highly Nonlinear Inverse Scattering Problems. IEEE Trans. Microw. Theory Tech. 2021. [Google Scholar] [CrossRef]
 Li, L.; Wang, L.G.; Teixeira, F.L.; Liu, C.; Nehorai, A.; Cui, T.J. DeepNIS: Deep Neural Network for Nonlinear Electromagnetic Inverse Scattering. IEEE Trans. Antennas Propag. 2019, 67, 1819–1825. [Google Scholar] [CrossRef] [Green Version]
 Lähivaara, T.; Yadav, R.; Link, G.; Vauhkonen, M. Estimation of Moisture Content Distribution in Porous Foam Using Microwave Tomography With Neural Networks. IEEE Trans. Comput. Imaging 2020, 6, 1351–1361. [Google Scholar] [CrossRef]
 Yadav, R.; Vauhkonen, M.; Link, G.; Betz, S.; Lähivaara, T. Microwave tomography for estimating moisture content distribution in porous foam using neural networks. In Proceedings of the 2020 14th European Conference on Antennas and Propagation (EuCAP), Denmark, Sweden, 15–20 March 2020; pp. 1–5. [Google Scholar] [CrossRef]
 Yadav, R.; Omrani, A.; Vauhkonen, M.; Link, G.; Lähivaara, T. Microwave Tomography for Moisture Level Estimation Using Bayesian Framework. In Proceedings of the 2021 15th European Conference on Antennas and Propagation (EuCAP), Düsseldorf, Germany, 22–26 March 2021; pp. 1–5. [Google Scholar] [CrossRef]
 Omrani, A.; Yadav, R.; Link, G.; Vauhkonen, M.; Lähivaara, T.; Jelonnek, J. A Combined Microwave Imaging Algorithm for Localization and Moisture Level Estimation in Multilayered Media. In Proceedings of the 2021 15th European Conference on Antennas and Propagation (EuCAP), Düsseldorf, Germany, 22–26 March 2021; pp. 1–5. [Google Scholar] [CrossRef]
 Balanis, C. Advanced Engineering Electromagnetics, 2nd ed.; Wiley: Hoboken, NJ, USA, 2012. [Google Scholar]
 Chew, W.C. Waves and Fields in Inhomogenous Media; IEEE Press: New York, NY, USA, 1995. [Google Scholar]
 Caorsi, S.; Gragnani, G.L.; Pastorino, M. Twodimensional microwave imaging by a numerical inverse scattering solution. IEEE Trans. Microw. Theory Tech. 1990, 38, 980–981. [Google Scholar] [CrossRef]
 Chew, W.; Wang, Y. Reconstruction of twodimensional permittivity distribution using the distorted Born iterative method. IEEE Trans. Med. Imaging 1990, 9, 218–225. [Google Scholar] [CrossRef]
 Sadeghi, S.; MohammadpourAghdam, K.; FarajiDana, R.; Burkholder, R.J. A DORTUniform Diffraction Tomography Algorithm for ThroughtheWall Imaging. IEEE Trans. Antennas Propag. 2020, 68, 3176–3183. [Google Scholar] [CrossRef]
 Janalizadeh, R.C.; Zakeri, B. A SourceType Best Approximation Method for Imaging Applications. IEEE Antennas Wirel. Propag. Lett. 2016, 15, 1707–1710. [Google Scholar] [CrossRef]
 Tai, C.T. Dyadic Green Functions in Electromagnetic Theory; IEEE: New York, NY, USA, 1994. [Google Scholar]
 Lindell, I.V. Methods for Electromagnetic Field Analysis; WileyIEEE Press: Hoboken, NJ, USA, 1992. [Google Scholar]
 Monk, P. Finite Element Methods for Maxwell’s Equations; Oxford University Press: Oxford, UK, 2003. [Google Scholar]
 Harrington, R. Timeharmonic Electromagnetic Fields; Electronics Series; McGrawHill: New York, NY, USA, 1961. [Google Scholar]
 Richmond, J. Scattering by a dielectric cylinder of arbitrary cross section shape. IEEE Trans. Antennas Propag. 1965, 13, 334–341. [Google Scholar] [CrossRef]
 Saad, Y.; Schultz, M.H. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comput. 1986, 7, 856–869. [Google Scholar] [CrossRef] [Green Version]
 Soldatov, S.; Kayser, T.; Link, G.; Seitz, T.; Layer, S.; Jelonnek, J. Microwave cavity perturbation technique for hightemperature dielectric measurements. In Proceedings of the 2013 IEEE MTTS International Microwave Symposium Digest (MTT), Seattle, WA, USA, 2–7 June 2013; pp. 1–4. [Google Scholar] [CrossRef]
 Rasmussen, C.; Williams, C. Gaussian Processes for Machine Learning; The MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
 Kingma, D.P.; Ba, J. Adam: A Method for Stochastic Optimization. arXiv 2014, arXiv:1412.6980. [Google Scholar]
 Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Corrado, G.S.; Davis, A.; Dean, J.; Devin, M.; et al. TensorFlow: LargeScale Machine Learning on Heterogeneous Systems. arXiv 2015, arXiv:1603.04467. [Google Scholar]
 Bucci, O.M.; Cardace, N.; Crocco, L.; Isernia, T. Degree of nonlinearity and a new solution procedure in scalar twodimensional inverse scattering problems. J. Opt. Soc. Am. A 2001, 18, 1832–1843. [Google Scholar] [CrossRef]
 Bevacqua, M.T.; Isernia, T. An Effective Rewriting of the Inverse Scattering Equations via Green’s Function Decomposition. IEEE Trans. Antennas Propag. 2021, 69, 4883–4893. [Google Scholar] [CrossRef]
 Abubakar, A.; van den Berg, P.; Semenov, S. A robust iterative method for Born inversion. IEEE Trans. Geosci. Remote Sens. 2004, 42, 342–354. [Google Scholar] [CrossRef]
 Kaipio, J.; Somersalo, E. Statistical inverse problems: Discretization, model reduction and inverse crimes. J. Comput. Appl. Math. 2007, 198, 493–504. [Google Scholar] [CrossRef] [Green Version]
 Ostadrahimi, M.; Mojabi, P.; Gilmore, C.; Zakaria, A.; Noghanian, S.; Pistorius, S.; LoVetri, J. Analysis of Incident Field Modeling and Incident/Scattered Field Calibration Techniques in Microwave Tomography. IEEE Antennas Wirel. Propag. Lett. 2011, 10, 900–903. [Google Scholar] [CrossRef]
 Gawlikowski, J.; Tassi, C.R.N.; Ali, M.; Lee, J.; Humt, M.; Feng, J.; Kruspe, A.; Triebel, R.; Jung, P.; Roscher, R.; et al. A Survey of Uncertainty in Deep Neural Networks. 2021. Available online: http://xxx.lanl.gov/abs/2107.03342 (accessed on 14 October 2021).
 Abdar, M.; Pourpanah, F.; Hussain, S.; Rezazadegan, D.; Liu, L.; Ghavamzadeh, M.; Fieguth, P.; Cao, X.; Khosravi, A.; Acharya, U.R.; et al. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Inf. Fusion 2021, 76, 243–297. [Google Scholar] [CrossRef]
${\overline{\mathit{a}}}_{\mathit{\theta}}$  ${\mathit{\delta}}_{{\mathit{a}}_{\mathit{\theta}}}$  ${\overline{\mathit{b}}}_{\mathit{\theta}}$  ${\mathit{\delta}}_{{\mathit{b}}_{\mathit{\theta}}}$  

${\u03f5}_{r}^{\prime}$  1.085  0.01591  0.01256  0.00062 
${\u03f5}_{r}^{\u2033}$  0.03021  0.0025  0.02249  0.0009 
Low Moisture  Moderate Moisture  

$\kappa $  0.9558  0.9361 
High Variation  Homogeneous  

$\kappa $  0.923  0.883 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Yadav, R.; Omrani, A.; Link, G.; Vauhkonen, M.; Lähivaara, T. Microwave Tomography Using Neural Networks for Its Application in an Industrial Microwave Drying System. Sensors 2021, 21, 6919. https://doi.org/10.3390/s21206919
Yadav R, Omrani A, Link G, Vauhkonen M, Lähivaara T. Microwave Tomography Using Neural Networks for Its Application in an Industrial Microwave Drying System. Sensors. 2021; 21(20):6919. https://doi.org/10.3390/s21206919
Chicago/Turabian StyleYadav, Rahul, Adel Omrani, Guido Link, Marko Vauhkonen, and Timo Lähivaara. 2021. "Microwave Tomography Using Neural Networks for Its Application in an Industrial Microwave Drying System" Sensors 21, no. 20: 6919. https://doi.org/10.3390/s21206919