Two-Step Contrast Source Learning Method for Electromagnetic Inverse Scattering Problems
Abstract
:1. Introduction
- (1)
- The proposed method enables CNNs to manage the entire imaging process without iterative procedures, thereby achieving near-real-time imaging.
- (2)
- Despite the inherent non-linearity between the scattered field and object permittivity, the introduction of CSs as intermediate variables in inversion techniques effectively mitigates this issue in EM-ISPs.
- (3)
- In the initial phase, integrating physical principles into the CS-Net training enhances noise resilience, incorporates prior physical knowledge, and expands the applicability of the learning algorithms.
- (4)
- In the first step, the initial imaging breakthrough allows for only rough imaging of weak scatterers while providing initial imaging of target scatterers with high contrast.
2. Problem Formulation
3. Theory and Methodology
3.1. Theoretical Background
3.2. Initial Guess (Step 1)
3.3. Fine Imaging (Step 2)
3.4. Image Evaluation
3.5. Computational Complexity
4. Numerical Example
4.1. Configuration of the Scattering System
4.2. Test Using MNIST Database with SNR = 25 dB
4.3. Test Using MNIST Database with SNR = 15 dB
4.4. Test Using Austria Profile with SNR = 25 dB
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
BA | Born approximation |
BIM | Born iteration method |
BN | batch normalization |
BP | back-propagation |
CG | conjugate gradient |
CNN | convolutional neural network |
CSI | contrast source inversion method |
CS | contrast source |
CS-Net | contrast source net |
DCS | dominant current scheme |
DIS | direct inversion scheme |
DNNs | deep neural networks |
DoI | domain of interest |
DT | diffraction tomography |
EM-ISPs | electromagnetic inverse scattering problems |
ENL | equivalent number of looks |
FFT | fast Fourier transform |
MNIST | the Modified National Institute of Standards and Technology |
MOM | method of moments |
MSE | mean square error |
PSNR | peak signal-to-noise ratio |
PSO | particle swarm optimization |
RA | Rytov approximation |
ReLU | rectified linear unit |
SNR | signal-to-noise ratio |
SOM | subspace-based optimization method |
SSIM | structural similarity index measurement |
SVD | singular value decomposition |
TM | transverse magnetic |
References
- Tang, F.; Ji, Y.; Zhang, Y.; Dong, Z.; Wang, Z.; Zhang, Q.; Zhao, B.; Gao, H. Drifting ionospheric scintillation simulation for L-band geosynchronous SAR. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens. 2023. [Google Scholar] [CrossRef]
- Kagiwada, H.; Kalaba, R.; Timko, S.; Ueno, S. Associate memories for system identification: Inverse problems in remote sensing. Math. Comput. Model. 1990, 14, 200–202. [Google Scholar] [CrossRef]
- Randazzo, A.; Ponti, C.; Fedeli, A.; Estatico, C.; D’Atanasio, P.; Pastorino, M.; Schettini, G. A two-step inverse-scattering technique in variable-exponent Lebesgue spaces for through-the-wall microwave imaging: Experimental results. IEEE Trans. Geosci. Remote Sens. 2021, 59, 7189–7200. [Google Scholar] [CrossRef]
- Nie, Z.; Yang, F.; Zhao, Y.; Zhang, Y. Variational Born iteration method and its applications to hybrid inversion. IEEE Trans. Geosci. Remote Sens. 2000, 38, 1709–1715. [Google Scholar]
- Van den Berg, P.M.; Abubakar, A. Contrast source inversion method: State of art. Prog. Electromagn. Res. 2001, 34, 189–218. [Google Scholar] [CrossRef]
- Sun, S.; Kooij, B.J.; Jin, T.; Yarovoy, A.G. Cross-correlated contrast source inversion. IEEE Trans. Antennas Propag. 2017, 65, 2592–2603. [Google Scholar] [CrossRef]
- Sun, S.; Dai, D.; Wang, X. A fast algorithm of cross-correlated contrast source inversion in homogeneous background media. IEEE Trans. Antennas Propag. 2023, 71, 4380–4393. [Google Scholar] [CrossRef]
- Chen, X. Subspace-based optimization method for solving inverse-scattering problems. IEEE Trans. Geosci. Remote Sens. 2009, 48, 42–49. [Google Scholar] [CrossRef]
- Zhong, Y.; Chen, X. Twofold subspace-based optimization method for solving inverse scattering problems. Inverse Probl. 2009, 25, 085003. [Google Scholar] [CrossRef]
- Pastorino, M.; Massa, A.; Caorsi, S. A microwave inverse scattering technique for image reconstruction based on a genetic algorithm. IEEE Trans. Instrum. Meas. 2000, 49, 573–578. [Google Scholar] [CrossRef]
- Yang, C.X.; Zhang, J.; Tong, M.S. A hybrid quantum-behaved particle swarm optimization algorithm for solving inverse scattering problems. IEEE Trans. Antennas Propag. 2021, 69, 5861–5869. [Google Scholar] [CrossRef]
- Chen, X. Computational Methods for Electromagnetic Inverse Scattering; John Wiley & Sons: Hoboken, NJ, USA, 2018. [Google Scholar]
- Wang, M.; Sun, S.; Dai, D.; Zhang, Y.; Su, Y. Coherence Factor-Based Polarimetric Diffraction Tomography for 3-D Inverse Scattering with a Sparse Planar Array. IEEE Trans. Geosci. Remote Sens. 2024, 62, 2002314. [Google Scholar] [CrossRef]
- Zhang, L.; Xu, K.; Song, R.; Ye, X.; Wang, G.; Chen, X. Learning-based quantitative microwave imaging with a hybrid input scheme. IEEE Sens. J. 2020, 20, 15007–15013. [Google Scholar] [CrossRef]
- Salucci, M.; Arrebola, M.; Shan, T.; Li, M. Artificial intelligence: New frontiers in real-time inverse scattering and electromagnetic imaging. IEEE Trans. Antennas Propag. 2022, 70, 6349–6364. [Google Scholar] [CrossRef]
- Wang, Y.; Zong, Z.; He, S.; Wei, Z. Multiple-space deep learning schemes for inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 2023, 61, 2000511. [Google Scholar] [CrossRef]
- Chiu, C.C.; Lee, Y.H.; Chen, P.H.; Shih, Y.C.; Hao, J. Application of Self-Attention Generative Adversarial Network for Electromagnetic Imaging in Half-Space. Sensors 2024, 24, 2322. [Google Scholar] [CrossRef]
- Wu, Z.; Zhao, F.; Zhang, M.; Huan, S.; Pan, X.; Chen, W.; Yang, L. Fast Near-Field Frequency-Diverse Computational Imaging Based on End-to-End Deep-Learning Network. Sensors 2022, 22, 9771. [Google Scholar] [CrossRef]
- Guo, M.F.; Zeng, X.D.; Chen, D.Y.; Yang, N.C. Deep-learning-based earth fault detection using continuous wavelet transform and convolutional neural network in resonant grounding distribution systems. IEEE Sens. J. 2017, 18, 1291–1300. [Google Scholar] [CrossRef]
- Khoshdel, V.; Ashraf, A.; LoVetri, J. Enhancement of multimodal microwave-ultrasound breast imaging using a deep-learning technique. Sensors 2019, 19, 4050. [Google Scholar] [CrossRef]
- Krizhevsky, A.; Sutskever, I.; Hinton, G.E. Imagenet classification with deep convolutional neural networks. Adv. Neural Inf. Process. Syst. 2012, 25. [Google Scholar] [CrossRef]
- Zhang, T.; Wang, Z.; Cheng, P.; Xu, G.; Sun, X. DCNNet: A distributed convolutional neural network for remote sensing image classification. IEEE Trans. Geosci. Remote Sens. 2023, 61, 5603618. [Google Scholar] [CrossRef]
- Guo, R.; Huang, T.; Li, M.; Zhang, H.; Eldar, Y.C. Physics-embedded machine learning for electromagnetic data imaging: Examining three types of data-driven imaging methods. IEEE Signal Process. Mag. 2023, 40, 18–31. [Google Scholar] [CrossRef]
- Xu, K.; Qian, Z.; Zhong, Y.; Su, J.; Gao, H.; Li, W. Learning-assisted inversion for solving nonlinear inverse scattering problem. IEEE Trans. Microw. Theory Tech. 2022, 71, 2384–2395. [Google Scholar] [CrossRef]
- Massa, A.; Marcantonio, D.; Chen, X.; Li, M.; Salucci, M. DNNs as applied to electromagnetics, antennas, and propagation—A review. IEEE Antennas Wirel. Propag. Lett. 2019, 18, 2225–2229. [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. 2018, 67, 1819–1825. [Google Scholar] [CrossRef]
- Chen, G.; Shah, P.; Stang, J.; Moghaddam, M. Learning-assisted multimodality dielectric imaging. IEEE Trans. Antennas Propag. 2019, 68, 2356–2369. [Google Scholar] [CrossRef]
- Wei, Z.; Chen, X. Deep-learning schemes for full-wave nonlinear inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 2018, 57, 1849–1860. [Google Scholar] [CrossRef]
- Wu, Z.; Peng, Y.; Wang, P.; Wang, W.; Xiang, W. A physics-induced deep learning scheme for electromagnetic inverse scattering. IEEE Trans. Microw. Theory Tech. 2024, 72, 927–947. [Google Scholar] [CrossRef]
- Xue, B.W.; Guo, R.; Li, M.K.; Sun, S.; Pan, X.M. Deep-learning-equipped iterative solution of electromagnetic scattering from dielectric objects. IEEE Trans. Antennas Propag. 2023, 71, 5954–5966. [Google Scholar] [CrossRef]
- Xue, F.; Guo, L.; Abbosh, A. Microwave imaging using cascaded convolutional neural networks. In Proceedings of the 2023 5th Australian Microwave Symposium (AMS), Melbourne, Australia, 16–17 February 2023; pp. 47–48. [Google Scholar]
- Wang, M.; Sun, S.; Dai, D.; Su, Y.; Wu, M. Quantitative diffraction tomography for weak scatterers based on aliasing modification of the multifrequency spatial spectrum. IEEE Trans. Geosci. Remote Sens. 2023, 61, 2002214. [Google Scholar] [CrossRef]
- Sanghvi, Y.; Kalepu, Y.; Khankhoje, U.K. Embedding deep learning in inverse scattering problems. IEEE Trans. Comput. Imaging 2019, 6, 46–56. [Google Scholar] [CrossRef]
- Yao, H.M.; Wei, E.; Jiang, L. Two-step enhanced deep learning approach for electromagnetic inverse scattering problems. IEEE Antennas Wirel. Propag. Lett. 2019, 18, 2254–2258. [Google Scholar] [CrossRef]
- Chen, X.; Wei, Z.; Maokun, L.; Rocca, P. A review of deep learning approaches for inverse scattering problems (invited review). Electromagn. Waves 2020, 167, 67–81. [Google Scholar] [CrossRef]
- Zhang, Y.; Lambert, M.; Fraysse, A.; Lesselier, D. Unrolled convolutional neural network for full-wave inverse scattering. IEEE Trans. Antennas Propag. 2022, 71, 947–956. [Google Scholar] [CrossRef]
- Xu, K.; Zhang, C.; Ye, X.; Song, R. Fast full-wave electromagnetic inverse scattering based on scalable cascaded convolutional neural networks. IEEE Trans. Geosci. Remote Sens. 2021, 60, 2001611. [Google Scholar] [CrossRef]
- Yao, H.M.; Ng, M.; Jiang, L. Deep Learning Electromagnetic Inversion Solver Based on Two-Step Framework for High-Contrast and Heterogeneous Scatterers. IEEE Trans. Antennas Propag. 2024. [Google Scholar] [CrossRef]
- Zhang, H.H.; Yao, H.M.; Jiang, L.; Ng, M. Enhanced two-step deep-learning approach for electromagnetic-inverse-scattering problems: Frequency extrapolation and scatterer reconstruction. IEEE Trans. Antennas Propag. 2022, 71, 1662–1672. [Google Scholar] [CrossRef]
- Si, A.; Dai, D.; Wang, M.; Fang, F. Two Steps Electromagnetic Quantitative Inversion Imaging Based on Convolutional Neural Network. In Proceedings of the 2024 5th International Conference on Geology, Mapping and Remote Sensing (ICGMRS), Wuhan, China, 12–14 April 2024; pp. 28–32. [Google Scholar]
- Gibson, W.C. The Method of Moments in Electromagnetics; Chapman and Hall/CRC: London, UK, 2021. [Google Scholar]
- Siddique, N.; Paheding, S.; Elkin, C.P.; Devabhaktuni, V. U-net and its variants for medical image segmentation: A review of theory and applications. IEEE Access 2021, 9, 82031–82057. [Google Scholar] [CrossRef]
- Ioffe, S.; Szegedy, C. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. arXiv 2015, arXiv:1502.03167. [Google Scholar]
- Huang, Y.; Song, R.; Xu, K.; Ye, X.; Li, C.; Chen, X. Deep learning-based inverse scattering with structural similarity loss functions. IEEE Sens. J. 2020, 21, 4900–4907. [Google Scholar] [CrossRef]
- Deng, L. The mnist database of handwritten digit images for machine learning research [best of the web]. IEEE Signal Process. Mag. 2012, 29, 141–142. [Google Scholar] [CrossRef]
- Anzengruber, S.W.; Ramlau, R. Convergence rates for Morozov’s discrepancy principle using variational inequalities. Inverse Probl. 2011, 27, 105007. [Google Scholar] [CrossRef]
Variable Name | Size | Meaning |
---|---|---|
The total internal fields data on domain S. | ||
The incident fields data in the DoI. | ||
The scattered fields data on domain S. | ||
The CS function in the DoI. | ||
The diagnonal matrix of the contrast function. | ||
The radiation operators (the DoI to the domain S). | ||
The radiation operators (the DoI to the DoI). |
Metric | (b) | (c) | (d) | (e) | (f) | |
---|---|---|---|---|---|---|
Test(1): first row digit “7” | SSIM | 0.2201 | 0.1064 | 0.2836 | 0.4003 | 0.7175↑ |
PSNR | 14.1030 | 11.8024 | 12.3382 | 18.2844 | 24.4498↑ | |
ENL | 0.3615 | 0.1303 | 0.1754 | 0.2353 | 0.1433↓ | |
Test(2): second row digit “1” | SSIM | 0.3532 | 0.1722 | 0.3311 | 0.5839 | 0.8924↑ |
PSNR | 22.9435 | 17.6328 | 14.1315 | 21.5712 | 30.3555↑ | |
ENL | 0.1390 | 0.0487 | 0.0767 | 0.1070 | 0.0760↓ | |
Test(3): third row digit “0” | SSIM | 0.2861 | 0.2054 | 0.3219 | 0.5883 | 0.7931↑ |
PSNR | 19.7904 | 11.9012 | 12.2882 | 22.8354 | 24.7904↑ | |
ENL | 0.5392 | 0.2267 | 0.2296 | 0.3488 | 0.2241↓ | |
Test(4): fourth row digit “4” | SSIM | 0.2369 | 0.1621 | 0.3483 | 0.4007 | 0.7147↑ |
PSNR | 17.9987 | 13.2725 | 14.0838 | 14.6851 | 19.2040↑ | |
ENL | 0.3461 | 0.2015 | 0.1606 | 0.2630 | 0.1520↓ | |
Test(5): fifth row digit “9” | SSIM | 0.6835 | 0.2170 | 0.3672 | 0.6761 | 0.8810↑ |
PSNR | 23.4289 | 15.2951 | 17.0325 | 19.4508 | 28.7113↑ | |
ENL | 0.2665 | 0.1842 | 0.2059 | 0.2910 | 0.1988↓ | |
Test(6): sixth row digit “6” | SSIM | 0.2442 | 0.1957 | 0.3692 | 0.3248 | 0.8093↑ |
PSNR | 19.4901 | 16.8602 | 17.4904 | 18.5999 | 21.6365↑ | |
ENL | 0.4229 | 0.3294 | 0.3005 | 0.4351 | 0.2793↓ |
Metric | (b) | (c) | (d) | (e) | (f) | |
---|---|---|---|---|---|---|
Test(1): first row digit “2” | SSIM | 0.5584 | 0.0837 | 0.1613 | 0.4605 | 0.7858↑ |
PSNR | 24.4427 | 9.1308 | 10.5866 | 18.2878 | 25.2928↑ | |
ENL | 0.3148 | 0.0950 | 0.2762 | 0.3246 | 0.2074↓ | |
Test(2): second row digit “9” | SSIM | 0.2649 | 0.2029 | 0.3932 | 0.3721 | 0.7834↑ |
PSNR | 17.6125 | 13.9164 | 15.1215 | 19.9382 | 24.2291↑ | |
ENL | 0.4853 | 0.1522 | 0.2116 | 0.3396 | 0.2251↓ | |
Test(3): third row digit “4” | SSIM | 0.1715 | 0.0825 | 0.2969 | 0.3782 | 0.6624↑ |
PSNR | 14.6102 | 13.2556 | 13.5544 | 19.4833 | 25.4906↑ | |
ENL | 0.3144 | 0.2008 | 0.1457 | 0.2521 | 0.1836↓ | |
Test(4): fourth row digit “7” | SSIM | 0.2460 | 0.0425 | 0.1987 | 0.3335 | 0.7619↑ |
PSNR | 23.0121 | 10.8452 | 10.3681 | 19.3680 | 27.3919↑ | |
ENL | 0.2694 | 0.1246 | 0.1618 | 0.2369 | 0.1318↓ | |
Test(5): fifth row digit “3” | SSIM | 0.2606 | 0.1185 | 0.2410 | 0.3281 | 0.6867↑ |
PSNR | 15.3602 | 12.0076 | 12.4106 | 17.9613 | 21.3712↑ | |
ENL | 0.4859 | 0.2035 | 0.2214 | 0.3627 | 0.2007↓ | |
Test(6): sixth row digit “1” | SSIM | 0.2030 | 0.0949 | 0.2745 | 0.3910 | 0.7850↑ |
PSNR | 18.8818 | 7.6736 | 5.4605 | 22.9497 | 24.5452↑ | |
ENL | 0.2282 | 0.2142 | 0.1063 | 0.1433 | 0.0936↓ |
Metric | (b) | (c) | (d) | (e) | (f) | |
---|---|---|---|---|---|---|
Austria: | SSIM | 0.5787 | 0.2834 | 0.4001 | 0.3960 | 0.8195↑ |
PSNR | 16.0104 | 15.9636 | 16.8300 | 17.4791 | 19.0318↑ | |
ENL | 0.5823 | 0.4490 | 0.3048 | 0.5658 | 0.4268↓ |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 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
Si, A.; Wang, M.; Fang, F.; Dai, D. Two-Step Contrast Source Learning Method for Electromagnetic Inverse Scattering Problems. Sensors 2024, 24, 5997. https://doi.org/10.3390/s24185997
Si A, Wang M, Fang F, Dai D. Two-Step Contrast Source Learning Method for Electromagnetic Inverse Scattering Problems. Sensors. 2024; 24(18):5997. https://doi.org/10.3390/s24185997
Chicago/Turabian StyleSi, Anran, Miao Wang, Fuping Fang, and Dahai Dai. 2024. "Two-Step Contrast Source Learning Method for Electromagnetic Inverse Scattering Problems" Sensors 24, no. 18: 5997. https://doi.org/10.3390/s24185997
APA StyleSi, A., Wang, M., Fang, F., & Dai, D. (2024). Two-Step Contrast Source Learning Method for Electromagnetic Inverse Scattering Problems. Sensors, 24(18), 5997. https://doi.org/10.3390/s24185997