volume | PIER Journals

Abstract

Although classical imaging is limited by the Rayleigh criterion, it has been demonstrated that subwavelength imaging of two point-like scatterers can be achieved with probing sensors arrays, even if the scatterers are located in the far field of the sensors. However, the role of noise is crucial to determine the resolution limit. This paper proposes a quantitative study of the influence of noise on the subwavelength resolution obtained with the DORT-MUSIC method. The DORT method, French acronym for decomposition of the time reversal operator, consists in studying the invariants of the time reversal operator. The method is combined here with the estimator MUSIC (MUltiple SIgnal Classification) to detect and image two close metallic wires. The microwaves measurements are performed between 2.6 GHz and 4 GHz. Two wires of λ/100 diameters separated by λ/6 are imaged and separated experimentally. To interpret this result in terms of noise level, the analytical expression of the eigenvectors of the time reversal operator perturbed by the noise is established. We then deduce the noise level above which the subwavelength resolution fails. Numerical simulations and experimental results validate the theoretical developments.

1. Cheney, M., "The linear sampling method and the MUSIC algorithm," Inverse Problems, Vol. 17, 591-595, 2001.
doi:10.1088/0266-5611/17/4/301 Google Scholar

2. Lev-Ari, H. and A. J. Devancy, "The time-reversal technique re-interpreted: Subspace-based signal processing for multi-static target location," Sensor Array and Multichannel Signal Processing Workshop, 2000, Proceedings of the 2000 IEEE, 509-513, 2000.
doi:10.1109/SAM.2000.878061 Google Scholar

3. Prada, C. and M. Fink, "Eigenmodes of the time reversal operator: A solution to selective focusing in multiple-target media," Wave Motion, Vol. 20, 151-163, 1994.
doi:10.1016/0165-2125(94)90039-6 Google Scholar

4. Kerbrat, E., R. K. Ing, C. Prada, D. Cassereau, and M. Fink, "The D. O. R. T. method applied to detection and imaging in plates using Lamb waves," Review of Progress in Quantitative Nondestructive Evaluation, 934-940, Ames, Iowa (USA), 2001. Google Scholar

5. Prada, C., M. Tanter, and M. Fink, "Flaw detection in solid with the D. O. R. T. method," Ultrasonics Symposium, Vol. 1, 679-683, 1997. Google Scholar

6. Kerbrat, E., D. Clorennec, C. Prada, D. Royer, D. Cassereau, and M. Fink, "Detection of cracks in a thin air-filled hollow cylinder by application of the DORT method to elastic components of the echo," Ultrasonics, Vol. 40, 715-720, 2002.
doi:10.1016/S0041-624X(02)00199-3 Google Scholar

7. Mordant, N., C. Prada, and M. Fink, "Highly resolved detection and selective focusing in a waveguide using the D. O. R. T. method," The Journal of the Acoustical Society of America, Vol. 105, 2634-2642, 1999.
doi:10.1121/1.426879 Google Scholar

8. Tortel, H., G. Micolau, and M. Saillard, "Decomposition of the time reversal operator for electromagnetic scattering," Journal of Electromagnetic Waves and Applications, Vol. 13, No. 5, 687-719, 1999.
doi:10.1163/156939399X01113 Google Scholar

9. Prada, C., S. Manneville, D. Spoliansky, and M. Fink, "Decomposition of the time reversal operator: Detection and selective focusing on two scatterers ," The Journal of the Acoustical Society of America, Vol. 99, 2067-2076, 1996.
doi:10.1121/1.415393 Google Scholar

10. Prada, C. and J.-L. Thomas, "Experimental subwavelength localization of scatterers by decomposition of the time reversal operator interpreted as a covariance matrix ," The Journal of the Acoustical Society of America, Vol. 114, 235-243, 2003.
doi:10.1121/1.1568759 Google Scholar

11. Devaney, A. J., "Super-resolution processing of multi-static data using time reversal and MUSIC,", http://www.ece.neu.edu/faculty/devaney/preprints/paper02n 00.pdf. Google Scholar

12. Lehman, S. K. and A. J. Devaney, "Transmission mode time-reversal super-resolution imaging," The Journal of the Acoustical Society of America, Vol. 113, 2742-2753, 2003.
doi:10.1121/1.1566975 Google Scholar

13. Miwa, T. and I. Arai, "Super-resolution imaging for point reflectors near transmitting and receiving array," IEEE Transactions on Antennas and Propagation, Vol. 52, 220-229, 2004.
doi:10.1109/TAP.2003.820975 Google Scholar

14. Baussard, A. and T. Boutin, "Time-reversal RAP-MUSIC imaging," Waves in Random and Complex Media, Vol. 18, 151-160, 2008.
doi:10.1080/17455030701481856 Google Scholar

15. Simonetti, F., "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 73, 036619-13, 2006. Google Scholar

16. Simonetti, F., "Pushing the boundaries of ultrasound imaging to unravel the subwavelength world," Proceedings of IEEE International Ultrasonics Symposium, 313-316, Vancouver, Canada, 2006.

17. Simonetti, F., M. Fleming, and E. A. Marengo, "Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements," J. Opt. Soc. Am. A, Vol. 25, 292-303, 2008.
doi:10.1364/JOSAA.25.000292 Google Scholar

18. De Rosny, J. and C. Prada, "Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave," Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol. 75, 048601-2, 2007. Google Scholar

19. Minonzio, J.-G., C. Prada, A. Aubry, and M. Fink, "Multiple scattering between two elastic cylinders and invariants of the time-reversal operator: Theory and experiment," The Journal of the Acoustical Society of America, Vol. 120, 875-883, 2006.
doi:10.1121/1.2217128 Google Scholar

20. Moura, J. M. F. and J. Yuanwei, "Detection by time reversal: Single antenna," IEEE Transactions on Signal Processing, Vol. 55, 187-201, 2007.
doi:10.1109/TSP.2006.882114 Google Scholar

21. Moura, J. M. F. and J. Yuanwei, "Time reversal imaging by adaptive interference canceling," IEEE Transactions on Signal Processing, Vol. 56, 233-247, 2008.
doi:10.1109/TSP.2007.906745 Google Scholar

22. Minonzio, J.-G., M. Davy, J. de Rosny, C. Prada, and M. Fink, "Theory of the time-reversal operator for the dielectric cylinder using separate transmit and received arrays," IEEE Transactions on Antennas and Propagation, August 2009. Google Scholar

23. Stewart, G. W., "Perturbation theory for the singular value decomposition," SVD and Signal Processing, II: Algorithms Analysis and Applications, 99-109, 1990. Google Scholar

24. Xu, Z., "Perturbation analysis for subspace decomposition with applications in subspace-based algorithms," IEEE Transactions on Signal Processing, Vol. 50, 2820-2830, 2002. Google Scholar

25. Zhenhua, L., "Direct perturbation method for reanalysis of matrix singular value decomposition," Applied Mathematics and Mechanics, Vol. 18, 471-477, 1997.
doi:10.1007/BF02453742 Google Scholar

26. Liu, J., X. Liu, and X. Ma, "First-order perturbation analysis of singular vectors in singular value decomposition," IEEE Transactions on Signal Processing, Vol. 56, 3044-3049, 2008.
doi:10.1109/TSP.2007.916137 Google Scholar

27. De Moor, B., "The singular value decomposition and long and short spaces of noisy matrices," IEEE Transactions on Signal Processing, Vol. 41, 2826-2838, 1993.
doi:10.1109/78.236505 Google Scholar

Mr. Matthieu Davy

Dr. Jean-Gabriel Minonzio

Dr. Julien de Rosny

Dr. Claire Prada

Prof. Mathias Fink