volume | PIER Journals

Abstract

This paper discusses the problem of choosing an appropriate direction of the test dipole used in linear sampling for the 2-dimensional inverse scattering problem of the transverse electric case. In particular, we propose two approaches, one purely mathematical and the other based on the physics theory of multipole expansion of the scattered magnetic field. It is shown that though the approaches are drawn from different perspectives, they perform similarly and show reasonable reconstruction for several interesting and difficult to reconstruct dielectric scatterers.

1. Colton, D. and Colton, "Using fundamental solutions in inverse scattering," Inverse Problems, Vol. 22, R49-R66, June 2006.
doi:10.1088/0266-5611/22/3/R01

2. Kirsch, A. and S. Ritter, "A linear sampling method for inverse scattering from an open arc," Inverse Problems, Vol. 16, 89-105, February 2000.
doi:10.1088/0266-5611/16/1/308

3. Cakoni, F., D. Colton, and H. Haddar, "The linear sampling method for anisotropic media," Journal of Computational and Applied Mathematics, Vol. 146, 285-299, September 2002.
doi:10.1016/S0377-0427(02)00361-8

4. Colton, D., H. Haddar, and P. Monk, "The linear sampling method for solving the electromagnetic inverse scattering problem," SIAM Journal on Scientific Computing, Vol. 24, 719-731, January 2002.
doi:10.1137/S1064827501390467

5. Cakoni, F. and D. Colton, "The linear sampling method for cracks," Inverse Problems, Vol. 19, 279-295, April 2003.
doi:10.1088/0266-5611/19/2/303

6. Collino, F., M. Fares, and H. Haddar, "Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems," Inverse Problems, Vol. 19, 1279-1298, December 2003.
doi:10.1088/0266-5611/19/6/004

7. Colton, D., H. Haddar, and M. Piana, " The linear sampling method in inverse electromagnetic scattering theory," Inverse Problems, Vol. 19, S105-S137, December 2003.
doi:10.1088/0266-5611/19/6/057

8. Arens, T., "Why linear sampling works," Inverse Problems, Vol. 20, 163-173, February 2004.
doi:10.1088/0266-5611/20/1/010

9. Cakoni, F., M. Fares, and H. Haddar, "Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects," Inverse Problems, Vol. 22, 845-867, June 2006.
doi:10.1088/0266-5611/22/3/007

10. Gebauer, B., M. Hanke, and C. Schneider, "Sampling methods for low-frequency electromagnetic imaging," Inverse Problems, Vol. 24, 15007-18, 2008.
doi:10.1088/0266-5611/24/1/015007

11. Hanke, M., "Why linear sampling really seems to work," Inverse Problems and Imaging, Vol. 2, 373-395, August 2008.
doi:10.3934/ipi.2008.2.373

12. Pelekanos, G., K. H. Leem, and V. Sevroglou, "A preconditioned linear sampling method in inverse acoustic obstacle scattering," Journal of Computational Analysis and Applications, Vol. 10, 453-464, October 2008.

13. Chen, Q., H. Haddar, A. Lechleiter, and P. Monk, "A sampling method for inverse scattering in the time domain," Inverse Problems, Vol. 26, 085001, 2010.
doi:10.1088/0266-5611/26/8/085001

14. Thanh, N. T. and M. Sini, "An analysis of the accuracy of the linear sampling method for an acoustic inverse obstacle scattering problem," Inverse Problems, Vol. 26, 015010, 2010.
doi:10.1088/0266-5611/26/1/015010

15. Zeng, F., F. Cakoni, and J. Sun, "An inverse electromagnetic scattering problem for a cavity," Inverse Problems, Vol. 27, 125002, 2011.
doi:10.1088/0266-5611/27/12/125002

16. Catapano, I., F. Soldovieri, and L. Crocco, "On the feasibility of the linear sampling method for 3D GPR surveys," Progress In Electromagnetics Research, Vol. 118, 185-203, 2011.
doi:10.2528/PIER11042704

17. Scapaticci, R., L. Di Donato, I. Catapano, and L. Crocco, "Feasibility study on microwave imaging for brain stroke monitoring," Progress In Electromagnetics Research B, 305-324, 2012.
doi:10.2528/PIERB12022006

18. Shelton, N. and K. F. Warnick, "Behavior of the regularized sampling inverse scattering method at internal resonance frequencies," Progress In Electromagnetics Research, Vol. 38, 29-45, 2003.
doi:10.2528/PIER02092502

19. Catapano, I., L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Transactions on Antennas and Propagation, Vol. 55, 1431-1436, May 2007.
doi:10.1109/TAP.2007.895563

20. Catapano, I., L. Crocco, and T. Isernia, "Linear sampling method: Physical interpretation and guidelines for a successful application," PIERS Proceedings, 1152-1156, Hangzhou, China, March 24–28, 2008.

21. Agarwal, K., X. Chen, and Y. Zhong, "A multipole-expansion based linear sampling method for solving inverse scattering problems," Optics Express, Vol. 18, 6366-6381, March 2010.
doi:10.1364/OE.18.006366

22. Agarwal, K. and X. Chen, "Multipoles-based linear sampling method: Impact of using multipole expansion," Journal of Physics: Conference Series, Vol. 290, 012001-1, 2011.
doi:10.1088/1742-6596/290/1/012001

23. Aramini, R., G. Caviglia, A. Massa, and M. Piana, "The linear sampling method and energy conservation," Inverse Problems, Vol. 26, 055004, 2010.
doi:10.1088/0266-5611/26/5/055004

24. Li, J., H. Liu, and Q. Wang, "Enhanced multilevel linear sampling methods for inverse scattering problems," Journal of Computational Physics, Vol. 257, 554-571, 2014.
doi:10.1016/j.jcp.2013.09.048

25. Li, J., H. Liu, and J. Zou, "Strengthened linear sampling method with a reference ball," SIAM Journal on Scientific Computing, Vol. 31, 4013-4040, 2009.
doi:10.1137/080734170

26. Li, J., H. Liu, H. Sun, and J. Zou, "Reconstructing acoustic obstacles by planar and cylindrical waves," Journal of Mathematical Physics, Vol. 53, 2012.

27. Brignone, M., G. Bozza, A. Randazzo, M. Piana, and M. Pastorino, "A hybrid approach to 3D microwave imaging by using linear sampling and ACO," IEEE Transactions on Antennas and Propagation, Vol. 56, 3224-3232, October 2008.
doi:10.1109/TAP.2008.929504

28. Chen, X. and K. Agarwal, "MUSIC algorithm for two-dimensional inverse problems with special characteristics of cylinders," IEEE Transactions on Antennas and Propagation, Vol. 56, 1808-1812, June 2008.
doi:10.1109/TAP.2008.923333

29. Chen, X. and Y. Zhong, "MUSIC electromagnetic imaging with enhanced resolution for small inclusions," Inverse Problems, Vol. 25, Article No. 015008, January 2009.

30. Rao, T. and X. Chen, "Analysis of the time-reversal operator for a single cylinder under twodimensional settings," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 15, 2153-2165, 2006.
doi:10.1163/156939306779322503

31. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, 7th edition, Dover Publications, New York, 1972.

Dr. Krishna Agarwal