A new approach for position and shape reconstruction of both penetrable and impenetrable objects from the measurements of the scattered fields is introduced and described. The approach takes advantage of the fact that for perfect electric conductors the induced currents are localized on the boundary, and equivalent sources also placed on the surface of the scatterers can be considered in the case of dielectric targets by virtue of the equivalence theorem. Starting from these considerations, a new inversion approach is formulated in order to retrieve the location and the boundary of unknown objects. Examples with both numerical and experimental data are given to demonstrate and assess the effectiveness of the method.
2. Bozza, G., M. Brignone, and M. Pastorino, "Application of the no-sampling linear sampling method to breast cancer detection," IEEE Trans. Biomed. Eng., Vol. 57, No. 10, 2525-2534, 2010.
3. Bozza, G., M. Brignone, M. Pastorino, M. Piana, and A. Randazzo, "A linear sampling approach to crack detection in microwave imaging," 2008 IEEE International Workshop on Imaging Systems and Techniques, 222-226, Crete, 2008.
4. Cakoni, F. and D. Colton, Qualitative Methods in Inverse Scattering Theory, Springer-Verlag, Berlin, Germany, 2006.
5. Agarwal, K. and X. Chen, "Applicability of MUSIC-type imaging in two-dimensional electromagnetic inverse problems," IEEE Trans. Antennas Propag., Vol. 56, No. 10, 3217-3223, 2008.
6. Zhong, Y. and X. Chen, "MUSIC imaging and electromagnetic inverse scattering of multiplescattering small anisotropic spheres," IEEE Trans. Antennas Propag., Vol. 55, No. 12, 3542-3549, 2007.
7. Iakovleva, E., S. Gdoura, D. Lesselier, and G. Perrusson, "Multistatic response matrix of a 3D inclusion in half space and MUSIC imaging," IEEE Trans. Antennas Propag., Vol. 55, No. 9, 2598-2609, 2007.
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.
9. Devaney, A. J., E. A. Marengo, and F. K. Gruber, "Time-reversal-based imaging and inverse scattering of multiply scattering point targets," J. Acoust. Soc. Am., Vol. 118, No. 5, 3129-3138, 2005.
10. Colton, D. and A. Kirsch, "A simple method for solving inverse scattering problems in the resonant region," Inverse Probl., Vol. 12, 383-393, 1996.
11. Colton, D., M. Piana, and R. Potthast, "A simple method using morozov’s discrepancy principle for solving inverse scattering problems," Inverse Probl., Vol. 13, 1477-1493, 1997.
12. Catapano, I., L. Crocco, and T. Isernia, "On simple methods for shape reconstruction of unknown scatterers," IEEE Trans. Antennas Propag., Vol. 55, No. 5, 1431-1436, 2007.
13. Kirsch, A., "Characterization of the shape of a scattering obstacle using the spectral data of the far-field operator," Inverse Probl., Vol. 14, 1489-1512, 1998.
14. Kirsch, A., "Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory," Inverse Probl., Vol. 15, 413-429, 1999.
15. Potthast, R., "A point source method for inverse acoustic and electromagnetic obstacle scattering problems," IMA J. Appl. Math., Vol. 61, No. 2, 119-140, 1998.
16. Litman, A., D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl., Vol. 14, No. 3, 685-706, 1998.
17. Dorn, O. and D. Lesselier, "Level set methods for inverse scattering," Inverse Probl., Vol. 22, No. 4, R67-R131, 2006.
18. Kleinman, R. E. and P. M. den Berg, "Two-dimensional location and shape reconstruction," Radio Science, Vol. 29, No. 4, 1157-1169, 1994.
19. Liseno, A. and R. Pierri, "Imaging perfectly conducting objects as support of induced currents: Kirchhoff approximation and frequency diversity," J. Opt. Soc. Am. A, Vol. 19, 1308-1318, 2002.
20. Shen, J., Y. Zhong, X. Chen, and L. Ran, "Inverse scattering problems of reconstructing perfectly electric conductors with TE illumination," IEEE Trans. Antennas Propag., Vol. 61, No. 9, 4713-4721, Sept. 2013.
21. Poli, L., G. Oliveri, and A. Massa, "Imaging sparse metallic cylinders through a local shape function Bayesian compressive sensing approach," J. Opt. Soc. Am. A, Vol. 30, No. 6, 1261-1272, 2013.
22. Stevanovic, M. N., L. Crocco, A. R. Djordjevic, and A. Nehorai, "Higher order sparse microwave imaging of PEC scatterers," IEEE Trans. Antennas Propag., Vol. 64, No. 3, 988-997, Mar. 2016.
23. Franceschetti, G., Electromagnetics: Theory, Techniques, and Engineering Paradigms, Springer Science & Business Media, 2013.
24. Donoho, D., "Compressed sensing," IEEE Trans. Inf. Theory, Vol. 52, No. 4, 1289-1306, 2006.
25. Massa, A., P. Rocca, and G. Oliveri, "Compressive sensing in electromagnetics — A review," IEEE Antennas and Propagation Magazine, Vol. 57, No. 1, 224-238, Feb. 2015.
26. Bevacqua, M., L. Crocco, L. Di Donato, T. Isernia, and R. Palmeri, "Exploiting field conditioning and sparsity for microwave imaging of non-weak buried targets," Radio Sci., 2016.
27. Shah, P., U. K. Khankhoje, and M. Moghaddam, "Inverse scattering using a joint L1 − L2 normbased regularization," IEEE Trans. Antennas Propag., Vol. 64, No. 4, 1373-1384, Apr. 2016.
28. Azghani, M., P. Kosmas, and F. Marvasti, "Microwave medical imaging based on sparsity and an iterative method with adaptive thresholding," IEEE Trans. Medical Imaging, Vol. 34, No. 2, 357-365, 2015.
29. Morabito, A. F., R. Palmeri, and T. Isernia, "A compressive-sensing-inspired procedure for array antenna diagnostics by a small number of phaseless measurements," IEEE Trans. Antennas Propag., Vol. 64, No. 7, 3260-3265, Jul. 2016.
30. Bevacqua, M., T. Isernia, L. Crocco, and L. Di Donato, "A (CS)2 approach to inverse scattering," 2014 IEEE Conference on Antenna Measurements & Applications (CAMA), 1-3, Nov. 16–19, 2014.
31. Hawes, M. B. and W. Liu, "Compressive sensing-based approach to the design of linear robust sparse antenna arrays with physical size constraint," IET Microwaves, Antennas & Propagation, Vol. 8, No. 10, 736-746, 2014.
32. Winters, D. W., B. D. Van Veen, and S. C. Hagness, "A sparsity regularization approach to the electromagnetic inverse scattering problem," IEEE Trans. Antennas Propag., Vol. 58, No. 1, 145-154, Jan. 2010.
33. Colton, D. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, Germany, 1998.
34. Bertero, M. and P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics, Bristol, UK, 1998.
35. Bucci, O. M. and T. Isernia, "Electromagnetic inverse scattering: Retrievable nformation and measurement strategies," Radio Sci., Vol. 32, 2123-2138, 1997.
36. Chen, S., D. Donoho, and M. Saunders, "Atomic decomposition by basis pursuit," SIAM J. Sci. Comput., Vol. 20, No. 1, 33-61, 1999.
37. Tibshirani, R., "Regression shrinkage and selection via the lasso," J. Roy. Stat. Soc. Ser., Vol. 58, No. 1, 267-288, 1996.
38. Liu, Y., P. You, C. Zhu, X. Tan, and Q. H. Liu, "Synthesis of sparse or thinned linear and planar arrays generating reconfigurable multiple real patterns by iterative linear programming," Progress In Electromagnetics Research, Vol. 155, 27-38, 2016.
39. Brancaccio, A., G. Leone, and R. Solimene, "Single-frequency subsurface remote sensing via a non-cooperative source," Journal of Electromagnetic Waves and Applications, Vol. 30, No. 9, 1-15, 2016.
40. Gennarelli, G., R. Solimene, F. Soldovieri, and M. G. Amin, "Three-dimensional through-wall sensing of moving targets using passive multistatic radars," IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, Vol. 9, No. 1, 141-148, Jan. 2016.
41. Bevacqua, M. T. and R. Scapaticci, "A compressive sensing approach for 3D breast cancer microwave imaging with magnetic nanoparticles as contrast agent," IEEE Transactions on Medical Imaging, Vol. 35, No. 2, 665-673, Feb. 2016.
42. Soldovieri, F., A. Brancaccio, G. Leone, and R. Pierri, "Shape reconstruction of perfectly conducting objects by multiview experimental data," IEEE Trans. on Geosci. and Remote Sens., Vol. 43, No. 1, 65-71, Jan. 2005.
43. Belkebir, K. and M. Saillard, "Special section: Testing inversion algorithms against experimental data," Inverse Probl., Vol. 7, 1565-2028, 2001.
44. CVX Research, Inc., CVX: Matlab software for disciplined convex programming, 2.0, http://cvxr.com/cvx, Apr. 2011.
45. Grant, M. and S. Boyd, "Graph implementations for non smooth convex programs," Lecture Notes in Control and Information Sciences, 95-110, Chapter Recent Advances in Learning and Control (a tribute to M. Vidyasagar), Springer, 2008.
46. Richmond, J., "Scattering by a dielectric cylinder of arbitrary cross section shape," IEEE Trans. Antennas Propag., Vol. 13, No. 3, 334-341, 1965.
47. Wirgin, A., "The inverse crime," ArXiv Mathematical Physics e-prints, Jan. 2004.