volume | PIER Journals

Abstract

An approach based on the use of the arithmetic of intervals and Interval Analysis for the solution of inverse scattering problems is presented and assessed. By exploiting the property of the Interval Analysis to find the global minimum of a functional in a n-dimensional space, the proposed approach adopts a branch and bound process to discard the regions of the solutions space not containing the global solution, while keeping those where a feasible solution is expected until a suitable converge criterion is reached. A representative set of results concerned with the reconstruction of circular dielectric objects within the first-order Born approximation are reported and discussed to show potentialities and current limitations of the proposed approach.

1. Bertero, M. and P. Boccacci, Introduction to Inverse Problems in Imaging, IOP Press, Bristol, U.K., 1998.
doi:10.1887/0750304359

2. Massa, A. and S. Caorsi, Special Issue on `Microwave Imaging and Inverse Scattering Techniques', Vol. 16, No. 2, 2003.
doi:ISSN 0920-5071

3. Lesselier, D., T. Habashy, and , "Special issue on `Electromagnetic imaging and inversion on the Earth's subsurface'," Inverse Probl., Vol. 16, No. 5, 2000.

4. Chew, W. C. and D. Lesseliser, "Special issue on `Electromagnetic characterization of buried obstacles'," Inverse Probl., Vol. 20, No. 6, 2004.

5. Zhang, X., H. Tortel, S. Ruy, and A. Litman, "Microwave imaging of soil water diffusion using the linear sampling method," IEEE Geosci. Remote Sens. Lett., Vol. 8, No. 3, 421-425, 2011.
doi:10.1109/LGRS.2010.2082490

6. Caorsi, S., A. Massa, and M. Pastorino, "A crack identification microwave procedure based on a genetic algorithm for nondestructive testing ," IEEE Trans. Antennas Propag., Vol. 49, No. 12, 1812-1820, 2001.
doi:10.1109/8.982464

7. Benedetti, M., Benedetti, M., M. Donelli, and A. Massa, "Multicrack detection in two-dimensional structures by means of GA-based strategies," IEEE Trans. Antennas Propag., Vol. 55, No. 1, 205-215, 2007.
doi:10.1109/TAP.2006.888399

8. Mudanyal, O., S. Yldz, O. Semerci, A. Yapar, and I. Akduman, "A microwave tomographic approach for nondestructive testing of dielectric coated metallic surfaces," IEEE Geosci. Remote Sens. Lett., Vol. 5, No. 2, 180-184, 2008.
doi:10.1109/LGRS.2008.915602

9. Ma, Y.-K., P.-S. Kin, and W.-K. Park, "Analysis of topological derivative function for a fast electromagnetic imaging of perfectly conducing cracks," Progress In Electromagnetics Research, Vol. 122, 311-325, 2012.
doi:10.2528/PIER11092901

10. Joh, Y.-D. and W.-K. Park, "Structural behavior of the Music-type algorithm for imaging perfectly conducting cracks," Progress In Electromagnetics Research, Vol. 138, 211-226, 2013.

11. Bindu, G. N., S. J. Abraham, A. Lonappan, V. Thomas, C. K. Aanandan, and K. T. Mathew, "Active microwave imaging for breast cancer detection," Progress In Electromagnetics Research, Vol. 58, 149-169, 2006.
doi:10.2528/PIER05081802

12. Zhang, H., S. Y. Tan, and H. S. Tan, "A novel method for microwave breast cancer detection," Progress In Electromagnetics Research, Vol. 83, 413-434, 2008.
doi:10.2528/PIER08062701

13. Zhou, H., Zhou, H., T. Takenaka, J. E. Johnson, and T. Tanaka, "Breast imaging model using microwaves and a time domain three dimensional reconstruction method," Progress In Electromagnetics Research, Vol. 93, 57-70, 2009.
doi:10.2528/PIER09033001

14. Henriksson, T., N. Joachimowicz, C. Conessa, and J.-C. Bolomey, "Quantitative microwave imaging for breast cancer detection using a planar 2.45 GHz system ," IEEE Trans. Instrum. Meas., Vol. 59, No. 10, 2691-2699, 2010.
doi:10.1109/TIM.2010.2045540

15. Ashtari, A., S. Noghanian, A. Sabouni, J. Aronsson, G. Thomas, and S. Pistorius, "Using a priori information for regularization in breast microwave image reconstruction," IEEE Trans. Biomed. Eng., Vol. 57, No. 9, 2197-2208, 2010.
doi:10.1109/TBME.2010.2051439

16. 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.
doi:10.1109/TBME.2010.2055059

17. Flores-Tapia, D., M. O'Halloran, and S. Pistorius, "A bimodal reconstruction method for breast cancer imaging," Progress In Electromagnetics Research, Vol. 118, 461-486, 2011.
doi:10.2528/PIER11050408

18. Hagness, S., E. Fear, and A. Massa, "Special cluster on Microwave medical imaging," IEEE Antennas Wireless Propagat. Lett., Vol. 11, 1592-1597, 2012.
doi:10.1109/LAWP.2013.2240569

20. Davy, M., T. Lepetit, J. de Rosny, C. Prada, and M. Fink, "Detection and imaging of human beings behind a wall using the Dort method," Progress In Electromagnetics Research, Vol. 110, 353-369, 2010.
doi:10.2528/PIER10091703

20. Zhang, W., "Through-the-wall target localization with time reversal Music method," Progress In Electromagnetics Research, Vol. 106, 75-89, 2010.
doi:10.2528/PIER10052408

21. Lu, T., K. Agarwal, Y. Zhong, and X. Chen, "Through-wall imaging: Application of subspace-based optimization method," Progress In Electromagnetics Research, Vol. 102, 351-366, 2010.
doi:10.2528/PIER10020903

22. Catapano, I. and L. Crocco, "A qualitative inverse scattering method for through-the-wall imaging," IEEE Geosci. Remote Sens. Lett., Vol. 7, No. 4, 685-689, 2010.
doi:10.1109/LGRS.2010.2045473

23. Burkholder, R. J. and K. E. Browne, "Coherence factor enhancement of through-wall radar images," IEEE Antennas Wireless Propag. Lett., Vol. 9, 842-845, 2010.
doi:10.1109/LAWP.2010.2069078

24. Soldovieri, F., F. Ahmad, and R. Solimene, "Validation of microwave tomographic inverse scattering approach via through-the-wall experiments in semicontrolled conditions," IEEE Geosci. IEEE Geosci., Vol. 8, No. 1, 123-127, 2011.

25. Dorn, O. and D. Lesselier, "Special issue on `Electromagnetic inverse problems: Emerging methods and novel applications'," Inverse Probl., Vol. 26, No. 7, 2010.

26. Bucci, O. M. and G. Franceschetti, "On the degrees of freedom of scattered fields," IEEE Trans. Antennas Propag., Vol. 37, No. 7, 918-926, 1989.
doi:10.1109/8.29386

27. Bucci, O. M. and T. Isernia, "Electromagnetic inverse scattering: Retrievable information and measurement strategies," Radio Sci., Vol. 32, No. 6, 2123-2137, 1997.
doi:10.1029/97RS01826

28. Isernia, T., V. Pascazio, and R. Pierri, "On the local minima in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sens., Vol. 39, No. 7, 1596-1607, 2001.
doi:10.1109/36.934091

29. D'Urso, M., M., T. Isernia, and A. F. Morabito, "On the solution of 2-D inverse scattering problems via source-type integral equations," IEEE Trans. Geosci. Remote Sens., Vol. 48, No. 3, 1186-1198, 2010.
doi:10.1109/TGRS.2009.2032175

30. Agarwal, K., L. Pan, and X. Chen, "Subspace-based optimization method for reconstruction of 2-D complex anisotropic dielectric objects ," IEEE Trans. Microw. Theory Tech., Vol. 58, No. 4, 1065-1074, 2010.
doi:10.1109/TMTT.2010.2042523

31. Estatico, C., G. Bozza, A. Massa, M. Pastorino, and A. Randazzo, "A two steps inexact-Newton method for electromagnetic imaging of dielectric structures from real data," Inverse Probl., Vol. 21, No. 6, 81-94, 2005.
doi:10.1088/0266-5611/21/6/S07

32. Barriere, P., J. Idier, Y. Goussard, and J. Laurin, "Fast solutions of the 2D inverse scattering problem based on a TSVD approximation of the internal field for the forward model," IEEE Trans. Antennas Propag., Vol. 58, 4015-4024, 2010.
doi:10.1109/TAP.2010.2078440

33. Li, J., X. Wang, and T. Wang, "On the validity of Born approximation," Progress In Electromagnetics Research, Vol. 107, 219-237, 2010.
doi:10.2528/PIER10070504

34. Poli, L., G. Oliveri, and A. Massa, "Microwave imaging within the first-order Born approximation by means of the contrast-field Bayesian compressive sensing," IEEE Trans. Antennas Propagat., Vol. 60, 2865-2879, 2012.
doi:10.1109/TAP.2012.2194676

35. Oliveri, G., L. Poli, P. Rocca, and A. Massa, "Bayesian compressive optical imaging within the Rytov approximation," Opt. Lett., Vol. 37, 1760-1762, 2012.
doi:10.1364/OL.37.001760

36. 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, 2010.
doi:10.1109/TAP.2009.2035997

37. Oliveri, G. and A Bayesian-compressive-, "A Bayesian-compressive-sampling-based inversion for imaging sparse scatterers ," IEEE Trans. Geosci. Remote Sens., Vol. 49, No. 10, 3993-4006, 2011.
doi:10.1109/TGRS.2011.2128329

38. Poli, L., G. Oliveri, P. Rocca, and A. Massa, "Bayesian compressive sensing approaches for the reconstruction of two-dimensional sparse scatterers under TE illuminations ," IEEE Trans. Geosci. Remote Sens., Vol. 51, 2920-2936, 2013.

39. Viani, F., L. Poli, G. Oliveri, F. Robol, and A. Massa, "Spars scatterers imaging through approximated multi-task compressive sensing strategies ," Microwave Opt. Technol. Lett., Vol. 55, 1553-1557, 2013.
doi:10.1002/mop.27612

40. Rekanos, I. T., "Neural-network-based inverse-scattering technique for online microwave medical imaging," IEEE Trans. Mag., Vol. 38, No. 2, 1061-1064, 2002.
doi:10.1109/20.996272

41. Massa, A., A. Boni, and M. Donelli, "A classification approach based on SVM for electromagnetic subsurface sensing," IEEE Trans. Geosci. Remote Sens., Vol. 43, No. 9, 2084-2093, 2005.
doi:10.1109/TGRS.2005.853186

42. Bermani, E., Bermani, E., A. Boni, A. Kerhet, and A. Massa, "Kernels evaluation of SVM-based estimators for inverse scattering problems ," Progress In Electromagnetics Research, Vol. 53, 167-188, 2005.
doi:10.2528/PIER04090801

43. Miller, E. L. and A. S. Willsky, "A multiscale, statistically based inversion scheme for linearized inverse scattering problems," IEEE Trans. Geosci. Remote Sens., Vol. 34, No. 2, 346-357, 1996.
doi:10.1109/36.485112

44. Caorsi, S., M. Donelli, and A. Massa, "Detection, location and imaging of multiple scatterers by means of the iterative multiscaling method ," IEEE Trans. Microw. Theory Tech., Vol. 52, No. 4, 1217-1228, 2004.
doi:10.1109/TMTT.2004.825699

45. Poli, L. and P. Rocca, "Exploitation of TE-TM scattering data for microwave imaging through the multi-scaling reconstruction strategy ," Progress In Electromagnetics Research, Vol. 99, 245-260, 2009.
doi:10.2528/PIER09101105

46. Rocca, P., M. Donelli, G. L. Gragnani, and A. Massa, "Iterative multi- resolution retrieval of non-measurable equivalent currents for imaging purposes ," Inverse Probl., Vol. 25, No. 5, 1-25, 2009.

47. Rocca, P., "Multi-resolution retrieval of non-measurable equivalent currents in microwave imaging problems --- Experimental assessment," Progress In Electromagnetics Research, Vol. 96, 267-285, 2009.
doi:10.2528/PIER09072004

48. Lee, Y.-S., C.-C. Chiu, and Y.-S. Lin, "Electromagnetic imaging for an imperfectly conducting cylinder buried in a three-layer structure by the genetic algorithm," Progress In Electromagnetics Research, Vol. 48, 27-44, 2004.
doi:10.2528/PIER03120304

49. Chen, X., K.-M. Huang, and X.-B. Xu, "Microwave imaging of buried inhomogeneous objects using parallel genetic algorithm combined with FDTD method ," Progress In Electromagnetics Research, Vol. 53, 283-298, 2005.
doi:10.2528/PIER04102902

50. Semnani, A. and M. Kamyab, "An enhanced method for inverse scattering problems using Fourier series expansion in conjunction with FDTD and PSO ," Progress In Electromagnetics Research, Vol. 76, 45-64, 2007.
doi:10.2528/PIER07061204

51. Huang, C.-H., C.-C. Chiu, C.-L. Li, and K.-C. Chen, "Time domain inverse scattering of a two-dimensional homogenous dielectric object with arbitrary shape by particle swarm optimization ," Progress In Electromagnetics Research, Vol. 82, 381-400, 2008.
doi:10.2528/PIER08031904

52. Semnani, A. and M. Kamyab, "Truncated cosine Fourier series expansion method for solving 2-D inverse scattering problems," Progress In Electromagnetics Research, Vol. 81, 73-97, 2008.
doi:10.2528/PIER07122404

53. 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 Trans. Antennas Propag., Vol. 56, No. 10, 3224-3224, 2008.
doi:10.1109/TAP.2008.929504

54. Rocca, P., M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, "Evolutionary optimization as applied to inverse scattering problems ," Inverse Probl., Vol. 25, 1-41, 2009.

55. Semnani, A., I. T. Rekanos, M. Kamyab, and T. G. Papadopoulos, "Two-dimensional microwave imaging based on hybrid scatterer representation and differential evolution ," IEEE Trans. Antennas Propag., Vol. 58, No. 10, 3289-3298, 2010.
doi:10.1109/TAP.2010.2055793

56. Moore, R., "Interval Analysis," Prentice-Hall, 1966.

57. Hansen, E., "On solving systems of equations using interval arithmetic," Mathematics of Computation, Vol. 22, No. 102, 374-384, 1968.
doi:10.1090/S0025-5718-1968-0229411-4

58. Neumaier, A., "Interval iteration for zeros of systems of equations," BIT, Vol. 25, No. 1, 256-273, 1985.
doi:10.1007/BF01935003

59. Saxena, G. and D. A. Lowther, "The use of interval mathematics in electromagnetic design," IEEE Trans. Mag., Vol. 37, No. 5, 3588-3591, 2001.
doi:10.1109/20.952668

60. Egiziano, L., P. Lamberti, G. Spagnuolo, and V. Tucci, "Robust design of electromagnetic systems based on interval Taylor extension applied to a multiquadric performance function," IEEE Trans. Mag., Vol. 44, No. 6, 1134-1137, 2008.
doi:10.1109/TMAG.2007.916163

61. Soares, G. L., A. Arnold-Bos, L. Jaulin, C. A. Maia, and J. A. Vasconcelos, "An interval-based target tracking approach for range-only multistatic radar," IEEE Trans. Mag., Vol. 44, No. 6, 1350-1353, 2008.
doi:10.1109/TMAG.2007.916286

62. Chew, W. C., Waves and Fields in Inhomogeneous Media,, IEEE Press, New York, 1995.

63. Slaney, M., A. C. Kak, and L. E. Larsen, "Limitations of imaging with first-order diffraction tomography," IEEE Trans. Microw. Theory Tech., Vol. 32, No. 8, 860-874, 1984.
doi:10.1109/TMTT.1984.1132783

64. Caorsi, S. and M. Pastorino, "Two-dimensional microwave imaging approach based on a genetic algorithm," IEEE Trans. Antennas Propag., Vol. 48, No. 3, 370-373, 2000.
doi:10.1109/8.841897

65. Richmond, J. H., "Scattering by a dielectric cylinder of arbitrary cross section shape," IEEE Trans. Antennas Propag., Vol. 13, No. 3, 334-341, 1965.
doi:10.1109/TAP.1965.1138427

66. Colton, D. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, Germany, 1992.
doi:10.1007/978-3-662-02835-3

67. Hansen, E., "Global optimization using interval analysis --- The multi-dimensional case," Numer. Math., Vol. 34, 247-270, 1980.
doi:10.1007/BF01396702

68. Rall, L. B., Computational Solution of Nonlinear Operator Equations, Wiley, New York, 1969.

69. Caorsi, S., M. Donelli, and A. Massa, "Analysis of the stability and robustness of the iterative multi-scaling approach for microwave imaging applications ," Radio Sci., Vol. 39, 1-17, 2004.

Dr. Paolo Rocca

Mr. Matteo Carlin

Dr. Luca Manica

Prof. Andrea Massa