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ADAPTING THE NORMALIZED CUMULATIVE PERIODOGRAM PARAMETER-CHOICE METHOD TO THE TIKHONOV REGULARIZATION OF 2-D/TM ELECTROMAGNETIC INVERSE SCATTERING USING BORN ITERATIVE METHOD

By P. Mojabi and J. LoVetri

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Abstract:
A new method of choosing the regularization parameter, originally developed for a general class of discrete ill-posed problems, is investigated for electromagnetic inverse scattering problems that are formulated using a penalty method. This so-called Normalized Cumulative Periodogram (NCP) parameter-choice method uses more than just the norm of the residual to determine the regularization parameter, and attempts to choose the largest regularization parameter that makes the residual resemble white noise. This is done by calculating the NCP of the residual vector for each choice of the regularization parameter, starting from large values and stopping at the first parameter which puts the NCP inside the Kolmogorov- Smirnov limits. The main advantage of this method, as compared, for example, to the L-curve and Generalized Cross Validation (GCV) techniques, is that it is computationally inexpensive and therefore makes it an appropriate technique for large-scale problems arising in inverse imaging. In this paper, we apply this technique, with some modification, to the Tikhonov-regularized functional arising in the 2-D Transverse Magnetic (TM) inverse electromagnetic problem, which is formulated via an integral equation and solved using the Born iterative method (BIM).

Citation:
P. Mojabi and J. LoVetri, "Adapting the Normalized Cumulative Periodogram Parameter-Choice Method to the Tikhonov Regularization of 2-D/TM Electromagnetic Inverse Scattering Using Born Iterative Method," Progress In Electromagnetics Research M, Vol. 1, 111-138, 2008.
doi:10.2528/PIERM08012401

References:
1. Semenov, S. Y., V. G. Posukh, A. E. Bulyshev, and T. C. Williams, "Microwave tomographic imaging of the heart in intact swine," Journal of Electromagnetic Waves a Applications, Vol. 20, 873-890, 2006.
doi:10.1163/156939306776149897

2. Guo, B., Y. Wang, and J. Li, "Microwave imaging via adaptive beamforming methods for breast cancer detection," Journal of Electromagnetic Waves and Applications, Vol. 20, No. 1, 53-63, 2006.
doi:10.1163/156939306775777350

3. Yan, L. P., K. M. Huang, and Liu C. J., "A noninvasive method for determining dielectric properties of layered tissues on human back," Journal of Electromagnetic Waves and Applications, Vol. 21, 1829-1843, 2007.

4. Huang, K., X. B. Xu, and L. P. Yan, "A new noninvasive method for determining the conductivity of tissue embedded in multilayer biological structure," Journal of Electromagnetic Waves and Applications, Vol. 16, 851-860, 2002.
doi:10.1163/156939302X00192

5. Davis, S. K., E. J. Bond, X. Li, S. C. Hagness, and B. D. van Veen, "Microwave imaging via space-time beamforming for early detection of breast cancer: Beamformer design in the frequency domain," Journal of Electromagnetic Waves and Applications , Vol. 17, No. 2, 357-381, 2003.
doi:10.1163/156939303322235860

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

7. Weedon, W. H., W. C. Chew, and P. E. Mayes, "A step-frequency radar imaging system for microwave nondestructive evaluation," Progress In Electromagnetics Research, Vol. 28, 121-146, 2000.
doi:10.2528/PIER99062501

8. Roger, A. and F. Chapel, "Iterative methods for inverse problems," Progress in Electromagnetics Research, Vol. 05, 423-454, 1991.

9. Van den Berg, P. M. and A. Abubakar, "Contrast source inversion: State of art," Progress in Electromagnetics Research, Vol. 34, 189-218, 2001.
doi:10.2528/PIER01061103

10. Habashy, T. M. and A. Abubakar, "A general framework for constraint minimization for the inversion of electromagnetic measurements," Progress in Electromagnetics Research, Vol. 46, 265-312, 2004.
doi:10.2528/PIER03100702

11. Wang, Y. M. and W. C. Chew, "An iterative solution of twodimensional electromagnetic inverse scattering problem," Int. J. Imaging Syst. Technol., Vol. 1, 100-108, 1989.
doi:10.1002/ima.1850010111

12. Chew, W. C. and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Transactions on Medical Imaging, Vol. 9, 218-225, 1990.
doi:10.1109/42.56334

13. Habashy, T. M. and R. J. Mitra, "On some inverse methods in electromagnetics," Journal of Electromagnetic Waves and Applications, Vol. 1, 25-58, 1987.

14. Rekanos, I. T., "Time-domain inverse scattering using Lagrange multipliers: An iterative FDTD-based optimization technique," Journal of Electromagnetic Waves and Applications, Vol. 17, No. 2, 271-289, 2003.
doi:10.1163/156939303322235824

15. Kleinman, R. E. and P. M. van den Berg, "A modified gradient method for two-dimensional problems in tomography," Journal of Computational and Applied Mathematics, Vol. 42, 17-35, 1992.
doi:10.1016/0377-0427(92)90160-Y

16. Takenaka, T., H. Jia, and T. Tanaka, "Microwave imaging of electrical property distributions by a forward-backward time-stepping method," Journal of Electromagnetic Waves and Applications, Vol. 14, No. 12, 1609-1626, 2000.
doi:10.1163/156939300X00383

17. Belkebir, K., S. Bonnard, F. Pezin, P. Sabouroux, and M. Saillard, "Validation of 2D inverse scattering algorithms from multifrequency experimental data ," Journal of Electromagnetic Waves and Applications, Vol. 14, No. 12, 1637-1667, 2000.
doi:10.1163/156939300X00437

18. Zaeytijd, J. D., A. Franchois, C. Eyraud, and J. M. Geffrin, "Fullwave three-dimensional microwave imaging with a regularized Gauss-Newton method — Theory and experiment ," IEEE Transactions on Antennas and Propagation, Vol. 55, No. 11, 2007.

19. Tikhonov, A. N. and V. Y. Arsenin, Solution of Ill-posed Problems , John Wiley & Sons, New York, 1977.

20. Hansen, P. C., Rank-deficient and Discrete Ill-posed Problems , SIAM, Philadelphia, 1998.

21. Abubakar, A., P. M. van den Berg, T. M. Habashy, and H. Braunisch, "A multiplicative regularization approach for deblurring problems," IEEE Transactions on Image Processing, Vol. 13, No. 11, 1524-1532, 2004.
doi:10.1109/TIP.2004.836172

22. Hansen, P. C., "Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank," SIAM J. Sci. Stat. Comput., Vol. 11, 503-518, 1990.

23. Kilmer, M. E. and D. P. O'Leary, "Choosing regularization parameters in iterative methods for ill-posed problems," SIAM J. Matrix. Anal. Appl., Vol. 22, 1204-1221, 2001.
doi:10.1137/S0895479899345960

24. O'Leary, D. P. and J. A. Simmons, "A bidiagonalizationregularization procedure for large scale discretization of ill-posed problems," SIAM J. Sci. Statist. Comput., Vol. 2, 474-489, 1981.

25. Morozov, V. A., Methods for Solving Incorrectly Posed Problems, Springer-Verlag, New York, 1984.

26. Hansen, P. C., "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Review, Vol. 34, 561-580, 1992.
doi:10.1137/1034115

27. Hansen, P. C. and D. P. O'leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM J. Sci. Comp., Vol. 14, 1487-1503, 1993.
doi:10.1137/0914086

28. Golub, G., M. Heath, and G. Wahba, "Generalized crossvalidation as a method for choosing a good ridge parameter," Technometrics, Vol. 21, 215-223, 1979.
doi:10.2307/1268518

29. Iwama, N., M. Yamaguchi, K. Hattori, and M. Hayakawa, "GCVaided linear reconstruction of the wave distribution function for the ground-based direction finding of magnetospheric VLF/ELF waves," Journal of Electromagnetic Waves and Applications, Vol. 9, No. 5-6, 757-782, 1995.
doi:10.1163/156939395X00901

30. Belge, M., M. E. Kilmer, and E. L. Miller, "Efficient determination of multiple regularization parameters in a generalized L-curve framework," Inverse Problems, Vol. 18, 1161-1183, 2002.
doi:10.1088/0266-5611/18/4/314

31. Hansen, P. C., M. E. Kilmer, and R. H. Kjeldsen, "Exploiting residual information in the parameter choice for discrete ill-posed problems ," BIT Numerical Mathematics, Vol. 46, 41-59, 2006.
doi:10.1007/s10543-006-0042-7

32. Hansen, P. C., "The discrete picard condition for discrete ill-posed problems," BIT, Vol. 30, 658-672, 1990.
doi:10.1007/BF01933214

33. Geffrin, J. M., P. Sabouroux, and C. Eyraud, "Free space experimental scattering database continuation: Experimental setup and measurement precision ," Inverse Problems, Vol. 21, S117-S130, 2005.
doi:10.1088/0266-5611/21/6/S09

34. Chew, W. C. and J. H. Lin, "A frequency-hopping approach for microwave imaging of large inhomogeneous body," IEEE Microwave and Guided Wave Letters, Vol. 5, No. 12, 1995.
doi:10.1109/75.481854

35. Guest Editors'Introduction, "Testing inversion algorithms against experimental data: Inhomogeneous targets," Inverse Problems, S1-S3, 2005.

36. Zha, H. and P. C. Hansen, "Regularization and the general Guass-Markov linear model," Math. Comp., Vol. 55, 613-624, 1990.
doi:10.2307/2008436

37. Born, M. and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999.

38. Richmond, J. H., "Scattering by a dielectric cylinder of arbitrary cross section shape," IEEE Trans. Antennas. Propag., Vol. 13, 334-341, 1965.
doi:10.1109/TAP.1965.1138427

39. Volakis, J. L. and K. Barkeshli, "Applications of the conjugate gradient FFT method to radiation and scattering," Progress In Electromagnetics Research, Vol. 05, 159-239, 1991.

40. Tran, T. V. and A. McCowen, "A unified family of FFT-based methods for dielectric scattering problems ," Journal of Electromagnetics Waves and Applications, Vol. 7, No. 5, 739-763, 1993.
doi:10.1163/156939393X00840

41. Peng, Z. Q. and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: Marching-on-in-frequency approach," Journal of Electromagnetics Waves and Applications, Vol. 8, No. 8, 973-972, 1994.
doi:10.1163/156939394X00704

42. Hansen, P. C., "Numerical tools for analysis and solution of fredholm integral equation of the first kind," Inverse Problems, Vol. 8, 849-872, 1992.
doi:10.1088/0266-5611/8/6/005

43. Engl, H. W., M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 2000.

44. Hansen, P. C., "Perturbation bounds for discrete Tikhonov regularization," Inverse Problems, Vol. 5, L41-L44, 1989.
doi:10.1088/0266-5611/5/4/002

45. Hansen, P. C., "Regularization, GSVD and truncated GSVD," BIT, Vol. 29, 491-594, 1989.
doi:10.1007/BF02219234

46. Fuller, W. A., Introduction to Statistical Time Series, Wiley, New York, 1976.


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