This paper presents a Fisher information based Bayesian approach to analysis and design of the regularization and preconditioning parameters used with gradient based inverse scattering algorithms. In particular, a one-dimensional inverse problem is considered where the permittivity and conductivity profiles are unknown and the input data consist of the scattered field over a certain bandwidth. A priori parameter modeling is considered with linear, exponential and arctangential parameter scalings and robust preconditioners are obtained by choosing the related scaling parameters based on a Fisher information analysis of the known background. The Bayesian approach and a principal parameter (singular value) analysis of the stochastic Cramer-Rao bound provide a natural interpretation of the regularization that is necessary to achieve stable inversion, as well as an indicator to predict the feasibility of achieving successful reconstruction in a given problem set-up. In particular, the Tikhonov regularization scheme is put into a Bayesian estimation framework. A time-domain least-squares inversion algorithm is employed which is based on a quasi-Newton algorithm together with an FDTD-electromagnetic solver. Numerical examples are included to illustrate and verify the analysis.
"A Priori Modeling for Gradient Based Inverse Scattering Algorithms," Progress In Electromagnetics Research B,
Vol. 16, 407-432, 2009. doi:10.2528/PIERB09060805
1. Baussard, A. and O. V. D. Premel, "A bayesian approach for solving inverse scattering from microwave laboratory-controlled data," Inverse Problems, Vol. 17, No. 6, 1659-1669, 2001. doi:10.1088/0266-5611/17/6/309
2. Bertero, M., "Linear inverse and ill-posed problems," Advances in Electronics and Electron Physics, Vol. 75, 1-120, 1989.
3. Bucci, O. M., L. Crocco, T. Isernia, and V. Pascazio, "Subsurface inverse scattering problems: Quantifying qualifying and achieving the available information," IEEE Trans. Geoscience and Remote Sensing, Vol. 39, No. 11, 2527-2538, Nov. 2001. doi:10.1109/36.964991
4. Chew, W. C. and Y. M.Wang, "Reconstruction of twodimensional permittivity distribution using the distorted born iterative method," IEEE Trans. Med. Imaging, Vol. 9, No. 2, 218-225, 1990. doi:10.1109/42.56334
5. Colton, D. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992.
6. Fhager, A., M. Gustafsson, S. Nordebo, and M. Persson, "A statistically based preconditioner for two-dimensional microwave tomography," Proceedings of the Second International Workshop on Computational Advances in Multi-sensor Adaptive Processing, 173-176, Oct. 2007. doi:10.1109/CAMSAP.2007.4497993
7. Fhager, A. and M. Persson, "Comparison of two image reconstruction algorithms for microwave tomography," Radio Sci., Vol. 40, RS3017, Jun. 2005. doi:10.1029/2004RS003105
8. Fletcher, R., Practical Methods of Optimization, John Wiley & Sons, Ltd., Chichester, 1987.
9. Goharian, M., M. Soleimani, and G. Moran, "A trust region subproblem for 3D electrical impedance tomography inverse problem using experimental data," Progress In Electromagnetics Research, Vol. 94, 19-32, 2009.
10. Greenbaum, A., Iterative Methods for Solving Linear Systems, SIAM Press, Philadelphia, 1997.
11. Gustafsson, M., "Wave Splitting in Direct and Inverse Scattering Problems,", PhD thesis, Lund University, Department of Electromagnetic Theory, P. O. Box 118, Lund S-22100, Sweden, 2000, http://www.eit.lth.se.
12. Gustafsson, M. and S. He, "An optimization approach to twodimensional time domain electromagnetic inverse problems," Radio Sci., Vol. 35, No. 2, 525-536, 2000. doi:10.1029/1999RS900091
13. Gustafsson, M. and S. Nordebo, "Cramer --- Rao lower bounds for inverse scattering problems of multilayer structures," Inverse Problems, Vol. 22, 1359-1380, 2006. doi:10.1088/0266-5611/22/4/014
14. 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
16. Kaipio, J. and E. Somersalo, Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.
17. Kay, S. M., Fundamentals of Statistical Signal Processing, Estimation Theory, Prentice-Hall, Inc., NJ, 1993.
18. Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations, SIAM Press, Philadelphia, 1995.
19. Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996.
20. Knapp, C. H. and G. C. Carter, "The generalized correlation method for estimation of time delay," IEEE Trans. Acoustics, Speech, and Signal Process., Vol. 24, No. 4, 320-327, 1976. doi:10.1109/TASSP.1976.1162830
21. Kristensson, G. and R. J. Krueger, "Direct and inverse scattering in the time domain for a dissipative wave equation. Part 1: Scattering operators," J. Math. Phys., Vol. 27, No. 6, 1667-1682, 1986. doi:10.1063/1.527083
22. Marengo, E. A. and R. W. Ziolkowski, "Nonradiating and minimum energy sources and their fields: Generalized source inversion theory and applications," IEEE Trans. Antennas Propagat., Vol. 48, No. 10, 1553-1562, Oct. 2000. doi:10.1109/8.899672
23. Nordebo, S., A. Fhager, M. Gustafsson, and M. Persson, "A systematic approach to robust preconditioning for gradient based inverse scattering algorithms," Inverse Problems, Vol. 24, No. 2, 025027, 2008. doi:10.1088/0266-5611/24/2/025027
24. Nordebo, S., M. Gustafsson, and B. Nilsson, "Fisher information analysis for two-dimensional microwave tomography," Inverse Problems, Vol. 23, 859-877, 2007. doi:10.1088/0266-5611/23/3/001
25. Nordebo, S., M. Gustafsson, and K. Persson, "Sensitivity analysis for antenna near-field imaging," IEEE Trans. Signal Process., Vol. 55, No. 1, 94-101, Jan. 2007. doi:10.1109/TSP.2006.885742
26. Nordebo, S. and M. Gustafsson, "Statistical signal analysis for the inverse source problem of electromagnetics," IEEE Trans. Signal Process., Vol. 54, No. 6, 2357-2361, Jun. 2006. doi:10.1109/TSP.2006.873503
27. Pierri, R., A. Liseno, and F. Soldovieri, "Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach," IEEE Trans. Antennas Propagat., Vol. 49, No. 9, 1333-1343, Sep. 2001. doi:10.1109/8.947025
28. Pierri, R. and F. Soldovieri, "On the information content of the radiated fields in the near zone over bounded domains," Inverse Problems, Vol. 14, No. 2, 321-337, 1998. doi:10.1088/0266-5611/14/2/008
29. Soleimani, M., C. N. Mitchell, R. Banasiak, R. Wajman, and A. Adler, "Four-dimensional electrical capacitance tomography imaging using experimental data," Progress In Electromagnetics Research, Vol. 90, 171-186, 2009. doi:10.2528/PIER09010202
30. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-di®erence Time-domain Method, Artech House, Boston, London, 2000.
31. Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, Society for Industrial and Applied Mathematics, Philadelphia, 2005.
32. Tipping, M. E., "Sparse Bayesian learning and the relevance vector machine," J. Mach. Learning Res., Vol. 1, No. 2, 211-244, 2001. doi:10.1162/15324430152748236
33. Van Trees, H. L., Detection, Estimation and Modulation Theory, Part I, John Wiley & Sons, Inc., New York, 1968.
34. Yu, Y. and L. Carin, "Three-dimensional Bayesian inversion with application to subsurface sensing," IEEE Trans. Geoscience and Remote Sensing, Vol. 45, No. 5, 1258-1270, 2007. doi:10.1109/TGRS.2007.894932
35. Yu, Y., B. Krishnapuram, and L. Carin, "Inverse scattering with sparse Bayesian vector regression," Inverse Problems, Vol. 20, No. 6, S217-S231, 2004. doi:10.1088/0266-5611/20/6/S13