Overview and classification of some regularization techniques for the Gauss-Newton inversion method applied to inverse scattering problems

IEEE Transactions on Antennas and Propagation

Volumn 57, Issue 9, 2009, Pages 2658-2665

  Mojabi, Puyan ^a   LoVetri, Joe ^a  

^a UNIVERSITY OF MANITOBA   (Canada)

Author keywords

Inverse scattering; Regularization

Indexed keywords

ELECTROMAGNETIC INVERSE SCATTERING PROBLEMS; GAUSS-NEWTON; INVERSE SCATTERING; INVERSE SCATTERING PROBLEMS; INVERSION METHODS; NONLINEAR DATA; PENALTY METHODS; PROJECTION METHOD; QUADRATIC FORM; REGULARIZATION; REGULARIZATION STRATEGIES; REGULARIZATION TECHNIQUE;

IMAGE RETRIEVAL; INVERSE PROBLEMS; NUMBER THEORY;

SCATTERING;

EID: 70349263705     PISSN: 0018926X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAP.2009.2027161     Document Type: Review

References (50)

1. M. Pastorino, S. Caorsi, and A. Massa, "A global optimization technique for microwave nondestructive evaluation," IEEE. Trans. Instrum. Meas., vol. 51, no. 4, pp. 666-673, Aug. 2002.
2. W. H. Weedon, W. C. Chew, and P. E. Mayes, "A step-frequency radar imaging system for microwave nondestructive evaluation," Progr. Electromagn. Res., vol. 28, pp. 121-146, 2000.
3. A. Abubakar, T. M. Habashy, V. L. Druskin, L. Knizhnerman, and D. Alumbaugh, "2.5D forward and inverse modeling for interpreting low-frequency electromagnetic measurements," Geophys., vol. 73, no. 4, pp. F165-F177, Jul.-Aug. 2008.
4. A. Abubakar and P. M. van den Berg, "Non-linear three-dimensional inversion of cross-well electrical measurements," Geophy. Prospect. vol. 48, pp. 109-134, 2000.
5. L. P. Song, C. Yu, and Q. H. Liu, "Through-wall imaging (TWI) by radar: 2-D tomographic results and analyses," IEEE Trans. Geosci. Remote Sensing, vol. 43, no. 12, pp. 2793-2798, Dec. 2005.
6. T. Rubaek, P. M. Meaney, P. Meincke, and K. D. Paulsen, "Nonlinear microwave imaging for breast-cancer screening using Gauss-Newton's method and the CGLS inversion algorithm," IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2320-2331, Aug. 2007.
7. S. Y. Semenov, R. H. Svenson, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, V. G. Posukh, A. Pavlovsky, P. N. Repin, A. N. Starostin, B. A. Voinov, M. Taran, G. P. Tastis, and V. Y. Baranov, "Three-dimensional microwave tomography: Initial experimental imaging of animals," IEEE Trans. Biomed. Eng., vol. 49, no. 1, pp. 55-63, Jan. 2002.
8. A. Abubakar, P. M. van den Berg, and J. J. Mallorqui, "Imaging of biomedical data using a multiplicative regularized contrast source inversion method," IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1761-1777, July 2002.
9. W. C. Chew and Y. M. Wang, "Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method," IEEE Trans. Med. Imaging, vol. 9, no. 2, pp. 218-225, 1990.
10. G. L.Wang, W. C. Chew, T. J. Cui, A. A. Aydiner, D. L.Wright, and D. V. Smith, "3D near-to-surface conductivity reconstruction by inversion of VETEM data using the distorted Born iterative method," Inverse Problems, vol. 20, pp. S195-S216, 2004.
11. N. Joachimowicz, C. Pichot, and J. P. Hugonin, "Inverse scattering: An iterative numerical method for electromagnetic imaging," IEEE Trans. Antennas Propag., vol. 39, no. 12, pp. 1742-1752, Dec. 1991.
12. N. Joachimowicz, J. J. Mallorqui, J. C. Bolomey, and A. Broquetas, "Convergence and stability assessment of Newton-Kantorovich reconstruction algorithms for microwave tomography," IEEE Trans. Med. Imag., vol. 17, pp. 562-569, Aug. 1998.
13. A. Franchois and C. Pichot, "Microwave imaging-Complex permittivity reconstruction with a Levenberg-Marquardt method," IEEE Trans. Antennas Propag., vol. 45, no. 2, pp. 203-215, Feb. 1997.
14. A. E. Souvorov, A. E. Bulyshev, S. Y. Semenov, R. H. Svenson, A. G. Nazarov, Y. E. Sizov, and G. P. Tastis, "Microwave tomography: A two dimensional Newton iterative scheme," IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1654-1659, Nov. 1998.
15. A. G. Tijhuis, K. Belkebir, and A. C. S. Litman, "Theoretical and computational aspects of 2-D inverse profiling," IEEE Trans. Geosci. Remote Sensing, vol. 39, no. 6, pp. 1316-1330, 2001.
16. J. D. Zaeytijd, A. Franchois, C. Eyraud, and J. Geffrin, "Full-wave three-dimensional microwave imaging with a regularized Gauss-Newton method-theory and experiment," IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3279-3292, Nov. 2007.
17. A. E. Bulyshev, A. E. Souvorov, S. Y. Semenov, R. H. Svenson, A. G. Nazarov, Y. E. Sizov, and G. P. Tastis, "Three dimensional microwave tomography. Theory and computer experiments in scalar approximation," Inverse Problems, vol. 16, pp. 863-875, 2000.
18. T. M. Habashy and A. Abubakar, "A general framework for constraint minimization for the inversion of electromagnetic measurements," Progr. Electromagn. Res., vol. 46, pp. 265-312, 2004.
19. A. Franchois and A. G. Tijhuis, "A quasi-Newton reconstruction algorithm for a complex microwave imaging scanner environment," Radio Sci., vol. 38, no. 2, 2003.
20. S. Barkeshli and R. G. Lautzenheiser, "An iterative method for inverse scattering problems based on an exact gradient search," Radio Sci. vol. 29, pp. 1119-1130, 1994.
21. H. Harada, D. J. N. Wall, T. Takenaka, and M. Tanaka, "Conjugate gradient method applied to inverse scattering problem," IEEE Trans. Antennas. Propag., vol. 43, no. 8, pp. 784-792, Aug. 1995.
22. R. E. Kleinman and P. M. van den Berg, "A modified gradient method for two-dimensional problem in tomography," J. Comput. Appl. Math., vol. 42, no. 1, pp. 17-35, 1992.
23. P. M. van den Berg and R. E. Kleinman, "A contrast source inversion method," Inverse Problems, vol. 13, no. 6, pp. 1607-1620, 1997.
24. S. Caorsi, A. Massa, and M. A. Pastorino, "A computational technique based on a real-coded genetic algorithm for microwave imaging purposes," IEEE Trans. Geosci. Remote Sensing, vol. 38, no. 4, pp. 1697-1708, 2000.
25. L. Garnero, A. Franchois, J. P. Hugonin, C. Pichot, and N. Joachimowicz, "Microwave imaging-complex permittivity reconstruction by simulated annealing," IEEE Trans. Microw. Theory Tech., vol. 39, no. 11, pp. 1801-1807, Nov. 1991.
26. E. Chong and S. Zak, An Introduction to Optimization. New York: Wiley Interscience, 2001.
27. D. H. Brandwood, "A complex gradient operator and its application in adaptive array theory," Proc. Inst. Elect. Eng., vol. 130, no. 1, pt. F and H, pp. 11-16, 1983.
28. A. van den Bos, "Complex gradient and Hessian," Proc. Inst. Elect. Eng. Vision, Image Signal Process., vol. 141, no. 6, pp. 380-383, 1994.
29. P. R. McGillivray and D. W. Oldenburg, "Methods for calculating Fréchet derivatives and sensitivities for the nonlinear inverse problem: A comparative study," Geophys. Prospect., vol. 38, pp. 499-524, 1990.
30. M. S. Zhdanov, Geophys. Inverse Theory and Regularization. Amsterdam, The Netherlands: Elsevier, 2002.
31. A. Abubakar and P. M. van den Berg, "Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects," J. Comput. Phys., vol. 195, pp. 236-262, 2004.
32. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. New York: Wiley, 1977.
33. P. C. Hansen, "Numerical tools for analysis and solution of Fredholm integral equations of the first kind," Inverse Problems, vol. 8, pp. 849-872, 1992.
34. P. C. Hansen and D. P. O'leary, "The use of the L-curve in the regularization of discrete ill-posed problems," SIAM J. Sci. Comput. vol. 14, no. 6, pp. 1487-1503, Nov. 1993.
35. P. R. Johnston and R. M. Gulrajani, "Selecting the corner in the L-curve approach to Tikhonov regularization," IEEE Trans. Biomed. Eng., vol. 47, no. 9, pp. 1293-1296, Sep. 2000.
36. G. H. Golub, M. Heath, and G. Wahba, "Generalized cross validation as a method for choosing a good ridge parameter," Technometrics, vol. 21, pp. 215-224, 1979.
37. A. Franchois, A. Joisel, C. Pichot, and J.-C. Bolomey, "Quantitative microwave imaging with a 2.45-GHz planar microwave camera," IEEE Trans. Med. Imaging, vol. 17, no. 4, pp. 550-561, Aug. 1998.
38. A. Bjork, Numerical Methods for Least Squares Problems. Philadelphia, PA: SIAM, 1996.
39. P. C. Hansen, Rank-Deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion. Philadelphia, PA: SIAM, 1998.
40. J. D. Shea, P. Kosmas, S. C. Hagness, and B. D. van Veen, "Three-dimensional microwave breast imaging: A bounded, multi-frequency inverse scattering solution on a uniform voxel mesh," presented at the XXIX General Assembly of Int. Union of Radio Sci., Chicago, IL, Aug. 2008.
41. J. Chung, J. G. Nagy, and D. P. O'leary, "A weighted-GCV method for Lanczos-hybrid regularization," Electron. Trans. Numer. Analysis, vol. 28, pp. 149-167, 2008.
42. M. E. Kilmer and D. P. O'leary, "Choosing regularization parameters in iterative methods for ill-posed problems," SIAM J. Matrix Anal. Appl., vol. 22, no. 4, pp. 1204-1221, 2001.
43. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. New York: Springer-Verlag, 1992.
44. R. Pierri, A. Liseno, and F. Soldovieri, "Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach," IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1333-1343, Sep. 2001.
45. P. C. Hansen, "The discrete Picard condition for discrete ill-posed problems," BIT, vol. 30, pp. 658-672, 1990.
46. P. C. Hansen, "Analysis of discrete ill-posed problems by means of the L-curve," SIAM Review, vol. 34, no. 4, pp. 561-580, Dec. 1992.
47. P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, "Deterministic edge-preserving regularization in computed imaging," IEEE Trans. Image Processing, vol. 6, no. 2, pp. 298-311, Feb. 1997.
48. G. H. Golub and C. F. V. Loan, Matrix Computations. Baltimore, MD: Johns Hopkins Univ. Press, 1996.
49. T. K. Jensen, "Stabilization Algorithms for Large-Scale Problems," Ph.D. dissertation, Tech. Univ. Denmark, Kongens Lyngby, Denmark, 2006.
50. J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1998.