volume | PIER Journals

Abstract

We discuss the problem of the reconstruction of the profile of an inhomogeneous object from scattered field data. Our starting point is the contrast source inversion method, where the unknown contrast sources and the unknown contrast are updated by an iterative minimization of a cost functional. We discuss the possibility of the presence of local minima of the nonlinear cost functional and under which conditions they can exist. Inspired by the successful implementation of the minimization of total variation and other edgepreserving algorithms in image restoration and inverse scattering, we have explored the use of these image-enhancement techniques as an extra regularization. The drawback of adding a regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation. Therefore, we first discuss the regularization as a multiplicative constraint and show that the weighting parameter is now completely prescribed by the error norm of the data equation and the object equation. Secondly, inspired by the edge-preserving algorithms, we introduce a new type of regularization, based on a weighted L2 total variation norm. The advantage is that the updating parameters in the contrast source inversion method can be determined explicitly, without the usual line minimization. In addition this new regularization shows excellent edge-preserving properties. Numerical experiments illustrate that the present multiplicative regularized inversion scheme is very robust, handling noisy as well as limited data very well, without the necessity of artificial regularization parameters.

1. Abubakar, A. and P. M. van den Berg, "Nonlinear inversion in electrode logging in a highly deviated formation with invasion using an oblique coordinate system," IEEE Trans. Geosci. Remote Sensing, Vol. 38, 25-38, 2000.
doi:10.1109/36.823898

2. Abubakar, A., P. M. van den Berg, and B. J. Kooij, "A conjugate gradient contrast source technique for 3D profile inversion," IEICE Trans. Electron., E83-C, 1864-1874, 2000.

3. Acar, R. and C. R. Vogel, "Analysis of bounded variation penalty methods for ill-posed problems," Inverse Problems, Vol. 10, 1217-1229, 1994.
doi:10.1088/0266-5611/10/6/003

4. Charbonnier, P., L. Blanc-F´eraud, G. Aubert, and M. Barlaud, "Deterministic edge-preserving regularization in computed imaging," IEEE Trans. Image Process., Vol. 6, 298-311, 1996.
doi:10.1109/83.551699

5. Blomgren, P., T. F. Chan, P. Mulet, and C. K. Wong, "Total variation image restorations: numerical methods and extensions," IEEE Proc. ICIP 97, 384-387, 1997.

6. Chan, T. F. and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. on Image Processing, Vol. 7, 370-375, 1998.
doi:10.1109/83.661187

7. Chew, W. C., "Complexity issues in inverse scattering problems," Proceedings of Antennas and Propagation Society, IEEE International Symposium, Vol. 3, 1627, 1999.

8. Colton, D., J. Coyle, and P. Monk, "Recent developments in inverse acoustic scattering theory," Siam Review, Vol. 42, 369-414, 2000.
doi:10.1137/S0036144500367337

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

10. Colton, D. and P. Monk, "The numerical solution of an inverse scattering problem for acoustic waves," IMA Journal of Applied Mathematics, Vol. 49, 162-184, 1992.
doi:10.1093/imamat/49.2.163

11. Dobson, D. C. and F. Santosa, "An image-enhancement technique for electrical impedance tomography," Inverse Problems, Vol. 10, 317-334, 1994.
doi:10.1088/0266-5611/10/2/008

12. Dobson, D. C. and F. Santosa, "Recovery of blocky images for noisy and blurred data," SIAM Journal of Applied Mathematics, Vol. 56, 1181-1198, 1996.
doi:10.1137/S003613999427560X

13. Dourthe, C., Ch. Pichot, J. Y. Dauvignac, L. Blanc-Feraud, and M. Barlaud, "Regularized bi-conjugate gradient algorithm for tomographic reconstruction of buried objects," IEICE Trans. Electron., Vol. E83-C, 1858-1863, 2000.

14. Habashy, T. M., M. L. Oristaglio, and A. T. de Hoop, "Simultaneous nonlinear reconstruction of two-dimensional permittivity and conductivity," Radio Science, Vol. 29, 1101-1118, 1994.
doi:10.1029/93RS03448

15. 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

16. Isernia, T., V. Pascazio, and R. Pierri, "On the local mimina in a tomographic imaging technique," IEEE Trans. Geosci. Remote Sensing, to appear.

17. Kleinman, R. E. and P. M. van den Berg, "A modified gradient method for two-dimensional problems in tomography," J. Computat. Appl. Math., Vol. 42, 17-35, 1992.
doi:10.1016/0377-0427(92)90160-Y

18. Kleinman, R. E. and P. M. van den Berg, "An extended range modified gradient technique for profile inversion," Radio Science, Vol. 28, 877-884, 1993.
doi:10.1029/93RS01076

19. Kleinman, R. E. and P. M. van den Berg, "Two-dimensional location and shape reconstruction," Radio Science, Vol. 29, 1157-1169, 1994.
doi:10.1029/93RS03445

20. Kohn, R. V. and A. McKenney, "Numerical implementation of a variational method for electrical impedance tomography," Inverse Problems, Vol. 6, 389-414, 1990.
doi:10.1088/0266-5611/6/3/009

21. Lesselier, D. and B. Duchene, "Wavefield inversion of objects in stratified environments. From backpropagation schemes to full solutions," Review of Radio Science, 1993–1996, R. Stone (ed.), 235-268, Oxford University Press, Oxford, 1996.

22. Lobel, P., L. Blanc-Feraud, Ch. Pichot, and M. Barlaud, "A new regularization scheme for inverse scattering," Inverse Problems, Vol. 13, 403-410, 1997.
doi:10.1088/0266-5611/13/2/013

23. Litman, A., D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a levelset," Inverse Problems, Vol. 14, 685-706, 1998.
doi:10.1088/0266-5611/14/3/018

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

25. Rudin, L., S. Osher, and C. Fatemi, "Nonlinear total variation based noise removal algorithm," Physica, Vol. 60D, 259-268, 1992.

26. Sabatier, P. C., "Past and future of inverse problems," J. Math. Phys., Vol. 41, 4082-4124, 2000.
doi:10.1063/1.533336

27. Belkebir, K. and A. G. Tijhuis, "Using multiple frequency information in the iterative solution of a two-dimensional non-linear inverse problem," Proc. PIERS 96: Progress In Electromagnetic Research Symposium, 353, Innsbruck, Austria, 1996.

28. Van den Berg, P. M., "Iterative computational techniques in scattering based upon the integrated square error criterion," IEEE Trans. Antennas and Propagation, Vol. 32, 1063-1071, 1981.
doi:10.1109/TAP.1984.1143213

29. Van den Berg, P. M., "Non-linear scalar inverse scattering: algorithms and applications," Scattering, R. Pike and P. C. Sabatier (eds.), Chapter 1.3.3., Academic Press, London, 2001, to appear.

30. Van den Berg, P. M., A. L. van Broekhoven, and A. Abubakar, "Extended contrast source inversion," Inverse Problems, Vol. 15, 1325-1344, 1999.
doi:10.1088/0266-5611/15/5/315

31. Van den Berg, P. M. and R. E. Kleinman, "A total variation enhanced modified gradient algorithm for profile reconstruction," Inverse Problems, Vol. 11, L5-L10, 1995.
doi:10.1088/0266-5611/11/3/002

32. Van den Berg, P. M. and R. E. Kleinman, "A contrast source inversion method," Inverse Problems, Vol. 13, 1607-1620, 1997.
doi:10.1088/0266-5611/13/6/013

33. Vogel, C. R. and M. E. Oman, "Iterative methods for total variation denoising," SIAM Journal of Scientific Computing, Vol. 17, 227-238, 1996.
doi:10.1137/0917016

34. Zhdanov, M. and G. Hursan, "3D electromagnetic inversion based on quasi-analytical approximation," Inverse Problems, Vol. 16, 1297-1322, 2000.
doi:10.1088/0266-5611/16/5/311

Dr. Peter Van den Berg

Dr. A. Abubakar