A modified gradient method for two-dimensional problems in tomography. (English) Zbl 0757.65133

Kleinman, R. E.; Van den Berg, P. M.

J. Comput. Appl. Math. 42, No. 1, 17-35 (1992).

A method for reconstruction of a bounded two-dimensional inhomogeneous object from a measured scattered field is presented. The governing domain integral equation is solved iteratively by a successive overrelaxation technique.
Convergence of this process is established for indices of refraction much larger than required for convergence of the Born approximation. Numerical results are presented for a number of representative two-dimensional objects.

Reviewer: W.Heinrichs (Düsseldorf)

MSC:

65R10	Numerical methods for integral transforms
65R30	Numerical methods for ill-posed problems for integral equations
44A12	Radon transform
92C55	Biomedical imaging and signal processing

Keywords:

tomography; scattering; gradient method; two-dimensional; inverse problem; profile inversion; reconstruction; successive overrelaxation; Convergence; Numerical results

References:

[1]	Chew, W. C.; Wang, Y. M., Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method, IEEE Trans. Medical Imaging, 9, 218-225 (1990)
[2]	Devaney, A. J., A filtered backprojection algorithm for diffraction tomography, Ultrasonic Imaging, 4, 336-360 (1982)
[3]	Groetsch, C. W., The Theory of Tikhonov Regularization of Fredholm Equations of the First Kind (1984), Pitman: Pitman London · Zbl 0545.65034
[4]	Habashy, T. M.; Chow, E. Y.; Dudley, D. G., Profile inversion using the renormalized source-type integral equation approach, IEEE Trans. Antennas and Propagation, AP-38, 668-682 (1990)
[5]	Habashy, T. M.; Mittra, R., On some inverse methods in electromagnetics, J. Electromagn. Waves Appl., 1, 25-58 (1987)
[6]	Hestenes, M., Conjugate Directions in Optimization (1980), Springer: Springer New York
[7]	Kleinman, R. E.; Roach, G. F.; Schuetz, L. S.; Shirron, J.; Berg, P. M.van den, An over-relaxation method for the iterative solution of integral equations in scattering problems, Wave Motion, 12, 2, 161-170 (1990) · Zbl 0706.73078
[8]	Kleinman, R. E.; Roach, G. F.; Berg, P. M.van den, A convergent Born series for large refractive indices, J. Opt. Soc. Amer. A, 7, 890-897 (1990)
[9]	Kleinman, R. E.; Berg, P. M.van den; Gmitro, A. F.; Idell, P. S.; La Haie, I. J., Profile inversion via successive overrelaxation, Digital Image Synthesis and Inverse Optics, 1351, 129-139 (1990), Proc. SPIE
[10]	Kleinman, R. E.; Berg, P. M.van den, Non-linearized approach to profile inversion, Internat. J. Imaging Systems and Technology, 2, 119-126 (1991)
[11]	Kleinman, R. E.; Berg, P. M.van den; Sarkar, T. K., Iterative methods for solving integral equations, Application of Conjugate Gradient Methods to Electromagnetics and Signal Analysis, 5 (1991), Elsevier: Elsevier New York, Chapter 3
[12]	Lesselier, D., Optimization techniques and inverse problems: Reconstruction of conductivity profiles in the time domain, IEEE Trans. Antennas and Propagation, AP-30, 59-65 (1982)
[13]	Petryshyn, W. V., On a general iterative method for the approximate solution of linear operator equations, Math. Comp., 17, 1-10 (1963) · Zbl 0111.31701
[14]	Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T., Numerical Recipes, The Art of Scientific Computing (1986), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0587.65003
[15]	Richmond, J. H., Scattering by a dielectric cylinder of arbitrary cross section shape, IEEE Trans. Antennas and Propagation, AP-13, 334-341 (1965)
[16]	Roger, A., Newton-Kantorovitch algorithm applied to an electromagnetic inverse scattering problem, IEEE Trans. Antennas and Propagation, AP-29, 232-238 (1981) · Zbl 0947.65503
[17]	Roger, A.; Maystre, D.; Cadilhac, M., On a problem of inverse scattering in optics: The dielectric inhomogeneous medium, J. Optics (Paris), 9, 83-90 (1978)
[18]	Tabbara, W.; Duchêne, B.; Pichot, Ch.; Lesselier, D.; Chommeloux, L.; Joachimowicz, N., Diffraction tomography: Contribution to the analysis of applications in microwaves and ultrasonics, Inverse Problems, 4, 305-331 (1988)
[19]	Tijhuis, A. G.; Worm, C.van der, Iterative approach to the frequency-domain scattering solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab, IEEE Trans. Antennas and Propagation, AP-32, 711-716 (1984)
[20]	Tobocman, W., Iterative inverse scattering method employing Gram-Schmidt orthogonalization, J. Comput. Phys., 64, 230-245 (1986) · Zbl 0609.65097
[21]	van den Berg, P. M., Iterative computational techniques in scattering based upon the integrated square error criterion, IEEE Trans. Antennas and Propagation, AP-32, 1063-1071 (1984)
[22]	van den Berg, P. M.; Kleinman, R. E., Iterative solution of integral equations in scattering problems, (Datta, S. K.; Achenbach, J. D.; Rajapakse, Y. S., Elastic Waves and Ultrasonic Nondestructive Evaluation (1990), North-Holland: North-Holland Amsterdam), 57-62 · Zbl 1008.74049
[23]	Varga, R. S., Matrix Iterative Analysis (1962), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
[24]	Wang, Y. M.; Chew, W. C., An iterative solution of two-dimensional electromagnetic inverse scattering problem, J. Imaging Systems, 1, 100-108 (1989)
[25]	Xiong, Z., Dreidimensionale geoelektromagnetische Umkehraufgaben, (Dissertation (1989), Univ. Göttingen)