## Second-order iterative approach to inverse scattering: numerical results

JOSA A, Vol. 17, Issue 5, pp. 874-880 (2000)

http://dx.doi.org/10.1364/JOSAA.17.000874

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### Abstract

We introduce an iterative algorithm for the reconstruction of dielectric profile functions from scattered field data, in which each step corresponds to the solution of a quadratic inversion problem. This means that, at each iteration, we perform a second-order approximation of the scattering operator connecting the unknown profile to the data about a reference profile function. This procedure is then compared with a linear iterative inversion algorithm, and it is pointed out that, within a prescribed class of profile functions, the linear iterative inversion does not converge to the actual solution, whereas the proposed approach does. This feature can be explained by reference not only to the improved approximation provided by the addition of a further term for profile functions of a larger norm but also to the different classes of functions that can be reconstructed by either the linear or the quadratic model. Numerical examples of profile reconstruction in the scalar two-dimensional geometry, with far-zone scattered field data at a fixed frequency, confirm the theoretical analysis.

© 2000 Optical Society of America

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(290.3200) Scattering : Inverse scattering

**History**

Original Manuscript: June 10, 1999

Revised Manuscript: December 6, 1999

Manuscript Accepted: January 6, 2000

Published: May 1, 2000

**Citation**

Rocco Pierri, Giovanni Leone, and Raffaele Persico, "Second-order iterative approach to inverse scattering: numerical results," J. Opt. Soc. Am. A **17**, 874-880 (2000)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-17-5-874

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### References

- D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
- L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).
- W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).
- J.-H. Lin, W. C. Chew, “Ultrasonic imaging by local shape function method with CGFFT,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 43, 956–969 (1996). [CrossRef]
- N. Joachimwitz, C. Pichot, J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991). [CrossRef]
- D. O. Batrakov, N. P. Zhuck, “Solution of a general inverse scattering problem using the distorted Born approximation and iterative technique,” Inverse Probl. 10, 39–54 (1994). [CrossRef]
- H. T. Lin, Y. W. Kiang, “Iterative solution of inverse scattering for a two-dimensional dielectric object,” Int. J. Imaging Syst. Technol. 7, 25–32 (1996). [CrossRef]
- O. S. Haddadin, E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 1485–1496 (1998). [CrossRef]
- A. Brancaccio, V. Pascazio, R. Pierri, “A quadratic model for inverse profiling: the one-dimensional case,” J. Electromagn. Waves Appl. 9, 673–696 (1995). [CrossRef]
- R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a quadratic model,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999). [CrossRef]
- C. Torres-Verdin, T. M. Habashy, “Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear approximation,” Radio Sci. 29, 1051–1079 (1994). [CrossRef]
- J. J. Stamnes, L. J. Gelius, I. Johansen, N. Sponheim, “Diffraction tomography applications in seismics and medicine,” in Inverse Problems in Scattering and Imaging, M. Bertero, E. R. Pike, eds. (Hilger, Bristol, UK, 1992).
- R. Pierri, R. Persico, R. Bernini, “Information content of Born field scattered by an embedded slab: multifrequency, multiview, and multivifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (1999). [CrossRef]
- M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electronic Physics, P. W. Hawks, ed. (Academic, New York, 1990), pp. 1–120.
- M. Bertero, G. A. Viano, F. Pasqualetti, L. Ronchi, G. Toraldo Di Francia, “The inverse scattering problem in the Born approximation and the number of degrees of freedom,” Opt. Acta 27, 1011–1024 (1980). [CrossRef]
- M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. 32, 860–874 (1984). [CrossRef]
- A. Brancaccio, G. Leone, R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998). [CrossRef]
- G. Leone, R. Persico, R. Pierri, “Inverse scattering under the distorted Born approximation for cylindrical geometries,” J. Opt. Soc. Am. A 16, 1779–1787 (1999); “Errata,” 16, 2310 (1999). [CrossRef]
- S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).
- H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
- C. W. Groetsch, Inverse Problems in Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).
- A. Brancaccio, G. Leone, R. Pierri, “Linear and quadratic inverse scattering for angularly varying circular cylinders,” J. Opt. Soc. Am. A 16, 2887–2895 (1999). [CrossRef]
- A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 25, 203–215 (1997). [CrossRef]
- R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997). [CrossRef]
- J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13, 334–341 (1965). [CrossRef]
- M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
- W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

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