## Quadratic distorted approximation for the inverse scattering of dielectric cylinders

JOSA A, Vol. 18, Issue 3, pp. 600-609 (2001)

http://dx.doi.org/10.1364/JOSAA.18.000600

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### Abstract

The nonlinear (quadratic) distorted approximation of the inverse scattering of dielectric cylinders is investigated, with the aim of pointing out the influence of the background medium. We refer to a canonical geometry consisting of a radially symmetric dielectric cylinder illuminated at a single frequency. We discuss how the spatial variations of those unknown dielectric profile functions that can be reconstructed by a stable inversion procedure are related to the permittivity of the background cylinder. First, results for the linear distorted approximation, obtained by means of the singular-value decomposition, are recalled and compared with the Born approximation. It turns out that the distorted model provides a smoother behavior of the singular values, and thus the inversion is more sensitive to the presence of uncertainties in the data. Furthermore, a stable inversion procedure can reconstruct only a very limited class of unknowns in correspondence with fast spatial variations related to the background permittivity and the excitation frequency. On the other hand, the quadratic model improves the approximation in the distorted case. This can be traced not only to the higher allowable level of permittivity but mainly to the fact that the model makes it possible to reconstruct different spatial features as the solution space changes. Numerical results show that the quadratic inversion performs better than the linear one for the same amount of uncertainty in the data.

© 2001 Optical Society of America

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(290.0290) Scattering : Scattering

**History**

Original Manuscript: May 18, 2000

Revised Manuscript: September 5, 2000

Manuscript Accepted: September 5, 2000

Published: March 1, 2001

**Citation**

Giovanni Leone, Adriana Brancaccio, and Rocco Pierri, "Quadratic distorted approximation for the inverse scattering of dielectric cylinders," J. Opt. Soc. Am. A **18**, 600-609 (2001)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-18-3-600

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### References

- R. Pierri, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multifrequency, multiview, and multifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (1999). [CrossRef]
- W. Tabbara, B. Duchene, Ch. Pichot, D. Lesselier, L. Chommeloux, N. Joachimowitz, “Diffraction tomography: contribution to the analysis of some applications in microwaves and ultrasonics,” Inverse Probl. 4, 305–331 (1988). [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]
- 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]
- P. Lobel, L. Blanc-Feraud, Ch. Pichot, M. Barlaud, “New regularization scheme for inverse scattering,” Inverse Probl. 13, 403–410 (1997). [CrossRef]
- I. T. Rekanos, T. V. Yioultsis, T. D. Tsiboukis, “Inverse scattering using the finite-element method and a nonlinear optimization technique,” IEEE Trans. Microwave Theory Tech. 47, 336–344 (1999). [CrossRef]
- J. Lin, W. Chew, “Solution of the three-dimensional electromagnetic inverse problem by the local shape function and the conjugate gradient fast Fourier transform methods,” J. Opt. Soc. Am. A 14, 3037–3045 (1997). [CrossRef]
- N. Joachimowicz, J. J. Mallorqui, J. Ch. Bolomey, A. Broquetas, “Convergence and stability assessment of Newton–Kantarovich reconstruction algorithms for microwave tomography,” IEEE Trans. Med. Imaging 17, 562–570 (1998). [CrossRef] [PubMed]
- B. J. Kooij, P. M. van den Berg, “Non linear inversion in TE scattering,” IEEE Trans. Microwave Theory Tech. 46, 1706–1712 (1998). [CrossRef]
- P. M. van den Berg, A. L. van Broekhoven, A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325–1344 (1999). [CrossRef]
- U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997). [CrossRef]
- W. C. Chew, Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imaging 9, 218–225 (1990). [CrossRef] [PubMed]
- W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J., 1995).
- R. F. Remis, P. M. van den Berg, “On the equivalence of the Newton–Kantorovich and distorted Born methods,” Inverse Probl. 16, L1–L4 (2000). [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). [CrossRef]
- R. Pierri, A. Brancaccio, “Imaging of a rotationally symmetric cylinder by a quadratic approach,” J. Opt. Soc. Am. A 14, 2777–2785 (1997). [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]
- G. Leone, A. Brancaccio, R. Pierri, “Linear and quadratic inverse scattering for angularly varying circular cylinders,” J. Opt. Soc. Am. A 16, 2887–2895 (1999). [CrossRef]
- E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931). [CrossRef]
- R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
- M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
- M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawks, ed. (Academic, London, 1989), pp. 1–120.
- D. Slepian, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV: Extensions to many dimensions; generalized prolate spheroidal functions,” Bell Syst. Tech. J. 43, 3009–3057 (1964). [CrossRef]
- The behavior of the singular values of a linear integral operator follows a pattern similar to the decay of the Fourier coefficients of a function [F. Smithies, “The eigenvalues and singular values of integral equations,” Proc. London Math. Soc. 43, 255–279 (1937)]. If it is analytical, they decay exponentially fast to zero with the order; if the derivative of some order is discontinuous, they decay at a rate that is faster the higher the order for which the derivative exists and is continuous; if it is singular, but integrable, the rate is smoother.
- R. Pierri, A. Brancaccio, F. De Blasio, “Multifrequency dielectric profile inversion for a cylindrically stratified medium,” IEEE Trans. Geosci. Remote Sens. 38, 1716–1724 (2000). [CrossRef]
- T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in the phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996). [CrossRef]
- H. E. Bussey, J. H. Richmond, “Scattering by a lossy dielectric circular cylindrical multilayer, numerical values,” IEEE Trans. Antennas Propag. AP-23, 723–725 (1975). [CrossRef]
- L. V. Kantorovic, G. P. Akilov, Analisi funzionale (Editori riuniti, Rome, 1980).
- R. Pierri, G. Leone, R. Persico, “A second order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000). [CrossRef]

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