Abstract

The local-minima question that arises in the framework of a quadratic approach to inverse-scattering problems is investigated. In particular, a sufficient condition for the absence of local minima is given, and some guidelines to ensure the reliability of the algorithm are outlined for the case of data not belonging to the range of the relevant quadratic operator. This is relevant also when an iterated solution procedure based on a quadratic approximation of the electromagnetic scattering at each step is considered.

© 2002 Optical Society of America

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References

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  1. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  2. R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second-order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
    [CrossRef]
  3. B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
    [CrossRef]
  4. A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
    [CrossRef]
  5. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J.1995).
  6. M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
    [CrossRef]
  7. 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]
  8. 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]
  9. A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
    [CrossRef]
  10. 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]
  11. R. Pierri, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).
  12. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
    [CrossRef]
  13. T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
    [CrossRef]
  14. 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]
  15. L. V. Kantorovich, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).
  16. In practical cases one has at one’s disposal a finite number of data, and therefore one should indeed refer to the Euclidean norm in the space of the N-dimensional complex column vectors CN.However, the rationale remains unchanged.
  17. Analogously to the norm, one can refer to the scalar products in L2(Σ ⊗ S)in the general operational case and should refer to the scalar product in CNin the case with a finite number of data, but the rationale is the same.

2001

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

2000

1999

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]

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]

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second-order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
[CrossRef]

1997

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

1996

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

1995

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]

1984

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Akilov, G. P.

L. V. Kantorovich, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).

Belkebir, A. C.

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

Brancaccio, A.

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]

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]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J.1995).

Colton, D.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

De Blasio, F.

R. Pierri, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).

de Hon, B. P.

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

Isernia, T.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

Jen, L.

A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

Kak, A. C.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Kantorovich, L. V.

L. V. Kantorovich, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).

Kooij, B. J.

B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
[CrossRef]

Kress, R.

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

Lambert, M.

B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
[CrossRef]

Larsen, L. E.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Lee, C. K.

A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

Leone, G.

Lesselier, D.

B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
[CrossRef]

Liseno, A.

R. Pierri, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).

Litman, A. C. S.

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

Pascazio, V.

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]

Persico, R.

Pierri, R.

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]

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]

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]

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second-order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[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, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).

Qing, A.

A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

Slaney, M.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Soldovieri, F.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

R. Pierri, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

Tijhuis, A. G.

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a second-order Born approximation,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
[CrossRef]

A. G. Tijhuis, A. C. Belkebir, A. C. S. Litman, B. P. de Hon, “Theoretical and computational aspects of two dimensional inverse profiling,” IEEE Trans. Geosci. Remote Sens. 39, 1316–1330 (2001).
[CrossRef]

A. Qing, C. K. Lee, L. Jen, “Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm,” IEEE Trans. Geosci. Remote Sens. 39, 665–676 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. Slaney, A. C. Kak, L. E. Larsen, “Limitations of imaging with first order diffraction tomography,” IEEE Trans. Microwave Theory Tech. MTT-32, 860–874 (1984).
[CrossRef]

Inverse Probl.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1547–1568 (1997).
[CrossRef]

J. Electromagn. Waves Appl.

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]

J. Opt. Soc. Am. A

Radio Sci.

T. Isernia, G. Leone, R. Pierri, F. Soldovieri, “On the local minima in phase reconstruction algorithms,” Radio Sci. 31, 1887–1899 (1996).
[CrossRef]

B. J. Kooij, M. Lambert, D. Lesselier, “Non-linear inversion of a buried object in transverse electric scattering,” Radio Sci. 34, 1361–1371 (1999).
[CrossRef]

Other

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, Piscataway, N.J.1995).

R. Pierri, F. Soldovieri, A. Liseno, F. De Blasio, “Dielectric profiles reconstruction via the quadratic approach in 2D geometry from multi-frequency and multi-frequency/multi-view data,” IEEE Trans. Geosci. Remote Sens. (to be published).

D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).

L. V. Kantorovich, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).

In practical cases one has at one’s disposal a finite number of data, and therefore one should indeed refer to the Euclidean norm in the space of the N-dimensional complex column vectors CN.However, the rationale remains unchanged.

Analogously to the norm, one can refer to the scalar products in L2(Σ ⊗ S)in the general operational case and should refer to the scalar product in CNin the case with a finite number of data, but the rationale is the same.

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Figures (1)

Fig. 1
Fig. 1

Behavior of a homogeneous third-degree polynomial. Solid curve, unperturbed polynomial; dashed curve, perturbed polynomial.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

χ(r)=1-r(r)b,
E=Einc+Ai(χE),
Es=Ae(χE),
EEinc
Es=Ae(χEinc)=A(χ).
EEinc+Ai(χEinc).
Es=A(χ)+B(χ, χ),
B(χ, χ)=Ae(χAi(χEinc)).
F(χ)=A(χ)+B(χ, χ)-ES-n2,
f(λ)
=A(χm+λξ)+B(χm+λξ, χm+λξ)-ES-n2.
 
f(λ)=aλ4+bλ3+cλ2+dλ+e,
a=B(ξ, ξ)2,
b=2 ReA(ξ)+B(ξ, χm)+B(χm, ξ),B(ξ, ξ),
c=2 Remen, B(ξ, ξ)+A(ξ)+B(ξ, χm)+B(χm, ξ)2,
d=2 Remen, A(ξ)+B(ξ, χm)+B(χm, ξ),
e=men2,
men=A(χm)+B(χm, χm)-ES-n
f(λ)=aλ4+bλ3+c0λ2,
c0=A(ξ)+B(ξ, χm)+B(χm, ξ)2.
f(λ)=4aλ3+3bλ2+2c0λ=λ(4aλ2+3bλ+2c0).
b2ac0=4(ReA(ξ)+B(χm, ξ)+B(ξ, χm),B(ξ, ξ))2B(ξ, ξ)2A(ξ)+B(χm, ξ)+B(ξ, χm)2<329.
g(r, ζ)=SdζΣ|g(r, ζ)|2dr1/2,
f(λ)=4aλ3+3bλ2+2cλ+d.
b2ac=4(ReA(ξ)+B(χm, ξ)+B(ξ, χmm),B(ξ, ξ))2B(ξ, ξ)2(2 Re(men, B(ξ, ξ))+A(ξ)+B(χm, ξ)+B(ξ, χm)2)<329.
Remen, B(ξ, ξ)-menB(ξ, ξ).
|men, B(ξ, ξ)|menB(ξ, ξ),
|men, B(ξ, ξ)||Re(men, B(ξ, ξ))|=-Re(men, B(ξ, ξ)).
B(ξ, ξ)2(2 Remen, B(ξ, ξ)+A(ξ)+B(χm, ξ)+B(ξ, χm)2)B(ξ, ξ)2(A(ξ)+B(χm, ξ)+B(ξ, χm)2-2menB(ξ, ξ)).
4(ReA(ξ)+B(χm, ξ)+B(ξ, χm), B(ξ, ξ))2B(ξ, ξ)2(A(ξ)+B(χm, ξ)+B(ξ, χm)2-2menB(ξ, ξ))<329.
4(ReA(ξ)+B(χm, ξ)+B(ξ, χm), B(ξ, ξ))2B(ξ, ξ)2A(ξ)+B(χm, ξ)+B(ξ, χm)2
×11-2menB(ξ, ξ)A(ξ)+B(χm, ξ)+B(ξ, χm)2  <329.
2menB(ξ, ξ)A(ξ)+B(χm, ξ)+B(ξ, χm)21.
11-x1+x,
4(ReA(ξ)+B(χm, ξ)+B(ξ, χm), B(ξ, ξ))2B(ξ, ξ)2A(ξ)+B(χm, ξ)+B(ξ, χm)2
+8men(ReA(ξ)+B(χm, ξ)+B(ξ, χm), B(ξ, ξ))2B(ξ, ξ)A(ξ)+B(χm, ξ)+B(ξ, χm)4
<329.
menA(ξ)+B(χm, ξ)+B(ξ, χm)22B(ξ, ξ).
8men(ReA(ξ)+B(χm, ξ)+B(ξ, χm), B(ξ, ξ))2B(ξ, ξ)A(ξ)+B(χm, ξ)+B(ξ, χm)4
8menB(ξ, ξ)A(ξ)+B(χm, ξ)+B(ξ, χm)24.

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