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

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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. L. V. Kantorovic, G. P. Akilov, Functional Analysis (Pergamon, New York, 1982).
  3. W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. R. Pierri, G. Leone, “Inverse scattering of dielectric cylinders by a quadratic model,” IEEE Trans. Geosci. Remote Sens. 37, 374–382 (1999).
    [CrossRef]
  11. 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]
  12. 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).
  13. 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]
  14. 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.
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).
  20. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  21. C. W. Groetsch, Inverse Problems in Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).
  22. 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]
  23. A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 25, 203–215 (1997).
    [CrossRef]
  24. R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
    [CrossRef]
  25. J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13, 334–341 (1965).
    [CrossRef]
  26. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  27. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

1999 (4)

1998 (2)

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]

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]

1997 (2)

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]

1996 (2)

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]

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]

1995 (1)

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]

1994 (2)

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]

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]

1991 (1)

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]

1984 (1)

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]

1980 (1)

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]

1965 (1)

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13, 334–341 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Akilov, G. P.

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

Batrakov, D. O.

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]

Bernini, R.

Bertero, M.

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. 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.

Brancaccio, A.

Chew, W. C.

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]

W. C. Chew, Waves and Fields in Inhomogeneous Media (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995).

Colton, D.

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

Ebbini, E. S.

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]

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Franchois, A.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 25, 203–215 (1997).
[CrossRef]

Gelius, L. J.

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).

Groetsch, C. W.

C. W. Groetsch, Inverse Problems in Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

Habashy, T. M.

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]

Haddadin, O. S.

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]

Hanke, M.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Hugonin, J. P.

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]

Joachimwitz, N.

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]

Johansen, I.

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).

Kak, A. C.

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]

Kantorovic, L. V.

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

Kiang, Y. W.

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]

Kress, R.

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

Larsen, L. E.

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]

Leone, G.

Lin, H. T.

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]

Lin, J.-H.

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]

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Dordrecht, The Netherlands, 1996).

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]

Pasqualetti, F.

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]

Persico, R.

Pichot, C.

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 25, 203–215 (1997).
[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]

Pierri, R.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Richmond, J. H.

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13, 334–341 (1965).
[CrossRef]

Ronchi, L.

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]

Slaney, M.

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]

Sponheim, N.

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).

Stamnes, J. J.

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).

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Tamburrino, A.

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Toraldo Di Francia, G.

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]

Torres-Verdin, C.

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]

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Dover, New York, 1996).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1987).

Viano, G. A.

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]

Zhuck, N. P.

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]

IEEE Trans. Antennas Propag. (3)

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]

A. Franchois, C. Pichot, “Microwave imaging—complex permittivity reconstruction with a Levenberg–Marquardt method,” IEEE Trans. Antennas Propag. 25, 203–215 (1997).
[CrossRef]

J. H. Richmond, “Scattering by a dielectric cylinder of arbitrary cross section shape,” IEEE Trans. Antennas Propag. 13, 334–341 (1965).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

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]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (2)

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]

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]

Int. J. Imaging Syst. Technol. (1)

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]

Inverse Probl. (2)

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]

R. Pierri, A. Tamburrino, “On the local minima problem in conductivity imaging via a quadratic approach,” Inverse Probl. 13, 1–22 (1997).
[CrossRef]

J. Electromagn. Waves Appl. (1)

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 (4)

Opt. Acta (1)

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]

Radio Sci. (1)

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]

Other (10)

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).

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.

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).

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).

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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Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Three-dimensional plot of the actual contrast function for the first example.

Fig. 3
Fig. 3

Three-dimensional plot of the contrast function reconstructed at the second iteration of the linear iterative algorithm for the example of Fig. 2.

Fig. 4
Fig. 4

Behavior of the reconstruction error by (a) the linear and (b) the quadratic iterative algorithms for the example of Fig. 2.

Fig. 5
Fig. 5

Three-dimensional plot of the contrast function reconstructed at the second iteration of the quadratic iterative algorithm for the example of Fig. 2.

Fig. 6
Fig. 6

Three-dimensional plot of the actual contrast function for the second example.

Fig. 7
Fig. 7

Three-dimensional plot of the linear reconstruction at the first four iterations for the example of Fig. 6.

Fig. 8
Fig. 8

Three-dimensional plot of the quadratic reconstruction at the first four iterations for the example of Fig. 6.

Equations (30)

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

χ(r)=ε(r)-εbεb.
Ai(f)=DGi(r,r)f(r)dr,rD,
Ae(f)=DGe(r, r)f(r)dr,rΣ,
E=Einc+Ai(χE),
Es=Ae(χE).
E=(I-Aiχ)-1Einc,
Es=Ae[χ(I-Aiχ)-1Einc],
Δχ=χ-χ0,
ES-ES0=F(Δχ),
FLχ0(Δχ)=A(Δχ),
FQχ0(Δχ)=A(Δχ)+B(Δχ, Δχ),
ES-ES0=FLχ0(Δχ).
ES-ES0=FQχ0(Δχ).
Φ(Δχ)=FQχ0(Δχ)-Es+ES02,
Δχ(θ)=a+i=1Nbi cos(iθ)+ci sin(iθ).
errnL,Q=100(a-a0)2+i=1N[(bi-b0i)2+(ci-c0i)2]a2+i=1N(bi2+ci2)1/2%,
χA(θ)=a+i=1Nbi cos(iθ)+ci sin(iθ),
χR(θ)=a0+i=1Nb0i cos(iθ)+c0i sin(iθ).
MEL,Q=100k=1D|EA(k)-EL,Q(k)|2k=1D|EA(k)|21/2%,
ΔE=E-E0,
E0=(I-Aiχ0)-1Einc,
ΔE=Ai(χE)-Ai(χ0E0)
=Ai(ΔχE0)+Ai(χ0ΔE)+Ai(ΔχΔE)
ΔEL=(I-Aiχ0)-1Ai(ΔχE0),
Es=Ae(χ0E0)+Ae(ΔχE0)+Ae(χ0ΔE)+Ae(ΔχΔE).
A(Δχ)=Ae(χ0ΔEL)+Ae(ΔχE0)=FLχ0(Δχ).
ΔE=ΔEL+ΔEQ,
ΔEQ=(I-Aiχ0)-1Ai(ΔχΔEL).
FQχ0(Δχ)=ES-ES0=A(Δχ)+B(Δχ, Δχ),
B(Δχ, Δχ)=Ae(ΔχΔEL)+Ae(χ0ΔEQ)

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