Abstract

We present an inversion procedure for electromagnetic scattering, based on the powerful and flexible technique called the coupled-dipole method combined with an optimization algorithm. This method permits us to realize imaging of dielectric objects whose dimensions are comparable with the incident wavelength and is shown to be efficient with corrupted data (scattered electric field). The feasibility of this method is shown in a synthetic example in which the scattered field is corrupted with Gaussian noise. Two methods are used to invert the scattered field to recover the refractive index of the medium: a conventional matrix inversion and an iterative method.

© 2000 Optical Society of America

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  1. M. Baribaud, “Microwave imagery: analytical method and maximum entropy method,” J. Phys. D 23, 269–288 (1990).
    [CrossRef]
  2. S. Caorsi, G. L. Gragnani, M. Pasorino, “Redundant electromagnetic data for microwave imaging of three dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
    [CrossRef]
  3. H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
    [CrossRef]
  4. E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  5. B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  6. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–625 (1951).
    [CrossRef]
  7. P. H. Van Cittert, “Zum einfluss der Spaltbreite auf die Intensitätsverteilung in Sprektrallinien,” Z. Phys. 65, 298–308 (1931).
    [CrossRef]
  8. G. Wahba, “Three topics in ill-posed problems,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, New York, 1987), pp. 37–51.
  9. O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).
  10. M. Hanke, “Accelarated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
    [CrossRef]
  11. C. Xu, I. Aissaoui, S. Jacquey, “Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor,” J. Opt. Soc. Am. A 11, 2804–2808 (1994).
    [CrossRef]
  12. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  13. B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  15. E. H. Moore, “On the reciprocal of the general algebraic matrix (abstract),” Bull. Am. Math. Soc. 26, 389, 394–395 (1920).
  16. R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
    [CrossRef]
  17. C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).
  18. K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).
  19. D. W. Marquardt, “An algorithm for least-squares of nonlinear parameters,” J. Soc. Appl. Ind. Math. 11, 431–441 (1963).
    [CrossRef]
  20. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Manuf. 9, 84–97 (1962).
    [CrossRef]
  21. B. C. Cook, “Least structure of photonuclear yield functions,” Nucl. Instrum. Methods 24, 256–268 (1963).
    [CrossRef]
  22. A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).
  23. A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).
  24. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Manuf. 10, 97–101 (1963).
    [CrossRef]

1995

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

1994

1991

M. Hanke, “Accelarated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

1990

M. Baribaud, “Microwave imagery: analytical method and maximum entropy method,” J. Phys. D 23, 269–288 (1990).
[CrossRef]

1988

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1974

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).

1973

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

1963

B. C. Cook, “Least structure of photonuclear yield functions,” Nucl. Instrum. Methods 24, 256–268 (1963).
[CrossRef]

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Manuf. 10, 97–101 (1963).
[CrossRef]

D. W. Marquardt, “An algorithm for least-squares of nonlinear parameters,” J. Soc. Appl. Ind. Math. 11, 431–441 (1963).
[CrossRef]

1962

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Manuf. 9, 84–97 (1962).
[CrossRef]

1955

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

1951

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–625 (1951).
[CrossRef]

1944

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

1931

P. H. Van Cittert, “Zum einfluss der Spaltbreite auf die Intensitätsverteilung in Sprektrallinien,” Z. Phys. 65, 298–308 (1931).
[CrossRef]

1920

E. H. Moore, “On the reciprocal of the general algebraic matrix (abstract),” Bull. Am. Math. Soc. 26, 389, 394–395 (1920).

Aissaoui, I.

Baribaud, M.

M. Baribaud, “Microwave imagery: analytical method and maximum entropy method,” J. Phys. D 23, 269–288 (1990).
[CrossRef]

Caorsi, S.

S. Caorsi, G. L. Gragnani, M. Pasorino, “Redundant electromagnetic data for microwave imaging of three dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

Cook, B. C.

B. C. Cook, “Least structure of photonuclear yield functions,” Nucl. Instrum. Methods 24, 256–268 (1963).
[CrossRef]

Draine, B. T.

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Flatau, P. J.

Gragnani, G. L.

S. Caorsi, G. L. Gragnani, M. Pasorino, “Redundant electromagnetic data for microwave imaging of three dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

Hanke, M.

M. Hanke, “Accelarated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

Harada, H.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jacquey, S.

Lanczos, C.

C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–625 (1951).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares of nonlinear parameters,” J. Soc. Appl. Ind. Math. 11, 431–441 (1963).
[CrossRef]

Moore, E. H.

E. H. Moore, “On the reciprocal of the general algebraic matrix (abstract),” Bull. Am. Math. Soc. 26, 389, 394–395 (1920).

Pasorino, M.

S. Caorsi, G. L. Gragnani, M. Pasorino, “Redundant electromagnetic data for microwave imaging of three dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

Pennypacker, C. R.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Penrose, R.

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

Phillips, D. L.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Manuf. 9, 84–97 (1962).
[CrossRef]

Purcell, E. M.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Strand, O. N.

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).

Takenaka, T.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Tanaka, M.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

Twomey, S.

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Manuf. 10, 97–101 (1963).
[CrossRef]

Van Cittert, P. H.

P. H. Van Cittert, “Zum einfluss der Spaltbreite auf die Intensitätsverteilung in Sprektrallinien,” Z. Phys. 65, 298–308 (1931).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wahba, G.

G. Wahba, “Three topics in ill-posed problems,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, New York, 1987), pp. 37–51.

Wall, D. J. N.

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

Xu, C.

Am. J. Math.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–625 (1951).
[CrossRef]

Astrophys. J.

E. M. Purcell, C. R. Pennypacker, “Scattering and absorption of light by non spherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, “The discrete dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Bull. Am. Math. Soc.

E. H. Moore, “On the reciprocal of the general algebraic matrix (abstract),” Bull. Am. Math. Soc. 26, 389, 394–395 (1920).

IEEE Trans. Antennas Propag.

S. Caorsi, G. L. Gragnani, M. Pasorino, “Redundant electromagnetic data for microwave imaging of three dimensional dielectric objects,” IEEE Trans. Antennas Propag. 42, 581–589 (1994).
[CrossRef]

H. Harada, D. J. N. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Trans. Antennas Propag. 43, 784–792 (1995).
[CrossRef]

J. Assoc. Comput. Manuf.

D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Manuf. 9, 84–97 (1962).
[CrossRef]

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Manuf. 10, 97–101 (1963).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

M. Baribaud, “Microwave imagery: analytical method and maximum entropy method,” J. Phys. D 23, 269–288 (1990).
[CrossRef]

J. Soc. Appl. Ind. Math.

D. W. Marquardt, “An algorithm for least-squares of nonlinear parameters,” J. Soc. Appl. Ind. Math. 11, 431–441 (1963).
[CrossRef]

Nucl. Instrum. Methods

B. C. Cook, “Least structure of photonuclear yield functions,” Nucl. Instrum. Methods 24, 256–268 (1963).
[CrossRef]

Numer. Math.

M. Hanke, “Accelarated Landweber iterations for the solution of ill-posed equations,” Numer. Math. 60, 341–373 (1991).
[CrossRef]

Proc. Cambridge Philos. Soc.

R. Penrose, “A generalized inverse for matrices,” Proc. Cambridge Philos. Soc. 51, 406–413 (1955).
[CrossRef]

Q. Appl. Math.

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).

Sov. Math. Dokl.

A. N. Tikhonov, “Solution of incorrectly formulated problems and the regularization method,” Sov. Math. Dokl. 4, 1035–1038 (1963).

A. N. Tikhonov, “Regularization of incorrectly posed problems,” Sov. Math. Dokl. 4, 1624–1627 (1963).

Z. Phys.

P. H. Van Cittert, “Zum einfluss der Spaltbreite auf die Intensitätsverteilung in Sprektrallinien,” Z. Phys. 65, 298–308 (1931).
[CrossRef]

Other

G. Wahba, “Three topics in ill-posed problems,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, New York, 1987), pp. 37–51.

C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (15)

Fig. 1
Fig. 1

Description of the acquisition geometry for the scattered field.

Fig. 2
Fig. 2

Singular values of the 171 dipoles simulation (ka = 1.0).

Fig. 3
Fig. 3

Recovered refractive index (real part) for 171 dipoles simulation, with use of SVD with all singular values, noise free (ka = 1.0).

Fig. 4
Fig. 4

Recovered refractive index (imaginary part) for 171 dipoles simulation, with use of SVD with all singular values, noise free (ka = 1.0).

Fig. 5
Fig. 5

Selection of the regularization factor for the zeroth-order regularized (DLS, damped least-squares) inversion of the 171 dipoles simulation.

Fig. 6
Fig. 6

Recovered refractive index (real part) for 171 dipoles simulation, with use of zeroth-order regularization, with selected regularization factor, noise free (ka = 1.0).

Fig. 7
Fig. 7

Recovered refractive index (imaginary part) for 171 dipoles simulation, with use of zeroth-order regularization, with selected regularization factor, noise free (ka = 1.0).

Fig. 8
Fig. 8

Recovered refractive index (real part) for 171 dipoles simulation, with use of zeroth order regularization, with selected regularization factor, noise free (ka = 2.0).

Fig. 9
Fig. 9

Recovered refractive index (imaginary part) for 171 dipoles simulation, with use of zeroth-order regularization, with selected regularization factor, noise free (ka = 2.0).

Fig. 10
Fig. 10

Recovered refractive index (real part) for 171 dipoles simulation, with use of zeroth order regularization, with selected regularization factor, noisy data (ka = 2.0).

Fig. 11
Fig. 11

Recovered refractive index (imaginary part) for 171 dipoles simulation, with use of zeroth-order regularization, with selected regularization factor, noisy data (ka = 2.0).

Fig. 12
Fig. 12

Recovered refractive index (real part) for 171 dipoles simulation, with Landweber’s iterative method, noise free data (ka = 2.0).

Fig. 13
Fig. 13

Recovered refractive index (imaginary part) for 171 dipoles simulation, with Landweber’s iterative method, noise free data (ka = 2.0).

Fig. 14
Fig. 14

Recovered refractive index (real part) for 171 dipoles simulation, with Landweber’s iterative method, noisy data (ka = 2.0).

Fig. 15
Fig. 15

Recovered refractive index (imaginary part) for 171 dipoles simulation, with Landweber’s iterative method, noisy data (ka = 2.0).

Equations (15)

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

α=r-1r+2a3,
pi=αiETri,  i=1,, N,
Eri=j=1,, NjiΠri-rjpj,
Πr=expikrrk2I-n  n+1-ikrr2×3n  n-I,
pi=αiE0ri+αij=1,, NjiΠri-rjpj, i=1,, N.
αi=pi2/E0ri+j=1,, NjiEjri·pi*.
E=Πp,
Φp=pTp+tTE-Πp,
p=ΠTΠΠT-1E.
Π=UΣVT,
Π+=VΣ-1UT,
E=Πp,
pk+1=pk-λFpk,
F=½ pTΠTΠp-pTΠTE=ΠTΠp-ΠTE;
pk+1=pk-λΠTΠpk-E.

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