Abstract

We discuss the characterization of two-dimensional targets based on their diffracted intensity. The target characterization is performed by minimizing an adequate cost functional, combined with a level-set representation if the target is homogeneous. One key issue in this minimization is the choice of an updating direction, which involves the gradient of the cost functional. This gradient can be evaluated using a fictitious field, the solution of an adjoint problem in which receivers act as sources with a specific amplitude. We explore the Born approximation for the adjoint field and compare various approaches for a wide variety of objects.

© 2006 Optical Society of America

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  1. V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
    [CrossRef] [PubMed]
  2. N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
    [CrossRef]
  3. G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).
  4. L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
    [CrossRef]
  5. T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
    [CrossRef]
  6. M. Lambert and D. Lesselier, "Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields," Inverse Probl. 16, 563-576 (2000).
    [CrossRef]
  7. F. James and M. Sepulveda, "Parameter identification for a model of chromatographic column," Inverse Probl. 10, 1299-1314 (1994).
    [CrossRef]
  8. A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
    [CrossRef]
  9. M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
    [CrossRef]
  10. B. Samet, "L'analyse asymtotique topologique pour les équations de Maxwell et applications," Ph.D. thesis (Université Paul Sabatier, 2004).
  11. S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys. 79, 12-49 (1988).
    [CrossRef]
  12. Z. Q. Peng and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: marching-on-in frequency approach," J. Electromagn. Waves Appl. 7, 739-763 (1993).
    [CrossRef]
  13. K. Belkebir, P. C. Chaumet, and A. Sentenac, "Influence of multiple scattering on three-dimensional imaging with optical diffraction tomography," J. Opt. Soc. Am. A 23, 586-595 (2006).
    [CrossRef]
  14. A. Litman, "Reconstruction by level sets of n-ary scattering obstacles," Inverse Probl. 21, S131-S152 (2005).
    [CrossRef]
  15. A. G. Tijhuis, "Angularly propagating waves in a radially inhomogeneous, lossy dielectric cylinder and their connection with natural modes," IEEE Trans. Antennas Propag. 34, 813-824 (1986).
    [CrossRef]
  16. J. Sokolowski and A. Zochowski, On Topological Derivative in Shape Optimization, Tech. Rep. RR-3170 (Institut National de Recherche en Informatique et Automatique, 1997).

2006 (1)

2005 (2)

A. Litman, "Reconstruction by level sets of n-ary scattering obstacles," Inverse Probl. 21, S131-S152 (2005).
[CrossRef]

M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
[CrossRef]

2004 (2)

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).

L. Crocco, M. D'Urso, and T. Isernia, "Inverse scattering from phaseless measurements of the total field on a closed curve," J. Opt. Soc. Am. A 21, 622-631 (2004).
[CrossRef]

2002 (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

2001 (1)

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

2000 (1)

M. Lambert and D. Lesselier, "Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields," Inverse Probl. 16, 563-576 (2000).
[CrossRef]

1998 (1)

A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
[CrossRef]

1997 (1)

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

1994 (1)

F. James and M. Sepulveda, "Parameter identification for a model of chromatographic column," Inverse Probl. 10, 1299-1314 (1994).
[CrossRef]

1993 (1)

Z. Q. Peng and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: marching-on-in frequency approach," J. Electromagn. Waves Appl. 7, 739-763 (1993).
[CrossRef]

1988 (1)

S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys. 79, 12-49 (1988).
[CrossRef]

1986 (1)

A. G. Tijhuis, "Angularly propagating waves in a radially inhomogeneous, lossy dielectric cylinder and their connection with natural modes," IEEE Trans. Antennas Propag. 34, 813-824 (1986).
[CrossRef]

Belkebir, K.

Chaumet, P. C.

Crocco, L.

Destouches, N.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

D'Urso, M.

Gbur, G.

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).

Giovannini, H.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Guérin, C. A.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Harada, H.

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Isernia, T.

James, F.

F. James and M. Sepulveda, "Parameter identification for a model of chromatographic column," Inverse Probl. 10, 1299-1314 (1994).
[CrossRef]

Lambert, M.

M. Lambert and D. Lesselier, "Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields," Inverse Probl. 16, 563-576 (2000).
[CrossRef]

Lauer, V.

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

Lequime, M.

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Lesselier, D.

M. Lambert and D. Lesselier, "Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields," Inverse Probl. 16, 563-576 (2000).
[CrossRef]

A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
[CrossRef]

Litman, A.

A. Litman, "Reconstruction by level sets of n-ary scattering obstacles," Inverse Probl. 21, S131-S152 (2005).
[CrossRef]

A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
[CrossRef]

Masmoudi, M.

M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
[CrossRef]

Osher, S.

S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys. 79, 12-49 (1988).
[CrossRef]

Peng, Z. Q.

Z. Q. Peng and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: marching-on-in frequency approach," J. Electromagn. Waves Appl. 7, 739-763 (1993).
[CrossRef]

Pommier, J.

M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
[CrossRef]

Samet, B.

M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
[CrossRef]

B. Samet, "L'analyse asymtotique topologique pour les équations de Maxwell et applications," Ph.D. thesis (Université Paul Sabatier, 2004).

Santosa, F.

A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
[CrossRef]

Sentenac, A.

Sepulveda, M.

F. James and M. Sepulveda, "Parameter identification for a model of chromatographic column," Inverse Probl. 10, 1299-1314 (1994).
[CrossRef]

Sethian, J. A.

S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys. 79, 12-49 (1988).
[CrossRef]

Sokolowski, J.

J. Sokolowski and A. Zochowski, On Topological Derivative in Shape Optimization, Tech. Rep. RR-3170 (Institut National de Recherche en Informatique et Automatique, 1997).

Takenaka, T.

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Tanaka, M.

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Tijhuis, A. G.

Z. Q. Peng and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: marching-on-in frequency approach," J. Electromagn. Waves Appl. 7, 739-763 (1993).
[CrossRef]

A. G. Tijhuis, "Angularly propagating waves in a radially inhomogeneous, lossy dielectric cylinder and their connection with natural modes," IEEE Trans. Antennas Propag. 34, 813-824 (1986).
[CrossRef]

Wall, J. N.

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Wolf, E.

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).

Zochowski, A.

J. Sokolowski and A. Zochowski, On Topological Derivative in Shape Optimization, Tech. Rep. RR-3170 (Institut National de Recherche en Informatique et Automatique, 1997).

IEEE Trans. Antennas Propag. (1)

A. G. Tijhuis, "Angularly propagating waves in a radially inhomogeneous, lossy dielectric cylinder and their connection with natural modes," IEEE Trans. Antennas Propag. 34, 813-824 (1986).
[CrossRef]

Inf. Sci. (N.Y.) (1)

G. Gbur and E. Wolf, "The information content of the scattered intensity in diffraction tomography," Inf. Sci. (N.Y.) 162, 3-20 (2004).

Inverse Probl. (5)

M. Lambert and D. Lesselier, "Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields," Inverse Probl. 16, 563-576 (2000).
[CrossRef]

F. James and M. Sepulveda, "Parameter identification for a model of chromatographic column," Inverse Probl. 10, 1299-1314 (1994).
[CrossRef]

A. Litman, D. Lesselier, and F. Santosa, "Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set," Inverse Probl. 14, 685-706 (1998).
[CrossRef]

M. Masmoudi, J. Pommier, and B. Samet, "The topological asymptotic expansion for the Maxwell equations and some applications," Inverse Probl. 21, 547-564 (2005).
[CrossRef]

A. Litman, "Reconstruction by level sets of n-ary scattering obstacles," Inverse Probl. 21, S131-S152 (2005).
[CrossRef]

J. Comput. Phys. (1)

S. Osher and J. A. Sethian, "Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations," J. Comput. Phys. 79, 12-49 (1988).
[CrossRef]

J. Electromagn. Waves Appl. (1)

Z. Q. Peng and A. G. Tijhuis, "Transient scattering by a lossy dielectric cylinder: marching-on-in frequency approach," J. Electromagn. Waves Appl. 7, 739-763 (1993).
[CrossRef]

J. Microsc. (1)

V. Lauer, "New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope," J. Microsc. 205, 165-176 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Microwave Opt. Technol. Lett. (1)

T. Takenaka, J. N. Wall, H. Harada, and M. Tanaka, "Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field," Microwave Opt. Technol. Lett. 14, 182-188 (1997).
[CrossRef]

Opt. Commun. (1)

N. Destouches, C. A. Guérin, M. Lequime, and H. Giovannini, "Determination of the phase of the diffracted field in the optical domain. Application to the reconstruction of surface profiles," Opt. Commun. 198, 233-239 (2001).
[CrossRef]

Other (2)

B. Samet, "L'analyse asymtotique topologique pour les équations de Maxwell et applications," Ph.D. thesis (Université Paul Sabatier, 2004).

J. Sokolowski and A. Zochowski, On Topological Derivative in Shape Optimization, Tech. Rep. RR-3170 (Institut National de Recherche en Informatique et Automatique, 1997).

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Figures (14)

Fig. 1
Fig. 1

Geometry of the problem. A two-dimensional cylinder with cross-section Ω and permittivity contrast χ ( x , y ) is radiated by an electromagnetic source located on a circle Γ. The scattered intensity is assumed to be available on Γ.

Fig. 2
Fig. 2

Initial guess using the topological asymptotic expansion results (a) with modulus-only data; (b) with modulus and phase data. The object under test HOMOCYL16 is constituted by two circular cylinders of contrast χ = 0.6 . Black circles in the images correspond to boundaries of actual cylinders.

Fig. 3
Fig. 3

Reconstructed contrast distribution using a conjugate-gradient method, for the HOMOCYL16 object. The updating direction d n involves a gradient derived from a solution of an adjoint problem. (a) Both the internal field and the adjoint field are computed accurately (FULLFULL case); (c) same as in (a) but the evaluation of the adjoint field assumes the Born approximation (FULLBORN case); (e) the Born approximation is assumed for both the internal field and for the adjoint field (BORNBORN case). Curves (b), (d), and (f) represent the evolution in logarithmic scale of the minimized cost functional with respect to the iteration steps for the reconstructions plotted in (a), (c), and (e), respectively.

Fig. 4
Fig. 4

Comparisons between the reconstructed contrast presented in Fig. 3 and the actual one along the x axis. Left column comparisons are presented along the line y = 0.3 λ , which corresponds to a cut along a diameter of the large cylinder of Fig. 3. The right column presents comparisons along the line y = 0.2 λ , which corresponds to a cut along a diameter of the small cylinder of Fig. 3. The solid curves correspond to the actual profiles, while the dotted curves correspond to the reconstructed ones. (a) and (b) correspond to Fig. 3a. (c) and (d) correspond to Fig. 3c. (e) and (f) correspond to Fig. 3e.

Fig. 5
Fig. 5

Same as in Fig. 3, but the inversion is performed using the level-set scheme described in Subsection 4C, where it is assumed that the permittivity contrast of targets under test is known.

Fig. 6
Fig. 6

Same as in Fig. 3, but with the object under test HOMOCYL20, which is constituted by circular cylinders of permittivity contrast χ = 1 .

Fig. 7
Fig. 7

Comparisons between the reconstructed contrast presented in Fig. 6 and the actual one along the x axis. Left column comparisons are presented along the line y = 0.3 λ , which corresponds to a cut along a diameter of the large cylinder of Fig. 6. The right column presents comparisons along the line y 0.15 λ , which corresponds to a cut along a diameter of the small cylinder of Fig. 6. The solid curves correspond to the actual profiles, while the dotted curves correspond to the reconstructed ones. (a) and (b) correspond to Fig. 6a; (c) and (d) correspond to Fig. 6c; (e) and (f) correspond to Fig. 6e.

Fig. 8
Fig. 8

Same as in Fig. 6, but the inversion is performed using the level-set scheme described in Subsection 4C.

Fig. 9
Fig. 9

Modulus of electromagnetic fields in the test domain D for a source located at ( 1.5 λ , 0 ) . (a) Incident field; (b) internal field of the object LUNEBERG; (c) internal field of the object INHOMOSIN.

Fig. 10
Fig. 10

Reconstruction of the inhomogeneous object LUNEBERG from noiseless data, using the conjugate-gradient method described in Subsection 3B. (a) FULLFULL case; (c) FULLBORN case; (e) BORNBORN case. The second column of the figure presents the evolution in logarithmic scale of the minimized cost functional versus iteration steps that correspond to images plotted in the first column.

Fig. 11
Fig. 11

Same as in Fig. 10 but with the object INHOMOSIN.

Fig. 12
Fig. 12

Comparisons between the reconstructed profiles and the actual one along a horizontal line y 0.15 λ for the LUNEBERG (first column) and INHOMOSIN objects (second column). The solid curves stand for the actual profiles while the dotted curves correspond to the reconstructed ones. (a), (b) FULLFULL; (c), (d), FULLBORN; and (e), (f) BORNBORN.

Fig. 13
Fig. 13

Same as in Figs. 10, 11 but with 10% additive noise in the data. The first column corresponds to the LUNEBERG object while the second column corresponds to the INHOMOSIN object.

Fig. 14
Fig. 14

Same as in Fig. 12 but with 10% additive noise.

Equations (28)

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E l i ( r ) = E l i ( r ) u z = P ω μ 0 4 H 0 ( 1 ) ( k b r r l ) u z ,
E l s ( r Γ ) = k 0 2 D χ ( r ) E l ( r ) G ( r , r ) d r ,
E l ( r D ) = E l i + k 0 2 D χ ( r ) E l ( r ) G ( r , r ) d r ,
E l s = K χ E l E l = E l i + G χ E l .
J ( χ ) = 1 2 l = 1 L w l E l obs E l s ( χ ) Γ 2 ,
J ( χ ) = 1 2 l = 1 L w l I l obs E l s ( χ ) 2 Γ 2 ,
L ( χ , E s , E , U s , U ) = l = 1 L { F ( E l s ) + U l s E l s K χ E l Γ + U l E l E l i G χ E l D } ,
P l = P l i + G χ P l , P l i = K t F ( E l s ) ¯ .
P l i = w l K t ( E l obs E l s ¯ ) .
P l i = 2 w l K t E l s ¯ ( I l obs E l s 2 ) .
J ( χ ) δ χ D = Re l = 1 L E l P l ¯ δ χ D .
J { χ = ( ε r ε b r ) 1 B ρ ( ξ ε b r ) 1 D × B ρ } J { χ = ( ξ ε b r ) 1 D } = ρ 2 Re ( ε r ε b r ) k 0 2 B ( l = 1 L E l P l ) + o ( ρ 3 ) ,
χ 0 ( r ) = η Re l = 1 L E l ( r ) P l ( r ) ,
χ n = χ n 1 + α n d n ,
d n = g n + γ n d n 1 , γ n = g n g n g n 1 D g n 1 D 2 ,
χ ( r ) = ε r ε b r ( r Ω ) , χ ( r ) = 0 ( r Ω ) ,
Ω = { r D such that ϕ ( r ) < 0 } .
J ( ϕ ) δ ϕ D = Re ( ε r ε b r ) δ ( ϕ ) ϕ l = 1 L E l P l ¯ δ ϕ D ,
ϕ t = J ( ϕ ) ,
I ̃ l obs ( r ) = ( 1 + b u ) I l obs ( r ) ,
F ( E s + δ E s ) = F ( E s ) + Re F ( E s ) δ E s Γ + o ( δ E s Γ ) .
L [ χ , E s ( χ ) , E ( χ ) , U s , U ] = J ( χ ) , U s , U .
J ( χ ) δ χ D = χ L ( χ , E s , E , U s , U ) δ χ D + E s L ( χ , E s , E , U s , U ) χ E s δ χ D + E L ( χ , E s , E , U s , U ) χ E δ χ D .
E s L ( χ , E s , E , U s , U ) δ E s Γ = 0 , δ E s ,
E L ( χ , E s , E , U s , U ) δ E D = 0 , δ E .
U l s = F ( E l s ) .
P l = G χ P l K t F ( E l s ) ¯ .
J ( χ ) δ χ D = χ L ( χ , E s , E , U s , U ) δ χ D = Re l = 1 L E l P l ¯ δ χ D ,

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