Abstract

The achievable depth resolution in reconstructing the permittivity profile of a dielectric strip under the Born approximation when data are collected in the Fresnel zone is studied. We consider a rectilinear measurement aperture and an orthogonal and centered rectilinear investigation domain. The roles of the aperture extent and frequency diversity are highlighted.

© 2002 Optical Society of America

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References

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  1. M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).
  2. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996).
  3. T. Habashy, E. Wolf, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1675 (1994).
    [CrossRef]
  4. 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]
  5. R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth reconstruction for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
    [CrossRef]
  6. A. den Dekker, A. van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A 14, 547–557 (1997).
    [CrossRef]
  7. C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  8. A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
    [CrossRef]
  9. E. Hille, J. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
    [CrossRef]
  10. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  11. R. Piestun, D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
    [CrossRef]
  12. R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).
  13. R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
    [CrossRef]
  14. E. N. Leith, K. D. Mills, P. P. Naulleau, D. S. Dilworth, I. Iglesias, H. S. Chen, “Generalized confocal imaging and synthetic aperture imaging,” J. Opt. Soc. Am. A 16, 2880–2886 (1999).
    [CrossRef]
  15. R. Pierri, G. Leone, R. Persico, “Second-order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
    [CrossRef]

2001 (4)

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth reconstruction for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

1997 (1)

1994 (1)

T. Habashy, E. Wolf, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1675 (1994).
[CrossRef]

1968 (1)

1961 (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

1931 (1)

E. Hille, J. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

Boccacci, P.

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

Brancaccio, A.

Chen, H. S.

den Dekker, A.

Dilworth, D. S.

Habashy, T.

T. Habashy, E. Wolf, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1675 (1994).
[CrossRef]

Harris, R. W.

Hille, E.

E. Hille, J. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Iglesias, I.

Kirsch, A.

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996).

Leith, E. N.

Leone, G.

Liseno, A.

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth reconstruction for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

Miller, D. A. B.

Mills, K. D.

Naulleau, P. P.

Persico, R.

R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).

R. Pierri, G. Leone, R. Persico, “Second-order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
[CrossRef]

Pierri, R.

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).

A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth reconstruction for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

R. Pierri, G. Leone, R. Persico, “Second-order iterative approach to inverse scattering: numerical results,” J. Opt. Soc. Am. A 17, 874–880 (2000).
[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]

Piestun, R.

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Rushforth, C. K.

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

Soldovieri, F.

R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth reconstruction for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
[CrossRef]

Solimene, R.

Tamarkin, J.

E. Hille, J. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

van den Bos, A.

Wolf, E.

T. Habashy, E. Wolf, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1675 (1994).
[CrossRef]

Acta Math. (1)

E. Hille, J. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. Pierri, A. Liseno, F. Soldovieri, “Shape reconstruction from PO multifrequency scattered fields via the singular value decomposition approach,” IEEE Trans. Antennas Propag. 49, 1333–1343 (2001).
[CrossRef]

Int. J. Electron. Commun. (AEÜ) (1)

A. Liseno, R. Pierri, F. Soldovieri, “Depth-resolving power in near zone: numerical results for a strip source,” Int. J. Electron. Commun. (AEÜ) 55, 100–108 (2001).
[CrossRef]

J. Mod. Opt. (1)

T. Habashy, E. Wolf, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1675 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Nuovo Cimento C (1)

R. Pierri, G. Leone, F. Soldovieri, R. Persico, “Electromagnetic inversion for subsurface applications under the distorted Born approximation,” Nuovo Cimento C 24, 245–261 (2001).

Other (2)

M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, Bristol, UK, 1998).

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlag, New York, 1996).

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Figures (5)

Fig. 1
Fig. 1

Illustration of the geometry of the problem.

Fig. 2
Fig. 2

Illustration of two longitudinally located point scatterers viewed under different angles.

Fig. 3
Fig. 3

Normalized amplitude of the regularized reconstruction of δ(z-zc). The investigation domain is confined between z0=100λmin, x0=0, and z1=150λmin and λmax=1.1λmin. Top, zc=110λmin; bottom, zc=140λmin.

Fig. 4
Fig. 4

Comparison of the singular values (top) and the regularized reconstructions of δ(z-zc) (bottom) for the multifrequency case with measurement at only one point of the observation aperture. The investigation domain is confined between z0=100λmin and z1=150λmin and λmax=1.1λmin. Solid curve, x0=0; dashed curve, x0=60λmin.

Fig. 5
Fig. 5

Comparison of the regularized reconstructions of δ(z-zc) for the multifrequency cases with measurement at only one point x0=0 (solid curve) and measurement on an aperture having semi-extent a=60λmin (dashed curve). The investigation domain is confined between z0=100λmin and z1=150λmin and λmax=1.1λmin.

Equations (31)

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

Es(x, β)=z0z11zexp(-j2βz)×exp-j βx22zχ(z)dz
|x|aβminββmax,
Es=Lχ
Rχ=n=0N-11σn Es, vnY un,
Rδ(z-zc)=n=0N-1δ(z-zc), unX un,
Es(x, β0)=(Lsfχ)(x, β0)
Es(x0, β)=(Lmfχ)(x0, β).
(Lsfχ)(x, β0)=z0z11zexp(-j2β0z)
×exp-j β0x22zχ(z)dz,|x|a,
(Lmfχ)(x0, β)=z0z11zexp(-j2βz)exp-j βx022zχ(z)dz,
βminββmax.
(Lisχ)(x, β0)=z0z11zexp(-jβ0z)exp-j β0x22zχ(z)dz,
|x|a.
LsfLsf=Lisexp(-jβ0z)exp(jβ0z)Lis=LisLis,
Δz=2λz2a2-2λz.
f, g=z0z1f(z)g*(z)α(z)dz
vn(β)=exp(-jβtm)λn(c) Φn[c, (β-βm)],
un(z)=jnσnexp[jβmw(z)]zα(z)2πλn(c)Δβ2Ω Φnc, w(z)Δβ2Ω,
σn=2πλn(c),
NDF1λmin-1λmax2(z1-z0)+x0221z1-1z0.
NDF=2 (λmax-λmin)λmaxλmin (z1-z0),
Δz=λminλmax2(λmax-λmin)=πΔβ.
βπ-βx24πz2
βmax-βminπ+βmina24πz02.
Lmfχ, EsY=χ, LmfEsX,
(Lmf Es)(z)=1zα(z)βminβmaxexp(j2βz)×expj βx022zEs(β)dβ,
z0zz1.
(LmfLmfvn)(x)=2πβminβmaxvn(s)exp[j(s-β)tm]×sin[Ω(s-β)]π(s-β)ds=σn2vn(β).
2π-ΔβΔβgn(βm+h) sin[Ω(h-k)]π(h-k)dh=σn2gn(βm+k),
vn(β)=exp(-jtmβ)λn Φn(c, (β-βm).
un(z)=1σn (Lmfvn)(z).

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