Abstract

We address the inverse scattering problem of estimating the resolution limits achievable in the reconstruction of a dielectric strip object within a two-dimensional and scalar geometry. The scattered field is observed over a bounded rectilinear domain located in the Fresnel zone, and a single-frequency multistatic–multiview configuration is considered. The analysis is performed by casting the problem as the inversion of the linearized scattering operator arising from the Born approximation and by means of its singular-value decomposition. Finally, the role of the geometrical parameters of the measurement configuration is highlighted.

© 2004 Optical Society of America

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References

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  1. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer-Verlang, New York, 1996).
  2. A. J. Dekker, A. van den Boss, “Resolution: a survey,” J. Opt. Soc. Am. A 14, 547–557 (1997).
    [CrossRef]
  3. 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]
  4. F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
    [CrossRef]
  5. A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. A 4, 189–199 (1987).
    [CrossRef]
  6. M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
    [CrossRef]
  7. A. Reigber, A. Moreira, “First demostration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
    [CrossRef]
  8. D. J. Daniels, Surface-Penetrating Radar (Radar, Sonar, Navigation and Avionics Series 6 (IEE Press, London, 1996).
  9. R. Pierri, A. Liseno, R. Solimene, F. Tartaglione, “In-depth resolution from multifrequency Born fields scattered by a dielectric strip in the Fresnel,” J. Opt. Soc. Am. A 19, 1234–1238 (2002).
    [CrossRef]
  10. R. Solimene, “Resolution in linear tomographic reconstruction,” Ph.D. thesis (Seconda Università di Napoli, Aversa, Italy, 2003).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  13. C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  14. R. Pierri, A. Liseno, F. Soldovieri, R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
    [CrossRef]

2002 (1)

2001 (1)

2000 (1)

A. Reigber, A. Moreira, “First demostration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
[CrossRef]

1999 (1)

1998 (1)

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

1997 (1)

1989 (1)

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

1987 (1)

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]

Bertero, M.

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

Brancaccio, A.

Casanove, M. J.

Chen, F. C.

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

Chew, W. C.

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[CrossRef]

Daniels, D. J.

D. J. Daniels, Surface-Penetrating Radar (Radar, Sonar, Navigation and Avionics Series 6 (IEE Press, London, 1996).

Dekker, A. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Harris, R. W.

Kirsch, A.

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

Lannes, A.

Leone, G.

Liseno, A.

Moreira, A.

A. Reigber, A. Moreira, “First demostration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
[CrossRef]

Pierri, 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]

Reigber, A.

A. Reigber, A. Moreira, “First demostration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
[CrossRef]

Roques, S.

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.

Solimene, R.

Tartaglione, F.

van den Boss, A.

Adv. Electron. Electron Phys. (1)

M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
[CrossRef]

Appl. Phys. Lett. (1)

F. C. Chen, W. C. Chew, “Experimental verification of super resolution in nonlinear inverse scattering,” Appl. Phys. Lett. 72, 3080–3082 (1998).
[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. Geosci. Remote Sens. (1)

A. Reigber, A. Moreira, “First demostration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Other (4)

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

D. J. Daniels, Surface-Penetrating Radar (Radar, Sonar, Navigation and Avionics Series 6 (IEE Press, London, 1996).

R. Solimene, “Resolution in linear tomographic reconstruction,” Ph.D. thesis (Seconda Università di Napoli, Aversa, Italy, 2003).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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Figures (3)

Fig. 1
Fig. 1

Relevant geometry. (a) Orthogonal domains, (b) parallel domains.

Fig. 2
Fig. 2

Parallel domains. z¯=110λ, a=10λ; Xo=10λ (solid curves), Xo=20λ (dashed curves). Top panel, normalized behavior of the singular values; bottom panel, normalized amplitude of the reconstruction of a pulse object located at xc=0.

Fig. 3
Fig. 3

Orthogonal domains. z0=100λ, z1=200λ; Xo=45λ (solid curves), Xo=60λ (dashed curves). Top panel, normalized behavior of the singular values; middle panel, normalized amplitude of the reconstruction of a pulse object located at zc=110λ; bottom panel, normalized amplitude of the reconstruction of a pulse object located at zc=170λ.

Equations (20)

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Es(xo, xS)=-aa1z¯exp-j2βz¯-j β(x-xo)22z¯-j β(x-xs)22z¯χ(x)dx,
Es(xo, xS)=z0z11zexp-2jβz-j β(xo2+xS2)2zχ(z)dz.
Es(η)=(Apχ)(η)=-aaexp(jβxη/z¯)exp(-jβx2/z¯)χ(x)dx,
Es(γ)=(Aoχ)(γ)=z0z1exp(-j2βz)zexp[-jβγ/(2z)]χ(z)dz,
upn(x)=ϕn(cp, x)[λn(cp)]1/2expj βx2z¯,
σpn=[λzλ¯n(cp)]1/2,
vpn(η)=jn[λn(cp)]1/2 [a/(2X0)]1/2ϕn×[cp, ηa/(2X0)],
uon=jnexp{jβ[2z+X02/2(1/z-tm)]}[λn(co)]1/2z ×(X02/Δ)1/2ϕn[co, (1/z-tm)X02Δ],
σon=[2λλn(co)]1/2,
von(γ)=ϕn(co, γ-X02)[λn(co)]1/2exp(-jβγtm/2),
Rδ(r-ric)=n=0Niuin*(ric)uin(r),
Δx=2a/Np,
Δz=(2z2Δ/No)/(1-2zΔ/No).
Δx=λz¯/(4Xo),
Δz=z2λ/(Xo2-zλ).
(Aof)(z)=02X021zexp(j2βz)exp[jβγ/(2z)]f(γ)dγ.
(AoAovon)(γ)=σon2vn(γ).
2λ-X02X02sin[βΔ/2(s-y)]π(s-y) gn(s)ds=σon2gn(y),
von(γ)=ϕn[co,(γ-X02)][λn(co)]1/2exp(-jβtmγ/2).
1/σon(Aovon)(z).

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