Abstract

The problem of determining the achievable resolution limits in the reconstruction of a current distribution is considered. The analysis refers to the one-dimensional, scalar case of a rectilinear, bounded electric current distribution when data are collected by measurement of the radiated field over a finite rectilinear observation domain located in the Fresnel zone, orthogonal and centered with respect to the source. The investigation is carried out by means of analytical singular-value decomposition of the radiation operator connecting data and unknown, which is made possible by the introduction of suitable scalar products in both the unknown and data spaces. This strategy permits the use of the results concerning prolate spheroidal wave functions described by B. R. Frieden [Progress in Optics Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam1971), p. 311.] For values of the space–bandwidth product much larger than 1, the steplike behavior of the singular values reveals that the inverse problem is severely ill posed. This, in turn, makes it mandatory to use regularization to obtain a stable solution and suggests a regularization scheme based on a truncated singular-value decomposition. The task of determining the depth-resolving power is accomplished with resort to Rayleigh’s criterion, and the effect of the geometrical parameters of the measurement configuration is also discussed.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Gunsay, B. D. Jeffs, “Point source localization in blurred images by a frequency-domain eigenvector-based method,” IEEE Trans. Image Process. 4, 1602–1612 (1995).
    [CrossRef]
  2. P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
    [CrossRef]
  3. A. Reigberg, A. Moreira, “First demonstration of airborne SAR tomography using multibaseline L-band data,” IEEE Trans. Geosci. Remote Sens. 38, 2142–2152 (2000).
    [CrossRef]
  4. A. N. Tikhonov, V. I. Arsenine, Solution to Ill-Posed Problems (Halsted, New York, 1977).
  5. D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Krieger, Malabar, Fla., 1992).
  6. C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).
  7. M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations (Mir, Moscow, 1976).
  8. D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer-Verlag, Berlin, 1992).
  9. M. Bertero, “Linear inverse and ill-posed problems,” Adv. Electron. Electron Phys. 75, 1–120 (1989).
    [CrossRef]
  10. R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).
  11. Hu, B. R. Frieden, “Restoration of longitudinal images,” Appl. Opt. 27, 414–418 (1988).
    [CrossRef] [PubMed]
  12. R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
    [CrossRef]
  13. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–64 (1961).
    [CrossRef]
  14. C. J. Bouwkamp, “On the spheroidal wavefunctions of zero order,” J. Math. Phys. 26, 79–92 (1957).
  15. B. R. Frieden, “Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions,” Progress in Optics Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.
  16. C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Wiesbaden, Germany, 1993).
  17. A. J. den Dekker, A. van den Bos, “Resolution: a survey,” J. Opt. Soc. Am. A 14, 547–557 (1997).
    [CrossRef]
  18. E. Hille, J. D. Tamarkin, “On the characteristic values of linear integral equations,” Acta Math. 57, 1–76 (1931).
    [CrossRef]
  19. C. K. Rushforth, R. W. Harris, “Restoration, resolution, and noise,” J. Opt. Soc. Am. 58, 539–545 (1968).
    [CrossRef]
  20. M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
    [CrossRef]
  21. G. Newsam, R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985).
    [CrossRef]
  22. G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164.
  23. E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
    [CrossRef]
  24. P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
    [CrossRef]
  25. S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
    [CrossRef]
  26. 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).
    [CrossRef]
  27. R. Pierri, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multi- frequency, multiview, and multifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (1999).
    [CrossRef]
  28. R. Pierri, A. Liseno, F. Soldovieri, “Shape diagnostics via a Kirchhoff inverse scattering in 2-d geometry,” presented at the International Conference on Antennas and Propagation, Davos, Switzerland, April 2000.
  29. A. Taylor, D. Lay, Introduction to Functional Analysis (Krieger, Malabar, Fla., 1980).
  30. Abdul J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [CrossRef]

2000 (2)

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

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

1999 (2)

1998 (1)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

1997 (1)

1995 (1)

M. Gunsay, B. D. Jeffs, “Point source localization in blurred images by a frequency-domain eigenvector-based method,” IEEE Trans. Image Process. 4, 1602–1612 (1995).
[CrossRef]

1993 (1)

1989 (2)

P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
[CrossRef]

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

1988 (1)

1985 (2)

G. Newsam, R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985).
[CrossRef]

M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

1977 (1)

Abdul J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[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]

1957 (1)

C. J. Bouwkamp, “On the spheroidal wavefunctions of zero order,” J. Math. Phys. 26, 79–92 (1957).

1955 (1)

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

1931 (1)

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

Arsenine, V. I.

A. N. Tikhonov, V. I. Arsenine, Solution to Ill-Posed Problems (Halsted, New York, 1977).

Baker, C. T. H.

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

Barakat, R.

Baratonia, C.

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

Bernini, R.

Bertero, M.

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

M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “On the spheroidal wavefunctions of zero order,” J. Math. Phys. 26, 79–92 (1957).

Colton, D.

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

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Krieger, Malabar, Fla., 1992).

De Mol, C.

M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

den Dekker, A. J.

Fellgett, P.

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Frieden, B. R.

Hu, B. R. Frieden, “Restoration of longitudinal images,” Appl. Opt. 27, 414–418 (1988).
[CrossRef] [PubMed]

B. R. Frieden, “Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions,” Progress in Optics Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.

Germain, P.

P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
[CrossRef]

Groetsch, C. W.

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

Gunsay, M.

M. Gunsay, B. D. Jeffs, “Point source localization in blurred images by a frequency-domain eigenvector-based method,” IEEE Trans. Image Process. 4, 1602–1612 (1995).
[CrossRef]

Harris, R. W.

Hille, E.

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

Hu,

Jeffs, B. D.

M. Gunsay, B. D. Jeffs, “Point source localization in blurred images by a frequency-domain eigenvector-based method,” IEEE Trans. Image Process. 4, 1602–1612 (1995).
[CrossRef]

Jerri, Abdul J.

Abdul J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Kiselev, A. I.

M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations (Mir, Moscow, 1976).

Kopp, L.

P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
[CrossRef]

Krasnov, M. L.

M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations (Mir, Moscow, 1976).

Kress, R.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Krieger, Malabar, Fla., 1992).

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

Lay, D.

A. Taylor, D. Lay, Introduction to Functional Analysis (Krieger, Malabar, Fla., 1980).

Leone, G.

Linfoot, E.

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Liseno, A.

R. Pierri, A. Liseno, F. Soldovieri, “Shape diagnostics via a Kirchhoff inverse scattering in 2-d geometry,” presented at the International Conference on Antennas and Propagation, Davos, Switzerland, April 2000.

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

Maguer, A.

P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
[CrossRef]

Makarenko, G. I.

M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations (Mir, Moscow, 1976).

Moreira, A.

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

Newsam, G.

Persico, R.

Pierri, R.

R. Pierri, R. Persico, R. Bernini, “Information content of the Born field scattered by an embedded slab: multi- frequency, multiview, and multifrequency–multiview cases,” J. Opt. Soc. Am. A 16, 2392–2399 (1999).
[CrossRef]

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).
[CrossRef]

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

R. Pierri, A. Liseno, F. Soldovieri, “Shape diagnostics via a Kirchhoff inverse scattering in 2-d geometry,” presented at the International Conference on Antennas and Propagation, Davos, Switzerland, April 2000.

Pike, R.

M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

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]

Prasad, S.

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

Reigberg, A.

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

Rushforth, C. K.

Scalas, E.

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, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

R. Pierri, A. Liseno, F. Soldovieri, “Shape diagnostics via a Kirchhoff inverse scattering in 2-d geometry,” presented at the International Conference on Antennas and Propagation, Davos, Switzerland, April 2000.

Solimene, R.

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

Tamarkin, J. D.

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

Taylor, A.

A. Taylor, D. Lay, Introduction to Functional Analysis (Krieger, Malabar, Fla., 1980).

Tikhonov, A. N.

A. N. Tikhonov, V. I. Arsenine, Solution to Ill-Posed Problems (Halsted, New York, 1977).

van den Bos, A.

Viano, G. A.

E. Scalas, G. A. Viano, “Resolving power and information theory in signal recovery,” J. Opt. Soc. Am. A 10, 991–996 (1993).
[CrossRef]

G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164.

Acta Math. (1)

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

Adv. Electron. Electron Phys. (1)

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

Appl. Opt. (1)

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. Acoust., Speech, Signal Process. (1)

P. Germain, A. Maguer, L. Kopp, “Comparison of resolving power of array processing methods by using an analytical criterion,” IEEE Trans. Acoust., Speech, Signal Process. 4, 2791–2794 (1989).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

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

IEEE Trans. Image Process. (1)

M. Gunsay, B. D. Jeffs, “Point source localization in blurred images by a frequency-domain eigenvector-based method,” IEEE Trans. Image Process. 4, 1602–1612 (1995).
[CrossRef]

Inverse Probl. (2)

R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

M. Bertero, C. De Mol, R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Probl. 1, 301–330 (1985).
[CrossRef]

J. Math. Phys. (1)

C. J. Bouwkamp, “On the spheroidal wavefunctions of zero order,” J. Math. Phys. 26, 79–92 (1957).

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Proc. IEEE (1)

Abdul J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Other (11)

G. A. Viano, “Fredholm integral equations of first kind and the method of correlogram,” in Inverse and Ill-Posed Problems, H. W. Engl, C. W. Groetsch, eds. (Academic, San Diego, Calif., 1987), pp. 151–164.

R. Pierri, A. Liseno, F. Soldovieri, “Shape diagnostics via a Kirchhoff inverse scattering in 2-d geometry,” presented at the International Conference on Antennas and Propagation, Davos, Switzerland, April 2000.

A. Taylor, D. Lay, Introduction to Functional Analysis (Krieger, Malabar, Fla., 1980).

B. R. Frieden, “Evaluation design and extrapolation methods for optical signals, based on use of the prolate functions,” Progress in Optics Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.

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

R. Pierri, C. Baratonia, A. Liseno, F. Soldovieri, R. Solimene, “Depth resolving power in Fresnel and near zone,” in Image Reconstruction from Incomplete Data, M. Fiddy, R. P. Millane, eds., Proc. SPIE4123 (2000).

A. N. Tikhonov, V. I. Arsenine, Solution to Ill-Posed Problems (Halsted, New York, 1977).

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Krieger, Malabar, Fla., 1992).

C. T. H. Baker, The Numerical Treatment of Integral Equations (Clarendon, Oxford, 1977).

M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations (Mir, Moscow, 1976).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Normalized behavior of the singular values of the relevant operator for a source with z0=100λ and z1=500λ and five observation domains of extents 2a=80λ (leftmost solid curve), 2a=90λ (dotted curve), 2a=100λ (dashed–dotted curve), 2a=110λ (dashed curve) and 2a=120λ (rightmost solid curve).

Fig. 3
Fig. 3

Amplitude behavior of the un’s normalized with respect to their maximum for a source with z0=100λ and z1=500λ, for an observation domain with 2a=120λ, with n=1, 6,11, 16.

Fig. 4
Fig. 4

Amplitude behavior of the vn’s normalized with respect to their maximum for a source with z0=100λ and z1=500λ, for an observation domain with 2a=120λ, with n=1,6,11,16.

Fig. 5
Fig. 5

Representation of the system for which the impulse response is calculated.

Fig. 6
Fig. 6

Amplitude normalized to its maximum of the regularized reconstruction of δ(z-zc) with zc=150λ, z0=100λ, z1=500λ and 2a=120λ. Comparison of the spread for the regularized reconstruction with the position of the zeros for u16 (dotted curve).

Fig. 7
Fig. 7

Amplitude normalized to its maximum of the regularized reconstruction of δ(z-zc) with zc=250λ, z0=100λ, z1=500λ, and 2a=120λ.

Fig. 8
Fig. 8

Amplitude normalized to its maximum of the regularized reconstruction of δ(z-zc) with zc=150λ, z0=100λ, z1=500λ, and 2a=80λ normalized to its maximum value.

Equations (51)

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

E(x)=-ωμ04z0z1H0(2)(βr)J(z)dz,-axa,
H0(2)(βr)(2/βπr)1/2exp(-jβr)exp(jπ/4).
E(x)=z0z11zexp(-jβz)exp-j(βx2)2zJ(z)dz
-axa,
f, g=t1t2f(t)g*(t)α(t)dt
f, gL2((t1, t2), α(t)),
L : JXEY,
Lun=σnvn,
Lvn=σnun.
E=L(J)=n=0+σnJ, unX vnJX.
vn(x)=Φn[c, x2-a2/2][λn(c)]1/2exp(-jπtmx2/λ),
un(z)=jnσn1z3/2exp(jβz)exp-j πa22λtm-1z×λa2Δt1/2Φnc, tm-1za22Δt,
σn=[2λλn(c)]1/2,
-x0x0Φn[c, s]exp(jωs)ds
=jn2πλn(c)x0Ω1/2Φnc, ωx0Ωω,
NDF=a22λ1z0-1z1.
J=n=0+1σn E, vnY un,
RJ=n=0NDF-11σn E, vnY un.
J(z)=z0z1δ(z-μ)J(μ)dμa.e.onz0zz1,
E, vnY=σnJ, unX;
[Rδ(z-zc)](z)=zcn=0NDF-1un*(zc)un(z),
limN+n=0NΦn[c, y]Φn[c, s]λn(c)=δ(s-y),
|s|x0,|y|x0.
s=tm-1za22Δt,
withs-a22, a22,z[z0, z1],
Δs=a2/NDF
si+1-si=1zi-1zi+Δzia22Δt=Δs,
Δzi=2λzi2a2-2λzi.
J(z)=exp(-jβz)z J(z),
E(x)=t1t0J^(t)exp(-jπx2t/λ)dt=jπx2λ,
LJ, EY=J, LEX,
(LE)(z)=exp(jβz)z3/2-aa|x|exp[j(βx2)/2z]E(x)dx.
(LLvn)(x)=2λ-aa|x|vn(x)×expj πtmλ (x2-x2)×sin[(π/λ)Δt(x2-x2)]π(x2-x2)dx=σn2vn(x),
2λ-a2/2a2/2gny+a221/2sin[(π/λ)Δt(y-h)]π(y-h) dy
=σn2gnh+a221/2,
vn(x)=exp(-jπtmx2/λ)λn Φnx2-a22.
un(z)=1σn (Lvn)(z).
τ : hYτh=|x|hL2(-a, a),
(τE)(x)=|x|1/2exp(-jπtmx2/λ)n=0E, vnY×Φn[c, x2-a2/2][λn(c)]1/2;x(-a, a).
f(s)=n=0E, vnYΦn[c, s][λn(c)]1/2 s,
f(s)n-a22Ωπf n πΩsincs-n πΩs-a22.
g(x)=fx2-a22x.
g(x)n-a22Ωπgn πΩ+a221/2×sincx2-a22-n πΩx.
E(x)exp(-jπtmx2/λ)n-a22Ωπexp(jπtmxn2/λ)×E(xn)sincx2-a22-n πΩx,
xn=n πΩ+a221/2.
Δxi=1xi+1+xiπΩ.
E(x)exp(-jπtmx2/λ)|n|a22Ωπexp(jπtmxn2/λ)×E(xn)sincx2-a22-n πΩ
x(-a, a),
f(s)|n|a22Ωπf n πΩsincs-n πΩ
s-a22, a22.
N=2 a22Ωπ=2cπ.

Metrics