Abstract

The singular value decomposition of the far-zone scattering operator for weak strip-like scattering objects is studied under multiple view and/or multiple frequency illuminations. The aim is to highlight how such diversities impact the number of degrees of freedom (NDF) of the scattering problem. When the angles of incidence and/or frequencies vary within discrete finite sets, the singular values are analytically determined. It is shown that they exhibit a multistep behavior. For the continuous case, upper and lower bounds are found, which allows us to obtain estimations for the NDF dependending on the parameters of the configuration.

© 2013 Optical Society of America

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References

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  1. F. Riesz and B. Nagy, Functional Analysis (Dover, 1990).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
    [CrossRef]
  7. R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
    [CrossRef]
  8. A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
    [CrossRef]
  9. A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes applied to axicons,” Opt. Express 12, 377–383 (2004).
    [CrossRef]
  10. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
  11. N. Magnoli and G. A. Viano, “On the eigenfunction expansions associated with Fredholm integral equation of first kind in the presence of noise,” J. Math. Anal. Appl. 197, 188–206 (1996).
    [CrossRef]
  12. R. A. Frazin, D. G. Fischer, and P. S. Carney, “Information content of the near field: two-dimensional samples,” J. Opt. Soc. Am. A 21, 1050–1057 (2004).
    [CrossRef]
  13. D. G. Fischer, R. A. Frazin, M. Asipauskas, and P. S. Carney, “Information content of the near field: three-dimensional samples,” J. Opt. Soc. Am. A 28, 296–306 (2011).
    [CrossRef]
  14. E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283–310 (2002).
    [CrossRef]
  15. F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
    [CrossRef]
  16. D. Slepian and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [CrossRef]
  17. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
    [CrossRef]
  18. L.-J. Deng, T.-Z. Huang, X.-L. Zhao, L. Zhao, and S. Wang, “Signal restoration combining Tikhonov regularization and multilevel method with thresholding strategy,” J. Opt. Soc. Am. A 30, 948–955 (2013).
    [CrossRef]
  19. E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
    [CrossRef]
  20. D. L. Marks, “A family of approximations spanning the Born and Rytov scattering series,” Opt. Express 14, 8837–8847 (2006).
    [CrossRef]
  21. D. G. Fischer, “The information content of weakly scattered fields: implications for the near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).
  22. R. Persico, “On the role of measurement configuration in contactless GPR data processing by means of linear inverse scattering,” IEEE Trans. Antennas Propag. 54, 2062–2071 (2006).
    [CrossRef]
  23. R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
    [CrossRef]
  24. R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A 18, 352–359 (2001).
    [CrossRef]
  25. R. Pierri, A. Liseno, R. Solimene, and F. Tartaglione, “In-depth resolution from multifrequency Born fields scattered by a dielectric strip in the Fresnel zone,” J. Opt. Soc. Am. A 19, 1234–1238 (2002).
    [CrossRef]
  26. R. Solimene, G. Leone, and R. Pierri, “Multistatic multiview resolution from Born fields for strips in Fresnel zone,” J. Opt. Soc. Am. A 21, 1402–1406 (2004).
    [CrossRef]
  27. A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
    [CrossRef]
  28. R. Solimene and R. Pierri, “Localization of a planar perfect-electric-conduting interface embedded in a half-space,” J. Opt. A 8, 10–16 (2006).
    [CrossRef]
  29. R. Solimene, R. Barresi, and G. Leone, “Localizing a buried planar perfect electric conducting interface by multi-view data,” J. Opt. A 10, 015010 (2008).
    [CrossRef]
  30. F. Gori, “Integral equations for incoherent imagery,” J. Opt. Soc. Am. 64, 1237–1243 (1974).
    [CrossRef]
  31. F. Gori and C. Palma, “On the eigenvalues of the sinc2 kernel, “ J. Phys. A 8, 1709–1719 (1975).
    [CrossRef]
  32. D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
    [CrossRef]
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    [CrossRef]

2013 (2)

2011 (1)

2010 (1)

R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
[CrossRef]

2008 (1)

R. Solimene, R. Barresi, and G. Leone, “Localizing a buried planar perfect electric conducting interface by multi-view data,” J. Opt. A 10, 015010 (2008).
[CrossRef]

2006 (3)

R. Persico, “On the role of measurement configuration in contactless GPR data processing by means of linear inverse scattering,” IEEE Trans. Antennas Propag. 54, 2062–2071 (2006).
[CrossRef]

R. Solimene and R. Pierri, “Localization of a planar perfect-electric-conduting interface embedded in a half-space,” J. Opt. A 8, 10–16 (2006).
[CrossRef]

D. L. Marks, “A family of approximations spanning the Born and Rytov scattering series,” Opt. Express 14, 8837–8847 (2006).
[CrossRef]

2004 (3)

2003 (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

2002 (2)

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283–310 (2002).
[CrossRef]

R. Pierri, A. Liseno, R. Solimene, and F. Tartaglione, “In-depth resolution from multifrequency Born fields scattered by a dielectric strip in the Fresnel zone,” J. Opt. Soc. Am. A 19, 1234–1238 (2002).
[CrossRef]

2001 (1)

2000 (3)

1998 (3)

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

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

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

1996 (1)

N. Magnoli and G. A. Viano, “On the eigenfunction expansions associated with Fredholm integral equation of first kind in the presence of noise,” J. Math. Anal. Appl. 197, 188–206 (1996).
[CrossRef]

1989 (2)

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

1985 (1)

1975 (1)

F. Gori and C. Palma, “On the eigenvalues of the sinc2 kernel, “ J. Phys. A 8, 1709–1719 (1975).
[CrossRef]

1974 (1)

1973 (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

1969 (1)

1961 (2)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Asipauskas, M.

Barakat, R.

Barresi, R.

R. Solimene, R. Barresi, and G. Leone, “Localizing a buried planar perfect electric conducting interface by multi-view data,” J. Opt. A 10, 015010 (2008).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

Brancaccio, A.

Bucci, O. M.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Burvall, A.

Carney, P. S.

De Micheli, E.

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283–310 (2002).
[CrossRef]

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

Deng, L.-J.

di Francia, G. T.

Donoho, D. L.

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

Fischer, D. G.

Franceschetti, G.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Frazin, R. A.

Friberg, A. T.

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes applied to axicons,” Opt. Express 12, 377–383 (2004).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Gori, F.

F. Gori and C. Palma, “On the eigenvalues of the sinc2 kernel, “ J. Phys. A 8, 1709–1719 (1975).
[CrossRef]

F. Gori, “Integral equations for incoherent imagery,” J. Opt. Soc. Am. 64, 1237–1243 (1974).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Guattari, G.

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Huang, T.-Z.

Karelin, M.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Landau, H. J.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

Leone, G.

Liseno, A.

Magnoli, N.

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

N. Magnoli and G. A. Viano, “On the eigenfunction expansions associated with Fredholm integral equation of first kind in the presence of noise,” J. Math. Anal. Appl. 197, 188–206 (1996).
[CrossRef]

Marks, D. L.

Martinsson, P.

A. Burvall, P. Martinsson, and A. T. Friberg, “Communication modes applied to axicons,” Opt. Express 12, 377–383 (2004).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Miller, D. A.

Miller, D. A. B.

Nagy, B.

F. Riesz and B. Nagy, Functional Analysis (Dover, 1990).

Newsam, G.

Palma, C.

F. Gori and C. Palma, “On the eigenvalues of the sinc2 kernel, “ J. Phys. A 8, 1709–1719 (1975).
[CrossRef]

Persico, R.

R. Persico, “On the role of measurement configuration in contactless GPR data processing by means of linear inverse scattering,” IEEE Trans. Antennas Propag. 54, 2062–2071 (2006).
[CrossRef]

Pierri, R.

Piestun, R.

Pollak, H. O.

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

D. Slepian and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Riesz, F.

F. Riesz and B. Nagy, Functional Analysis (Dover, 1990).

Slepian, D.

D. Slepian and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

Soldovieri, F.

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

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

Solimene, R.

Somaraju, R.

R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
[CrossRef]

Stark, P. B.

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

Tartaglione, F.

Thaning, A.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Trumpf, J.

R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
[CrossRef]

Viano, G. A.

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283–310 (2002).
[CrossRef]

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

N. Magnoli and G. A. Viano, “On the eigenfunction expansions associated with Fredholm integral equation of first kind in the presence of noise,” J. Math. Anal. Appl. 197, 188–206 (1996).
[CrossRef]

Wang, S.

Zhao, L.

Zhao, X.-L.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Slepian and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[CrossRef]

H. J. Landau and H. O. Pollak, “Prolate spheroidal wave function, Fourier analysis and uncertainty II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. Persico, “On the role of measurement configuration in contactless GPR data processing by means of linear inverse scattering,” IEEE Trans. Antennas Propag. 54, 2062–2071 (2006).
[CrossRef]

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

IEEE Trans. Inf. Theory (1)

R. Somaraju and J. Trumpf, “Degrees of freedom of a communication channel: using DOF singular values,” IEEE Trans. Inf. Theory 56, 1560–1573 (2010).
[CrossRef]

Inverse Probl. (1)

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

J. Integral Equ. Appl. (1)

E. De Micheli and G. A. Viano, “Metric and probabilistic information associated with Fredholm integral equations of the first kind,” J. Integral Equ. Appl. 14, 283–310 (2002).
[CrossRef]

J. Math. Anal. Appl. (1)

N. Magnoli and G. A. Viano, “On the eigenfunction expansions associated with Fredholm integral equation of first kind in the presence of noise,” J. Math. Anal. Appl. 197, 188–206 (1996).
[CrossRef]

J. Mod. Opt. (1)

D. G. Fischer, “The information content of weakly scattered fields: implications for the near-field imaging of three-dimensional structures,” J. Mod. Opt. 47, 1359–1374 (2000).

J. Opt. A (3)

R. Solimene and R. Pierri, “Localization of a planar perfect-electric-conduting interface embedded in a half-space,” J. Opt. A 8, 10–16 (2006).
[CrossRef]

R. Solimene, R. Barresi, and G. Leone, “Localizing a buried planar perfect electric conducting interface by multi-view data,” J. Opt. A 10, 015010 (2008).
[CrossRef]

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

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

R. Pierri, A. Liseno, R. Solimene, and F. Tartaglione, “In-depth resolution from multifrequency Born fields scattered by a dielectric strip in the Fresnel zone,” J. Opt. Soc. Am. A 19, 1234–1238 (2002).
[CrossRef]

R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
[CrossRef]

A. Brancaccio, G. Leone, and R. Pierri, “Information content of Born scattered fields: results in the circular cylindrical case,” J. Opt. Soc. Am. A 15, 1909–1917 (1998).
[CrossRef]

G. Newsam and 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]

R. A. Frazin, D. G. Fischer, and P. S. Carney, “Information content of the near field: two-dimensional samples,” J. Opt. Soc. Am. A 21, 1050–1057 (2004).
[CrossRef]

R. Solimene, G. Leone, and R. Pierri, “Multistatic multiview resolution from Born fields for strips in Fresnel zone,” J. Opt. Soc. Am. A 21, 1402–1406 (2004).
[CrossRef]

D. G. Fischer, R. A. Frazin, M. Asipauskas, and P. S. Carney, “Information content of the near field: three-dimensional samples,” J. Opt. Soc. Am. A 28, 296–306 (2011).
[CrossRef]

D. A. Miller, “How complicated must an optical component be?” J. Opt. Soc. Am. A 30, 238–251 (2013).
[CrossRef]

L.-J. Deng, T.-Z. Huang, X.-L. Zhao, L. Zhao, and S. Wang, “Signal restoration combining Tikhonov regularization and multilevel method with thresholding strategy,” J. Opt. Soc. Am. A 30, 948–955 (2013).
[CrossRef]

J. Phys. A (1)

F. Gori and C. Palma, “On the eigenvalues of the sinc2 kernel, “ J. Phys. A 8, 1709–1719 (1975).
[CrossRef]

Opt. Commun. (1)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Opt. Express (2)

SIAM J. Appl. Math. (1)

D. L. Donoho and P. B. Stark, “Uncertainty principles and signal recovery,” SIAM J. Appl. Math. 49, 906–931 (1989).
[CrossRef]

SIAM J. Math. Anal. (1)

E. De Micheli, N. Magnoli, and G. A. Viano, “On the regularization of Fredholm integral equations of the first kind,” SIAM J. Math. Anal. 29, 855–877 (1998).
[CrossRef]

Other (2)

F. Riesz and B. Nagy, Functional Analysis (Dover, 1990).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).

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Figures (12)

Fig. 1.
Fig. 1.

Pictorial view of the scattering configuration considered in this paper.

Fig. 2.
Fig. 2.

Frequency band Ωm associated to the mth angle of view.

Fig. 3.
Fig. 3.

Case of two angles of incidence Ωi={umax,umax}. The top panel gives a pictorial view of the two frequency bands. In the bottom panel the singular value of the scattering operator for Ωo=[1,1] and a=20π/k0 is displayed. As can be seen, the knee indeed occurs at an index twice the one of the single view case.

Fig. 4.
Fig. 4.

Case of three angles of incidence Ωi={umax,0,umax}. The top panel gives a pictorial view of the frequency bands that now overlap. The bottom panel shows how the spectrum of the kernel can be still given in terms of disjoint bands.

Fig. 5.
Fig. 5.

Singular value behavior for six angles of incidence, Ωo=[1,1] and a=20π/k0. As can be seen, the expected five steps are well evident.

Fig. 6.
Fig. 6.

Singular value behavior of Ai for the case of continuous views. 2M=10, Ωo=[1,1], and a=20π/k0. Green and blue lines represent the square root of the eigenvalues of AiAi˜ and AiAi^, respectively.

Fig. 7.
Fig. 7.

Illustration of how to rearrange the frequency bands to obtain Eq. (3).

Fig. 8.
Fig. 8.

Singular value behavior of Af for the case of M=3 frequencies k0min, 1.5k0min and 2k0min, k0min=2πm1, and a=20π/k0min.

Fig. 9.
Fig. 9.

Pictorial view of the Fourier spectrum K(u) of the kernel function in Eq. (15). Note that for simplicity umax=1.

Fig. 10.
Fig. 10.

Singular value behavior of Af for the case of continuous frequencies. 2M=12, Ω=[1,1], and Ωk0=[2k0,2k0]. k0=2πm1 and a=20π/k0. Green and blue lines represent the square root of the eigenvalues of AfAf˜ and AfAf^, respectively.

Fig. 11.
Fig. 11.

Pictorial view of the Fourier spectrum K(u) of the kernel function in Eq. (20).

Fig. 12.
Fig. 12.

Singular value behavior of Aif for the case of continuous frequencies. 2M=12, Ω=[1,1], Ωk0=[k0,1.5k0], k0=2πm1, and a=20π/k0. Green and blue lines represent the square root of the eigenvalues of AfAf˜ and AfAf^, respectively.

Equations (22)

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

Aif:χ(x)LI2ES(uo,ui,k0)=aaexp[jk0(uoui)x]χ(x)dxL(Ωo×Ωi×Ωk0)2
S=α1PIBΩ1PI+α2PIBΩ2PI,
{{un[S]}={un1}{un2}{λn[S]}={α1λn1}{α2λn2},
K:f(x)LI2Kf(x)LI2
λn[K^]λn[K]λn[K˜]n,
K˜f(x)=m=1mK˜mPIBΩmPIf(x),
K^f(x)=m=1MK^mPIBΩmPIf(x)
AiAi=m=1M2πk0PIBΩmPI,
AiAi=2πk0PIBΩ1PI+2πk0PIBΩ2PI.
AiAi=4πk0PIBΩ˜1PI+2πk0PIBΩ˜2PI+2πk0PIBΩ˜3PI
AiAif(x)=aa4π2k02sin2[k0umax(xy)]π2(xy)2f(y)dy.
{N(τth,c)(m1)[2cM/π]K˜m<τthN(τth,c)2m[2cM/π]τth<K^mN(τth,c)[2c/π]K˜M>τthandmM.
AfAf=m=1M2πk0mPIBΩmPI,
AfAf=2πm=131k0mPIBΩ1PI+m=2M2πl=mM1k0l(PIBΩ˜mPI+PIBΩ^mPI),
AfAff(x)=aak0mink0max2πk0sin[k0umax(xy)]π(xy)dk0f(y)dy.
K(u)={2πln(k0max/k0min)|u|k0minumax2πln(k0maxumax/|u|)k0minumax|u|k0maxumax0elsewhere..
AfAf˜=2πln(k0max/k0min)PIBΩ0PI+m=1MK˜m(PIBΩ˜mPI+PIBΩ^mPI),
AfAf^=2πln(k0max/k0min)PIBΩ0PI+m=1MK^m(PIBΩ˜mPI+PIBΩ^mPI),
{N(τth,c)N0+2m[2cM/π]K˜m<τthN(τth,c)N0+2m[2cM/π]τth<K^mN(τth,c)[2c/π]K˜M>τthandmM.
AifAiff(x)=aak0mink0max×4π2k02[sin[k0umax(xy)]π(xy)]2dk0f(y)dy,
K(u)={4πumaxln(k0max/k0min)+2π|u|(1/k0max1/k0min)|u|2k0minumax4πumaxln(2k0maxumax/|u|)+2π|u|(1/k0max2umax/|u|)2k0minumax|u|2k0maxumax0elsewhere..
{N(τth,c)(m1)[2cM/π]K˜m<τthN(τth,c)2m[2cM/π]τth<K^mN(τth,c)[2c/π]K˜M>τthandmM.

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