Abstract

The ambiguity involved in reconstructing an image from limited Fourier data is removed using a new technique that incorporates prior knowledge of the location of regions containing small-scale features of interest. The prior discrete Fourier transform (PDFT) method for image reconstruction incorporates prior knowledge of the support, and perhaps general shape, of the object function being reconstructed through the use of a weight function. The new approach extends the PDFT by allowing different weight functions to modulate the different spatial frequency components of the reconstructed image. The effectiveness of the new method is tested on one- and two-dimensional simulations.

© 2010 Optical Society of America

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References

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  1. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998).
    [CrossRef]
  2. C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).
  3. M. Bertero, “Sampling theory, resolution limits and inversion methods,” in Inverse Problems in Scattering and Imaging, Malvern Physics Series, M.Bertero and E.R.Pike, eds. (Adam Hilger, 1992), pp. 71-94.
  4. R. W. Gerchberg, “Super-restoration through error energy reduction,” Opt. Acta 21, 709-720 (1974).
    [CrossRef]
  5. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735-742 (1975).
    [CrossRef]
  6. J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, 1967.
  7. J. P. Burg, “The relationship between maximum entropy spectra and maximum likelihood spectra,” Geophysics 37, 375-376 (1972).
    [CrossRef]
  8. J. P. Burg, “Maximum entropy spectral analysis,” Ph.D. thesis (Stanford University, 1975).
  9. C. L. Byrne and R. M. Fitzgerald, “Reconstruction from partial information, with applications to tomography,” SIAM J. Appl. Math. 42, 933-940 (1982).
    [CrossRef]
  10. C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481-1487 (1983).
    [CrossRef]
  11. C. L. Byrne and R. M. Fitzgerald, “Spectral estimators that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math. 44, 425-442 (1984).
    [CrossRef]
  12. C. L. Byrne and M. A. Fiddy, “Estimation of continuous object distributions from limited Fourier magnitude measurements,” J. Opt. Soc. Am. A 4, 112-117 (1987).
    [CrossRef]
  13. C. L. Byrne and M. A. Fiddy, “Image as power spectral; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
    [CrossRef]
  14. T. J. Hall, A. M. Darling, and M. A. Fiddy, “Image compression and restoration incorporating prior knowledge,” Opt. Lett. 7, 467-468 (1982).
    [CrossRef] [PubMed]
  15. H. M. Shieh, C. L. Byrne, and M. A. Fiddy, “Image reconstruction: a unifying model for resolution enhancement and data extrapolation. Tutorial,” J. Opt. Soc. Am. A 23, 258-266 (2006).
    [CrossRef]
  16. H. M. Shieh and M. A. Fiddy, “Accuracy of extrapolated data as a function of prior knowledge and regularization,” Appl. Opt. 45, 3283-3288 (2006).
    [CrossRef] [PubMed]
  17. A. M. Darling, T. J. Hall, and M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowledge,” J. Opt. Soc. Am. 73, 1466-1469 (1983).
    [CrossRef]
  18. M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
    [CrossRef]
  19. M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.
  20. M. E. Testorf and M. A. Fiddy, “Automated target morphing based on a linear spectral estimation technique,” in OSA Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America2001), pp. 39-41.
  21. H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
    [CrossRef]
  22. H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Iterative image reconstruction using prior knowledge,” J. Opt. Soc. Am. A 23, 1292-1300 (2006).
    [CrossRef]

2006 (3)

2002 (1)

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

2001 (1)

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
[CrossRef]

1988 (1)

C. L. Byrne and M. A. Fiddy, “Image as power spectral; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
[CrossRef]

1987 (1)

1984 (1)

C. L. Byrne and R. M. Fitzgerald, “Spectral estimators that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math. 44, 425-442 (1984).
[CrossRef]

1983 (2)

1982 (2)

T. J. Hall, A. M. Darling, and M. A. Fiddy, “Image compression and restoration incorporating prior knowledge,” Opt. Lett. 7, 467-468 (1982).
[CrossRef] [PubMed]

C. L. Byrne and R. M. Fitzgerald, “Reconstruction from partial information, with applications to tomography,” SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Super-restoration through error energy reduction,” Opt. Acta 21, 709-720 (1974).
[CrossRef]

1972 (1)

J. P. Burg, “The relationship between maximum entropy spectra and maximum likelihood spectra,” Geophysics 37, 375-376 (1972).
[CrossRef]

Bertero, M.

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

M. Bertero, “Sampling theory, resolution limits and inversion methods,” in Inverse Problems in Scattering and Imaging, Malvern Physics Series, M.Bertero and E.R.Pike, eds. (Adam Hilger, 1992), pp. 71-94.

Boccacci, P.

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

Burg, J. P.

J. P. Burg, “The relationship between maximum entropy spectra and maximum likelihood spectra,” Geophysics 37, 375-376 (1972).
[CrossRef]

J. P. Burg, “Maximum entropy spectral analysis,” Ph.D. thesis (Stanford University, 1975).

J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, 1967.

Byrne, C. L.

H. M. Shieh, C. L. Byrne, and M. A. Fiddy, “Image reconstruction: a unifying model for resolution enhancement and data extrapolation. Tutorial,” J. Opt. Soc. Am. A 23, 258-266 (2006).
[CrossRef]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Iterative image reconstruction using prior knowledge,” J. Opt. Soc. Am. A 23, 1292-1300 (2006).
[CrossRef]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
[CrossRef]

C. L. Byrne and M. A. Fiddy, “Image as power spectral; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
[CrossRef]

C. L. Byrne and M. A. Fiddy, “Estimation of continuous object distributions from limited Fourier magnitude measurements,” J. Opt. Soc. Am. A 4, 112-117 (1987).
[CrossRef]

C. L. Byrne and R. M. Fitzgerald, “Spectral estimators that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math. 44, 425-442 (1984).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481-1487 (1983).
[CrossRef]

C. L. Byrne and R. M. Fitzgerald, “Reconstruction from partial information, with applications to tomography,” SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).

Darling, A. M.

Fiddy, M. A.

H. M. Shieh, C. L. Byrne, and M. A. Fiddy, “Image reconstruction: a unifying model for resolution enhancement and data extrapolation. Tutorial,” J. Opt. Soc. Am. A 23, 258-266 (2006).
[CrossRef]

H. M. Shieh and M. A. Fiddy, “Accuracy of extrapolated data as a function of prior knowledge and regularization,” Appl. Opt. 45, 3283-3288 (2006).
[CrossRef] [PubMed]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Iterative image reconstruction using prior knowledge,” J. Opt. Soc. Am. A 23, 1292-1300 (2006).
[CrossRef]

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
[CrossRef]

C. L. Byrne and M. A. Fiddy, “Image as power spectral; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
[CrossRef]

C. L. Byrne and M. A. Fiddy, “Estimation of continuous object distributions from limited Fourier magnitude measurements,” J. Opt. Soc. Am. A 4, 112-117 (1987).
[CrossRef]

A. M. Darling, T. J. Hall, and M. A. Fiddy, “Stable, noniterative object reconstruction from incomplete data using a priori knowledge,” J. Opt. Soc. Am. 73, 1466-1469 (1983).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481-1487 (1983).
[CrossRef]

T. J. Hall, A. M. Darling, and M. A. Fiddy, “Image compression and restoration incorporating prior knowledge,” Opt. Lett. 7, 467-468 (1982).
[CrossRef] [PubMed]

M. E. Testorf and M. A. Fiddy, “Automated target morphing based on a linear spectral estimation technique,” in OSA Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America2001), pp. 39-41.

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.

Fitzgerald, R. M.

C. L. Byrne and R. M. Fitzgerald, “Spectral estimators that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math. 44, 425-442 (1984).
[CrossRef]

C. L. Byrne, R. M. Fitzgerald, M. A. Fiddy, T. J. Hall, and A. M. Darling, “Image restoration and resolution enhancement,” J. Opt. Soc. Am. 73, 1481-1487 (1983).
[CrossRef]

C. L. Byrne and R. M. Fitzgerald, “Reconstruction from partial information, with applications to tomography,” SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, “Super-restoration through error energy reduction,” Opt. Acta 21, 709-720 (1974).
[CrossRef]

Hall, T. J.

McGahan, R. V.

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

Semichaevsky, A. V.

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.

Shieh, H. M.

Testorf, M. E.

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Iterative image reconstruction using prior knowledge,” J. Opt. Soc. Am. A 23, 1292-1300 (2006).
[CrossRef]

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
[CrossRef]

M. E. Testorf and M. A. Fiddy, “Automated target morphing based on a linear spectral estimation technique,” in OSA Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America2001), pp. 39-41.

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.

Appl. Opt. (1)

Geophysics (1)

J. P. Burg, “The relationship between maximum entropy spectra and maximum likelihood spectra,” Geophysics 37, 375-376 (1972).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735-742 (1975).
[CrossRef]

Inverse Probl. (1)

C. L. Byrne and M. A. Fiddy, “Image as power spectral; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

R. W. Gerchberg, “Super-restoration through error energy reduction,” Opt. Acta 21, 709-720 (1974).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “Automated target morphing applied to objects in cluttered backgrounds,” in Proc. SPIE 4792, 78-89 (2002).
[CrossRef]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, “Incorporation of prior information in surface imaging applications,” in Proc. SPIE 4491, 336-345 (2001).
[CrossRef]

SIAM J. Appl. Math. (2)

C. L. Byrne and R. M. Fitzgerald, “Spectral estimators that extend the maximum entropy and maximum likelihood methods,” SIAM J. Appl. Math. 44, 425-442 (1984).
[CrossRef]

C. L. Byrne and R. M. Fitzgerald, “Reconstruction from partial information, with applications to tomography,” SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

Other (7)

J. P. Burg, “Maximum entropy spectral analysis,” Ph.D. thesis (Stanford University, 1975).

J. P. Burg, “Maximum entropy spectral analysis,” presented at the 37th Annual Society of Exploration Geophysicists Meeting, Oklahoma City, 1967.

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

C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).

M. Bertero, “Sampling theory, resolution limits and inversion methods,” in Inverse Problems in Scattering and Imaging, Malvern Physics Series, M.Bertero and E.R.Pike, eds. (Adam Hilger, 1992), pp. 71-94.

M. E. Testorf, A. V. Semichaevsky, R. V. McGahan, and M. A. Fiddy, “A shape-driven target signature algorithm applied to targets in cluttered backgrounds,” in Proceedings of the DoD Workshop on Detection and Classification of Difficult Targets (U.S. Army Aviation and Missile Command, 2002), pp. 57-69.

M. E. Testorf and M. A. Fiddy, “Automated target morphing based on a linear spectral estimation technique,” in OSA Signal Recovery and Synthesis, OSA Technical Digest Series (Optical Society of America2001), pp. 39-41.

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Figures (18)

Fig. 1
Fig. 1

Example of reconstructing a one-dimensional object function (solid lines) by the DFT and the PDFT: (a) DFT estimate (dotted line), (b) prior function (dashed line), and (c) PDFT estimate (dotted line) with the prior in (b).

Fig. 2
Fig. 2

Example of reconstructing a one-dimensional object function (solid lines) by our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) estimate by our proposed method (dotted line) with χ L in (a) for complex exponentials of frequencies { 10 , , 0 , , 10 } and χ S in (b) for otherwise.

Fig. 3
Fig. 3

Example of reconstructing a one-dimensional object function (solid lines) using a properly located but poorly sized window (too wide) as χ S in our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) best estimate (dotted line) with window functions χ L in (a) and χ S in (b).

Fig. 4
Fig. 4

Example of reconstructing a one-dimensional object function (solid lines) using a properly located but improperly sized window (too narrow) as χ S in our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) best estimate (dotted line) with window functions χ L in (a) and χ S in (b).

Fig. 5
Fig. 5

Example of reconstructing a one-dimensional object function (solid lines) using a properly sized but improperly located window as χ S ( χ S cannot cover the small portion of interest completely) in our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) the best estimate (dotted line) with window functions χ L in (a) and χ S in (b).

Fig. 6
Fig. 6

Example of reconstructing a one-dimensional object function (solid lines) using a properly sized but improperly located window as χ S ( χ S does not cover the small portion of interest at all) in our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) best estimate (dotted line) with window functions χ L in (a) and χ S in (b).

Fig. 7
Fig. 7

Example of reconstructing a one-dimensional object function (solid lines) by the DFT and the PDFT: (a) DFT estimate (dotted line), (b) prior function (dashed line), and (c) PDFT estimate (dotted line) with the prior in (b).

Fig. 8
Fig. 8

Example of reconstructing a one-dimensional object function (solid lines) by our proposed method: (a) flat window function χ L (dashed line), (b) flat window function χ S (dashed line), and (c) the estimate by our proposed method (dotted line) with χ L in (a) for complex exponentials of frequencies { 7 , , 0 , , 7 } and χ S in (b) for otherwise.

Fig. 9
Fig. 9

Norms of the image estimate by our proposed method with respect to k th for the example in Fig. 2.

Fig. 10
Fig. 10

Norms of the image estimate by our proposed method with respect to k th for the example in Fig. 8.

Fig. 11
Fig. 11

Example of reconstructing a two-dimensional object function by the DFT and the PDFT: (a) object, (b) DFT estimate, (c) prior function, and (c) PDFT estimate with the prior in (c).

Fig. 12
Fig. 12

Example of reconstructing a two-dimensional object function by our proposed method: (a) flat window function χ L , (b) flat window function χ S , and (c) estimate by our proposed method with χ L in (a) for complex exponentials of frequencies { ( α m , β n ) | α m 2 + β n 2 13 , for m = 1 , 2 , , M and n = 1 , 2 , , N } and χ s in (b) for otherwise.

Fig. 13
Fig. 13

Example of reconstructing a two-dimensional object function by our proposed method: (a) flat window function χ L , (b) flat window function χ S , and (c) estimate by our proposed method with χ L in (a) for complex exponentials of frequencies { ( α m , β n ) | α m 2 + β n 2 13 , for m = 1 , 2 , , M and n = 1 , 2 , , N } and χ s in (b) for otherwise.

Fig. 14
Fig. 14

Example of reconstructing a two-dimensional object function by our proposed method: (a) flat window function χ L , (b) flat window function χ S , and (c) estimate by our proposed method with χ L in (a) for complex exponentials of frequencies { ( α m , β n ) | α m 2 + β n 2 13 , for m = 1 , 2 , , M and n = 1 , 2 , , N } and χ s in (b) for otherwise.

Fig. 15
Fig. 15

Example of reconstructing the same object in Fig. 2 from noisy data (signal-to-noise ratio is 199.8422) by our proposed method: (a) flat window function χ L (dashed lines), (b) flat window function χ S (dashed line), and (c) estimate by our proposed method (dotted line) with the regularization value = 0.09 , χ L in (a) for complex exponentials of frequencies { 10 , , 0 , , 10 } , and χ S in (b) for otherwise.

Fig. 16
Fig. 16

Example of reconstructing the same object as in Fig. 2 from the same noisy data as in Fig. 15 by our proposed method: (a) window function χ L (dashed line) having smooth shapes around the boundaries of χ S , (b) smooth-shaped window function χ S (dashed line), and (c) estimate by our proposed method (dotted line) with the regularization value = 0.09 , χ L in (a) for complex exponentials of frequencies { 10 , , 0 , , 10 } , and χ S in (b) for otherwise.

Fig. 17
Fig. 17

Example of reconstructing the same object in Fig. 12 from noisy data (signal-to-noise ratio is 200.1675) by our proposed method: (a) flat window function χ L , (b) flat window function χ S , and (c) estimate by our proposed method with the regularization value = 0.5 , χ L in (a) for complex exponentials of frequencies { ( α m , β n ) | α m 2 + β n 2 13 , for m = 1 , 2 , , M and n = 1 , 2 , , N }, and χ s in (b) for otherwise.

Fig. 18
Fig. 18

Example of reconstructing the same object as in Fig. 12 from the same noisy data as in Fig. 17 by our proposed method: (a) the window function χ L (dashed line) having smooth shapes around the boundaries of χ S , (b) the smooth-shaped window function χ S (dashed line), and (c) the estimate by our proposed method with the regularization value = 0.5 , χ L in (a) for complex exponentials of frequencies { ( α m , β n ) | α m 2 + β n 2 13 , for m = 1 , 2 , , M and n = 1 , 2 , , N }, and χ s in (b) for otherwise.

Equations (19)

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

F ( k n ) = π π f ( x ) exp ( j x k n ) d x ,
f ̂ ( x ) = n = 1 N a n b n ( x ) ,
f ̂ ( x ) = 1 2 π n = 1 N F ( k n ) exp ( j x k n ) ,
f ̂ PDFT ( x ) = w ( x ) n = 1 N a n exp ( j x k n ) ;
F ( k m ) = n = 1 N a n W ( k m k n ) ,
f ̂ PDFT ( x ) = χ Ω ( x ) n = 1 N λ n 1 ( u n F ) U n ( x ) ,
U n ( x ) = m = 1 M ( u n ) m exp ( j x k m ) ,
Ω U n ( x ) U m ( x ) ¯ d x = λ n u m u n = 0
Ω | U n ( x ) | 2 d x = λ n .
f ̂ ( x ) = n = 1 N a n w n ( x ) exp ( j x k n ) ,
F ( k m ) = n = 1 N a n W n ( k m k n ) ,
f ̂ PDFT ( x ) = χ L ( x ) n = 1 N a n exp ( j x k n ) + χ S ( x ) n = 1 N a n exp ( j x k n ) ,
χ Ω ( x ) = χ L ( x ) + χ S ( x ) .
f ̂ ( x ) = χ L ( x ) n | k n L F a n exp ( j x k n ) + χ S ( x ) n | k n H F a n exp ( j x k n ) .
L exp ( j x k n ) exp ( j x k m ) d x = S exp ( j x k m ) exp ( j x k n ) d x = 0
F ( α m , β n ) = π π π π f ( x , y ) exp ( j x α m j y β n ) d x d y
f ̂ ( x , y ) = m = 1 M n = 1 N a m , n w m , n ( x , y ) exp ( j x α m + j y β n ) ,
F ( α p , β q ) = m = 1 M n = 1 N a m , n W m , n ( α p α m , β q β n )
w m , n ( x , y ) = { χ L , for α m 2 + β n 2 k th χ S , otherwise } ,

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