Abstract

The prior discrete Fourier transform (PDFT) is a linear spectral estimator that provides a solution that is both data consistent and of minimum weighted norm through the use of a suitably designed Hilbert space. The PDFT has been successfully used in imaging applications to improve resolution and overcome the nonuniqueness associated with having only finitely many spectral measurements. With the use of an appropriate prior function, the resolution of the reconstructed image can be improved dramatically. We explore the ways in which some significant parameters affect the PDFT estimate. A relationship between estimated spectral values, prior knowledge, and regularization was examined. It allows one to assess the reliability of the estimated spectral values for a given choice of prior estimate and provides a means for optimizing PDFT-based estimators.

© 2006 Optical Society of America

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References

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  1. C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with applications to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
    [CrossRef]
  2. 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]
  3. 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]
  4. 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]
  5. R. W. Gerchberg, "Super-resolution through error energy reduction," Opt. Acta 21, 709-720 (1974).
    [CrossRef]
  6. A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
    [CrossRef]
  7. M. Bertero, "Sampling theory, resolution limits and inversion methods," in Inverse Problems in Scattering and Imaging, M. Bertero and E. Pike, eds. (Adam Hilger, 1992), pp. 71-94.
  8. C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
    [CrossRef] [PubMed]
  9. H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, 1998).
    [PubMed]
  10. H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).
  11. C. L. Byrne, Signal Processing: A Mathematical Approach (Peters, Wellesley, Mass., 2005).

1993 (1)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
[CrossRef] [PubMed]

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 (1)

1982 (1)

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 bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

1974 (1)

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

Bertero, M.

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

Byrne, C. L.

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
[CrossRef] [PubMed]

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 (Peters, Wellesley, Mass., 2005).

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).

Darling, A. M.

Fiddy, M. A.

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, 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]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).

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-resolution through error energy reduction," Opt. Acta 21, 709-720 (1974).
[CrossRef]

Hall, T. J.

Papoulis, A.

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

Shieh, H. M.

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).

Stark, H.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, 1998).
[PubMed]

Testorf, M. E.

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).

Yang, Y.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, 1998).
[PubMed]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, "A new algorithm in spectral analysis and bandlimited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
[CrossRef]

IEEE Trans. Image Process. (1)

C. L. Byrne, "Iterative image reconstruction algorithms based on cross-entropy minimization," IEEE Trans. Image Process. 2, 96-103 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

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

SIAM J. Appl. Math. (2)

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

Other (4)

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

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics (Wiley, 1998).
[PubMed]

H. M. Shieh, C. L. Byrne, M. E. Testorf, and M. A. Fiddy, "Incorporation of prior information in surface imaging applications," in Subsurface and Surface Sensing Technologies and Applications III, C. Nguyen, ed., Proc. SPIE 4491, 336-345 (2001).

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

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Figures (8)

Fig. 1
Fig. 1

Absolute values of P c (k)H u n for n = 1, 2, … , 15 whose corresponding eigenvalues are (a) 1.00e + 00, (b) 9.99e − 01, (c) 9.82e − 01, (d) 8.29e − 01, (e) 4.02e − 01, (f) 7.82e − 02, (g) 6.83e − 03, (h) 3.52e − 04, (i) 1.21e − 05, (j) 2.91e − 07, (k) 4.85e − 09, (l) 5.53e − 11, (m) 4.12e −13, (n) 1.72e − 15, (o) 4.28e − 8. Vertical dashed line, boundary of the data support (k = NΔ k ).

Fig. 2
Fig. 2

PDFT estimation with a rectangular window prior function that has a width of 3 (a) in the spatial domain and (b) in the spectral domain.

Fig. 3
Fig. 3

DFT estimation of an object function whose true width is π∕2.

Fig. 4
Fig. 4

PDFT estimation with a rectangular window prior function that has a width of 1.8 (a) in the spatial domain and (b) in the spectral domain.

Fig. 5
Fig. 5

PDFT estimation with a prior function that accurately coincides with the true object's structures (a) in the spatial domain and (b) in the spectral domain.

Fig. 6
Fig. 6

PDFT estimation with a rectangular window prior function that has a width of 1.5 (a) in the spatial domain and (b) in the spectral domain.

Fig. 7
Fig. 7

PDFT estimation with a rectangular window prior function that has a width of 1.5 (ε = 0.1) (a) in the spatial domain and (b) in the spectral domain.

Fig. 8
Fig. 8

PDFT estimation with a rectangular window prior function that has a width of 1.5 (ε = 0.3) (a) in the spatial domain and (b) in the spectral domain.

Equations (20)

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f ^ ( r ) = n = N N a n x n ( r ) ,
d n = f ( r ) x n ( r ) ¯ d r , n = N , , N .
| f ( r ) f ^ ( r ) | 2 p ( r ) d r
f ^ ( r ) = p ( r ) n = N N a n exp ( j r k n ) .
d n = m = N N a m P ( k n k m )
F ( n Δ k ) = m = N N a m P ( n Δ k m Δ k )
= P ( n Δ k ) + E ( n Δ k )
P c ( k ) = [ P ( N Δ k k ) , , P ( N Δ k k ) ] T ,
F = [ F ( N Δ k ) , , F ( N Δ k ) ] T ,
a = [ a N , , a N ] T ,
E = [ E ( N Δ k ) , , E ( N Δ k ) ] T ,
F = P a = P c ( 0 ) + E ,
a = δ = [ 0 , , 0 , 1 , 0 , , 0 ] T ,
P c ( 0 ) = P δ ,
a = δ + P 1 E .
F ^ ( k ) = n = N N a n P ( k n Δ k )
= P c ( k ) H a
= P ( k ) + P c ( k ) H b ,
P 1 = U Λ 1 U H ,
λ n - 1 P c ( k ) H u n u n H E , n = 1 , 2 , , 2 N + 1 ;

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