Abstract

We consider the problem of reconstructing an object function f(r) from finitely many linear functional values. In our main application, the function f(r) is a tomographic image, and the data are integrals of f(r) along thin strips. Because the data are limited, resolution can be enhanced through the inclusion of prior knowledge. One way to do that, a generalization of the prior discrete Fourier transform (PDFT) method, was suggested in 1982 [SIAM J. Appl. Math. 42, 933 (1982)] but was found to be difficult to implement for the tomography problem, and that application was not pursued. Recent advances in approximating the PDFT make it possible to achieve the desired resolution enhancement in an easily implemented procedure.

© 2008 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. 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]
  4. 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]
  5. C. L. Byrne and R. M. Fitzgerald, “A unifying model for spectrum estimation,” in Proceedings of the RADC Spectrum Estimation Workshop, In-House Report RADC-TR-79-63 (Rome Air Development Center, Griffiss Air Force Base, 1979), pp. 157-162.
  6. C. L. Byrne, B. M. Levine, and J. Dainty, “Stable estimation of the probability density function of intensity from photon frequency counts,” J. Opt. Soc. Am. A 1, 1132-1135 (1984).
    [CrossRef]
  7. 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]
  8. 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]
  9. C. L. Byrne and M. A. Fiddy, “Image as power spectra; reconstruction as a Wiener filter approximation,” Inverse Probl. 4, 399-409 (1988).
    [CrossRef]
  10. C. L. Byrne, Signal Processing: a Mathematical Approach (A K Peters, 2005).
  11. R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
    [CrossRef] [PubMed]

2006

2005

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

1988

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

1987

1984

C. L. Byrne, B. M. Levine, and J. Dainty, “Stable estimation of the probability density function of intensity from photon frequency counts,” J. Opt. Soc. Am. A 1, 1132-1135 (1984).
[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]

1983

1982

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

1979

C. L. Byrne and R. M. Fitzgerald, “A unifying model for spectrum estimation,” in Proceedings of the RADC Spectrum Estimation Workshop, In-House Report RADC-TR-79-63 (Rome Air Development Center, Griffiss Air Force Base, 1979), pp. 157-162.

1970

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
[CrossRef] [PubMed]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
[CrossRef] [PubMed]

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]

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

C. L. Byrne and M. A. Fiddy, “Image as power spectra; 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, B. M. Levine, and J. Dainty, “Stable estimation of the probability density function of intensity from photon frequency counts,” J. Opt. Soc. Am. A 1, 1132-1135 (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 and R. M. Fitzgerald, “A unifying model for spectrum estimation,” in Proceedings of the RADC Spectrum Estimation Workshop, In-House Report RADC-TR-79-63 (Rome Air Development Center, Griffiss Air Force Base, 1979), pp. 157-162.

Dainty, J.

Darling, A. M.

Fiddy, M. A.

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]

C. L. Byrne and R. M. Fitzgerald, “A unifying model for spectrum estimation,” in Proceedings of the RADC Spectrum Estimation Workshop, In-House Report RADC-TR-79-63 (Rome Air Development Center, Griffiss Air Force Base, 1979), pp. 157-162.

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
[CrossRef] [PubMed]

Hall, T. J.

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
[CrossRef] [PubMed]

Levine, B. M.

Shieh, H. M.

Testorf, M. E.

Inverse Probl.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Theor. Biol.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471-481(1970).
[CrossRef] [PubMed]

SIAM J. Appl. Math.

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

C. L. Byrne and R. M. Fitzgerald, “A unifying model for spectrum estimation,” in Proceedings of the RADC Spectrum Estimation Workshop, In-House Report RADC-TR-79-63 (Rome Air Development Center, Griffiss Air Force Base, 1979), pp. 157-162.

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

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Figures (1)

Fig. 1
Fig. 1

Image reconstruction from 90-angle projection data, 151 sampled values for each angle. (a)  Object function, (b) estimate by the FBP, (c) estimate by the ART after one iteration using the data access order of random permutation, (d) prior function, (e) estimate by the DPDFT after one iteration using the data access order of random permutation and the prior in (d).

Equations (31)

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

d n = S f ( r ) h n ( r ) ¯ d r .
( H a ) ( r ) = a 1 h 1 ( r ) + + a N h N ( r ) .
d = H H a
f 0 ( r ) = ( H a ) ( r )
f 2 2 = f 0 2 2 + u 2 2 .
T f , h 2 = f , T h 2 .
f , h T = T 1 f , T 1 h 2 .
d n = f , t n T ,
d n = f , h n 2 = T f , T h n T .
T f , h T = f , T * h T .
d n = f , T * T h n T .
d n = f , T T h n T .
f 1 = m = 1 N c m T T h m .
d n = f 1 , T T h n T
= m = 1 N c m T T h m , T T h n T .
d n = m = 1 N c m T h m , T h n 2 .
d n = S f ( x ) e i ω n x d x ,
f 0 ( x ) = m = 1 N a m e i ω m x ,
d n = m = 1 N a m π π e i ( ω m ω n ) x d x .
f 0 ( x ) = n = 1 N d n e i n x .
d n = π π f ( x ) χ X ( x ) e i n x d x ,
f 0 ( x ) = χ X ( x ) m = 1 N a m e i m x ,
d n = 2 m = 1 N a m sin X ( m n ) m n .
0 < ϵ p ( x ) E < + ,
f , h T = π π f ( x ) h ( x ) ¯ p ( x ) 1 d x .
d n = π π f ( x ) p ( x ) e i n x p ( x ) 1 d x ,
f 1 ( x ) = m = 1 N c m p ( x ) e i m x = p ( x ) m = 1 N c m e i m x ,
d n = m = 1 N c m π π p ( x ) e i ( m n ) x d x .
d n = j = 1 J f j h j n ¯ .
f w = j = 1 J | f j | 2 p j 1
RMSEOFTD = 1 N D | o n r n | 2 ,

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