Abstract

A method is proposed to reconstruct signals from incomplete data. The method, which can be interpreted both as a discrete implementation of the so-called prior discrete Fourier transform (PDFT) spectral estimation technique and as a variant of the algebraic reconstruction technique, allows one to incorporate prior information about the reconstructed signal to improve the resolution of the signal estimated. The context of diffraction tomography and image reconstruction from samples of the far-field scattering amplitude are used to explore the performance of the method. On the basis of numerical computations, the optimum choice of parameters is determined empirically by comparing image reconstructions of the noniterative PDFT algorithm and the proposed iterative scheme.

© 2006 Optical Society of America

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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, "Image as power spectral; reconstruction as a Wiener filter approximation," Inverse Probl. 4, 399-409 (1988).
    [CrossRef]
  5. 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]
  6. C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, 2005).
  7. S. Kaczmarz, "Angenaherte auflosung von system linearer gleichungen," Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A 35, 355-357 (1937).
  8. 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]
  9. K. Tanabe, "Projection method for solving a singular system of linear equations and its applications," Numer. Math. 17, 203-214 (1971).
    [CrossRef]
  10. R. Gordon, "A tutorial on ART," IEEE Trans. Nucl. Sci. NS-21, 78-93 (1974).
  11. G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273-294 (1976).
    [CrossRef] [PubMed]
  12. G. T. Herman and H. K. Tuy, "Image reconstruction from projections: an approach from mathematical analysis," in Basic Methods of Tomography and Inverse Problems., P.C.Sabatier, ed. (Hilger, 1987), pp. 1-124.
  13. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).
  14. G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imaging 12, 600-609 (1993).
    [CrossRef] [PubMed]
  15. H. Guan and R. Gordon, "A projection access order for speedy convergence of ART (algebraic reconstruction technique): a multilevel scheme for computed tomography," Phys. Med. Biol. 39, 2005-2022 (1994).
    [CrossRef] [PubMed]
  16. G. T. Herman, "A relaxation method for reconstructing objects from noisy X-rays," Math. Program. 8, 1-19 (1975).
    [CrossRef]
  17. G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
    [CrossRef]
  18. M. R. Trummer, "Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation," Computing 26, 189-195 (1981).
    [CrossRef]
  19. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
    [CrossRef]
  20. P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
    [CrossRef]

2006 (1)

1994 (1)

H. Guan and R. Gordon, "A projection access order for speedy convergence of ART (algebraic reconstruction technique): a multilevel scheme for computed tomography," Phys. Med. Biol. 39, 2005-2022 (1994).
[CrossRef] [PubMed]

1993 (1)

G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imaging 12, 600-609 (1993).
[CrossRef] [PubMed]

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]

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]

1981 (2)

M. R. Trummer, "Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation," Computing 26, 189-195 (1981).
[CrossRef]

P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
[CrossRef]

1978 (1)

G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
[CrossRef]

1976 (1)

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273-294 (1976).
[CrossRef] [PubMed]

1975 (1)

G. T. Herman, "A relaxation method for reconstructing objects from noisy X-rays," Math. Program. 8, 1-19 (1975).
[CrossRef]

1974 (1)

R. Gordon, "A tutorial on ART," IEEE Trans. Nucl. Sci. NS-21, 78-93 (1974).

1971 (1)

K. Tanabe, "Projection method for solving a singular system of linear equations and its applications," Numer. Math. 17, 203-214 (1971).
[CrossRef]

1970 (1)

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]

1937 (1)

S. Kaczmarz, "Angenaherte auflosung von system linearer gleichungen," Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A 35, 355-357 (1937).

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]

Bertero, M.

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

Boccacci, P.

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

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]

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 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.

Eggermont, P.

P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
[CrossRef]

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]

Gordon, R.

H. Guan and R. Gordon, "A projection access order for speedy convergence of ART (algebraic reconstruction technique): a multilevel scheme for computed tomography," Phys. Med. Biol. 39, 2005-2022 (1994).
[CrossRef] [PubMed]

R. Gordon, "A tutorial on ART," IEEE Trans. Nucl. Sci. NS-21, 78-93 (1974).

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]

Guan, H.

H. Guan and R. Gordon, "A projection access order for speedy convergence of ART (algebraic reconstruction technique): a multilevel scheme for computed tomography," Phys. Med. Biol. 39, 2005-2022 (1994).
[CrossRef] [PubMed]

Hall, T. J.

Herman, G. T.

G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imaging 12, 600-609 (1993).
[CrossRef] [PubMed]

P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
[CrossRef]

G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
[CrossRef]

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273-294 (1976).
[CrossRef] [PubMed]

G. T. Herman, "A relaxation method for reconstructing objects from noisy X-rays," Math. Program. 8, 1-19 (1975).
[CrossRef]

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]

G. T. Herman and H. K. Tuy, "Image reconstruction from projections: an approach from mathematical analysis," in Basic Methods of Tomography and Inverse Problems., P.C.Sabatier, ed. (Hilger, 1987), pp. 1-124.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

Kaczmarz, S.

S. Kaczmarz, "Angenaherte auflosung von system linearer gleichungen," Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A 35, 355-357 (1937).

Lent, A.

P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
[CrossRef]

G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
[CrossRef]

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273-294 (1976).
[CrossRef] [PubMed]

Lutz, P. H.

G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
[CrossRef]

Meyer, L. B.

G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imaging 12, 600-609 (1993).
[CrossRef] [PubMed]

Shieh, H. M.

Tanabe, K.

K. Tanabe, "Projection method for solving a singular system of linear equations and its applications," Numer. Math. 17, 203-214 (1971).
[CrossRef]

Trummer, M. R.

M. R. Trummer, "Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation," Computing 26, 189-195 (1981).
[CrossRef]

Tuy, H. K.

G. T. Herman and H. K. Tuy, "Image reconstruction from projections: an approach from mathematical analysis," in Basic Methods of Tomography and Inverse Problems., P.C.Sabatier, ed. (Hilger, 1987), pp. 1-124.

Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A (1)

S. Kaczmarz, "Angenaherte auflosung von system linearer gleichungen," Bull. Int. Acad. Pol. Sci. Lett., Cl. Sci. Math. Nat., Ser. A 35, 355-357 (1937).

Commun. ACM (1)

G. T. Herman, A. Lent, and P. H. Lutz, "Relaxation methods for image reconstruction," Commun. ACM 21, 152-158 (1978).
[CrossRef]

Comput. Biol. Med. (1)

G. T. Herman and A. Lent, "Iterative reconstruction algorithms," Comput. Biol. Med. 6, 273-294 (1976).
[CrossRef] [PubMed]

Computing (1)

M. R. Trummer, "Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation," Computing 26, 189-195 (1981).
[CrossRef]

IEEE Trans. Med. Imaging (1)

G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imaging 12, 600-609 (1993).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (1)

R. Gordon, "A tutorial on ART," IEEE Trans. Nucl. Sci. NS-21, 78-93 (1974).

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. (1)

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

J. Theor. Biol. (1)

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]

Linear Algebr. Appl. (1)

P. Eggermont, G. T. Herman, and A. Lent, "Iterative algorithms for large partitioned linear systems, with applications to image reconstruction," Linear Algebr. Appl. 40, 37-67 (1981).
[CrossRef]

Math. Program. (1)

G. T. Herman, "A relaxation method for reconstructing objects from noisy X-rays," Math. Program. 8, 1-19 (1975).
[CrossRef]

Numer. Math. (1)

K. Tanabe, "Projection method for solving a singular system of linear equations and its applications," Numer. Math. 17, 203-214 (1971).
[CrossRef]

Phys. Med. Biol. (1)

H. Guan and R. Gordon, "A projection access order for speedy convergence of ART (algebraic reconstruction technique): a multilevel scheme for computed tomography," Phys. Med. Biol. 39, 2005-2022 (1994).
[CrossRef] [PubMed]

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 and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

G. T. Herman and H. K. Tuy, "Image reconstruction from projections: an approach from mathematical analysis," in Basic Methods of Tomography and Inverse Problems., P.C.Sabatier, ed. (Hilger, 1987), pp. 1-124.

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, 1980).

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

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Figures (8)

Fig. 1
Fig. 1

Illustration of the Fourier space mapping of the simulated data.

Fig. 2
Fig. 2

Reconstruction of an object of compact support: (a) the object, (b) the discrete Fourier transform estimate, (c) circular flat top used as the prior function, (d) the PDFT estimate, (e) the DPDFT estimate after six iterations, (f) the DPDFT estimate after 15 iterations.

Fig. 3
Fig. 3

RMSEs between the original and the reconstructed image for simulated noiseless Fourier data: (a) the RMSEs for the PDFT and the DPDFT after 1–20 iterations, (b) same as in (a) applying the RPS reordering, (c) same as in (a) applying the Herman–Meyer reordering.

Fig. 4
Fig. 4

Reconstruction of an object with compact support using the DPDFT with reordering and relaxation: (a) one-iteration DPDFT estimate with RPS reordering, (b) one-iteration DPDFT estimate with HMS reordering, (c) one-iteration DPDFT estimate with the relaxation parameter 0.07, (d) three-iteration DPDFT estimate with the relaxation parameter 0.09.

Fig. 5
Fig. 5

Impact of the relaxation parameter on the RMSEs: (a) the RMSEs for the PDFT and the DPDFT after three iterations with no reordering, (b) same for the DPDFT after one iteration with RPS reordering, (c) same for the DPDFT after one iteration with HMS reordering.

Fig. 6
Fig. 6

Reconstruction of an object of compact support from the noisy data: (a) the PDFT estimate with regularization value of κ = 0.5 , (b) the DPDFT estimate after eight iterations, (c) the DPDFT estimate after 20 iterations, (d) the DPDFT estimate after ten iterations with regularization value of ϵ = 0.7 , (e) the DPDFT estimate after one iteration with regularization value of ϵ = 0.7 and RPS reordering, (f) the DPDFT estimate after one iteration with regularization value of ϵ = 0.7 and HMS reordering.

Fig. 7
Fig. 7

RMSEs between object and image computed from simulated noisy data: (a) the DPDFT estimate without regularization, (b) the DPDFT estimate with regularization value of ϵ = 0.7 .

Fig. 8
Fig. 8

RMSEs between object and image for simulated noisy data: (a) RPS reordering and regularization value of ϵ = 0.7 , (b) HMS reordering and regularization value of ϵ = 0.7 .

Equations (21)

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

d m = f ( r ) g m ( r ) ¯ d r .
P m n = p ( r ) g n ( r ) g m ( r ) ¯ d r ,
f ̂ P D F T ( r ) = p ( r ) n = 1 M a n g n ( r ) .
h 2 = h ( r ) 2 p 1 ( r ) d r .
d m = n = 1 N A m n f n ,
f p 2 = n = 1 N f n 2 p n .
f ̂ D P D F T = W 1 A ( AW 1 A ) 1 d ,
f ̂ P D F T ( r ) = p ( r ) m = 1 M a m exp ( j r k m ) ,
d = Pa ,
F ( α m , β m ) = f ( x , y ) exp [ j ( x α m + y β m ) ] d x d y ,
F ( α m , β m ) u = U U v = V V f ( u Δ x , v Δ y ) exp [ j ( u Δ x α m + v Δ y β m ) ] Δ x Δ y .
Af = AW 1 2 W 1 2 f = Bg .
g n k + 1 = g n k + B ¯ m n ( d m i = 1 N B m i g i k ) i = 1 N B m i 2 ,
f n k + 1 = f n k + p n A ¯ m n ( d m i = 1 N A m i f i k ) i = 1 N p i A m i 2 .
f n k + 1 = f n k + λ k p n A ¯ m n ( d m i = 1 N A m i f i k ) i = 1 N p i A m i 2 ,
R M S E = 1 N n = 1 N o n r n 2 ,
S ( A T ) = { v v = n = 1 N α n ( A T ) n c o l , for some arbitrary real numbers α n } ,
[ A ϵ I ] ( f v ) = d .
f n k + 1 = f n k + p n A ¯ m n ( d m i = 1 N A m i f i k ϵ v m k ϵ 2 + i = 1 N p i A m i 2 ) ,
v m k + 1 = v m k + ϵ ( d m i = 1 N A m i f i k ϵ v m k ϵ 2 + i = 1 N p i A m i 2 ) ,
v j k + 1 = v j k for j mod ( k , M ) + 1 .

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