Abstract

We consider the problem of reconstructing a function f with bounded support S from finitely many values of its Fourier transform F. Although f cannot be band limited since it has bounded support, it is typically the case that f can be modeled as the restriction to S of a σ-band-limited function, say g. Our reconstruction method is based on such a model for f. Of particular interest is the effect of the choice of σ>0 on the resolution.

© 2006 Optical Society of America

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References

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  1. J. P. Burg, "Maximum entropy spectral analysis," presented at The 37th Annual Meeting of the Society of Exploration Geophysicists (Oklahoma City, Oklahoma, 1967).
  2. J. P. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
    [CrossRef]
  3. C. L. Byrne, Signal Processing: A Mathematical Approach (AK Peters, Ltd., 2005).
  4. C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with application to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
    [CrossRef]
  5. 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]
  6. 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]
  7. 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]
  8. D. Widder, Advanced Calculus (Dover, 1989).
  9. P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
    [CrossRef] [PubMed]

2006 (1)

1998 (1)

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[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]

1982 (1)

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

1972 (1)

J. P. Burg, "The relationship between maximum entropy spectra and maximum likelihood spectra," Geophysics 37, 375-376 (1972).
[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," presented at The 37th Annual Meeting of the Society of Exploration Geophysicists (Oklahoma City, Oklahoma, 1967).

Byrne, C.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[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]

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 and R. M. Fitzgerald, "Reconstruction from partial information, with application to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

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

deVries, D.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

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 and R. M. Fitzgerald, "Reconstruction from partial information, with application to tomography," SIAM J. Appl. Math. 42, 933-940 (1982).
[CrossRef]

Glick, S.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

King, M.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

Pan, T.-S.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

Pretorius, P.

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

Shieh, H. M.

Widder, D.

D. Widder, Advanced Calculus (Dover, 1989).

Geophysics (1)

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

J. Opt. Soc. Am. A (2)

Phys. Med. Biol. (1)

P. Pretorius, M. King, T.-S. Pan, D. deVries, S. Glick, and C. Byrne, "Reducing the influence of the partial volume effect on SPECT activity quantitation with 3D modeling of spatial resolution in iterative reconstruction," Phys. Med. Biol. 43, 407-420 (1998).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (2)

C. L. Byrne and R. M. Fitzgerald, "Reconstruction from partial information, with application 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 (3)

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

J. P. Burg, "Maximum entropy spectral analysis," presented at The 37th Annual Meeting of the Society of Exploration Geophysicists (Oklahoma City, Oklahoma, 1967).

D. Widder, Advanced Calculus (Dover, 1989).

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Figures (7)

Fig. 1
Fig. 1

DFT estimate from 17 computed Fourier data sampled at frequencies { 8 , , 8 } .

Fig. 2
Fig. 2

Estimate by Eq. (3) with τ = π , σ = 8 , and { x 1 , x 2 , , x 17 } sampled at a constant interval over the ranges (a) [ 3 , 3 ] , (b) [ 2 , 2 ] , (c) [ 1 , 1 ] , (d) [ π 4 , π 4 ] , (e) [ 0.6 , 0.6 ] , (f) [ 3 , 1 ] . In each case the condition number of the matrix A in Eq. (5) is 2.968, 1.680 × 10 4 , 6.382 × 10 9 , 4.059 × 10 11 , 3.846 × 10 13 , 1.208 × 10 13 .

Fig. 3
Fig. 3

Same as Fig. 2 but with τ = 2 and the condition number of the matrix A in Eq. (5) as (a) 1.441 × 10 7 , (b) 4.119 × 10 10 , (c) 1.204 × 10 16 , (d) 1.393 × 10 16 , (e) 5.624 × 10 16 , and (f) 1.286 × 10 16 .

Fig. 4
Fig. 4

Same as Fig. 2 but with τ = 1 and the condition number of the matrix A in Eq. (5) as (a) 7.730 × 10 16 , (b) 3.479 × 10 16 , (c) 7.162 × 10 16 , (d) 3.961 × 10 16 , (e) 5.301 × 10 16 , (f) 2.514 × 10 17 .

Fig. 5
Fig. 5

Same as Fig. 2 but with τ = 0.7 and the condition number of the matrix A in Eq. (5) as (a) 7.402 × 10 17 , (b) 7.684 × 10 16 , (c) 5.695 × 10 16 , (d) 9.364 × 10 16 , (e) 1.575 × 10 17 , and (f) 2.109 × 10 17 .

Fig. 6
Fig. 6

Estimate by Eq. (3) with τ = 1 and (a) σ = 13 with { x 1 , x 2 , , x 17 } sampled at a constant interval over the range [ 3 , 3 ] , (b) σ = 17 with { x 1 , x 2 , , x 17 } over the range [ 2 , 2 ] , (c) σ = 18 and { x 1 , x 2 , , x 17 } over the range [ 1 , 1 ] , (d) σ = 13 and { x 1 , x 2 , , x 17 } over the range [ π 4 , π 4 ] . In each case the condition number of the matrix A in Eq. (5) is 5.931 × 10 14 , 3.024 × 10 12 , 2.055 × 10 14 , 1.830 × 10 16 , respectively.

Fig. 7
Fig. 7

Reconstruction of a two-dimensional function from its limited Fourier values with (a) the object function, (b) the DFT estimate, (c) the estimate by Eq. (8) with τ x = τ y = 1.5 , σ α = σ β = 4 , and [ ( x n , y n ) n = 1 , 2 , , 81 ] sampled on a regular grid over the range 3 x 3 , 3 y 3 , (d) the same as (c) but with σ α = σ β = 6 .

Equations (14)

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F ( ω n ) = f ( x ) exp ( j x ω n ) d x ,
g ( x ) = n = 1 N a n h n ( x ) .
f ̂ ( x ) = n = 1 N a n sin [ σ ( x x n ) ] σ ( x x n ) ,
F ̂ ( ω m ) = F ( ω m ) , for m = 1 , 2 , , N ,
F ( ω m ) = n = 1 N a n A m n
A m n = τ τ sin [ σ ( x x n ) ] σ ( x x n ) exp ( j x ω m ) d x .
F ( α n , β n ) = f ( x , y ) exp ( j x α n j y β n ) d x d y .
f ̂ ( x , y ) = n = 1 N a n sin [ σ α ( x x n ) ] σ α ( x x n ) sin [ σ β ( y y n ) ] σ β ( y y n ) ,
F ( α m , β m ) = n = 1 N a n A m n B m n
A m n = τ x τ x sin [ σ α ( x x n ) ] σ α ( x x n ) exp ( j x α m ) d x
B m n = τ y τ y sin [ σ β ( y y n ) ] σ β ( y y n ) exp ( j y β m ) d y .
Si ( x ) = x sin ( t ) t d t ,
Ci ( x ) = x cos ( t ) t d t .
A m n = D τ x τ x sin [ σ α ( x x n ) ] π ( x x n ) exp [ j ( x x n ) α m ] d x = D 2 π [ Si ( h 1 ) Si ( h 2 ) + Si ( h 3 ) Si ( h 4 ) ] + j D 2 π [ Ci ( h 1 ) Ci ( h 2 ) Ci ( h 3 ) + Ci ( h 4 ) ] ,

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