Abstract

Coded aperture imaging is analyzed in Fourier space and the conditions for obtaining artifact-free 3-D images are obtained. It is deduced that an infinite square array of coding apertures will obey these conditions. A finite array is considered and it is shown that, after a certain coordinate transformation has been performed, the finite aperture acts to bandlimit the spatial frequencies in the image. This result is used to deduce a sampling theorem for coded apertures which places limits on the artifact-free 3-D information that may be obtained. It is thus deduced that 3-D information with a resolution greater than the limits placed here may only be obtained by extrapolating the data to larger viewing angles.

© 1987 Optical Society of America

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References

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  1. H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).
  2. N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
    [Crossref]
  3. E. E. Fenimore, T. M. Cannon, D. B. van Hulsteyn, P. Lee, “Uniformly Redundant Array Imaging of Laser Driven Compressions: Preliminary Results,” Appl. Opt. 18, 945 (1979).
    [Crossref] [PubMed]
  4. K. A. Nugent, B. Luther-Davies, “Penumbral Imaging of High Energy X-Rays from Laser-Produced Plasmas,” Opt. Commun. 49, 393 (1984).
    [Crossref]
  5. N. M. Ceglio, L. W. Coleman, “Spatially Solved α Emission from Laser Fusion Targets,” Phys. Rev. Lett. 39, 20 (1977).
    [Crossref]
  6. K. A. Nugent, B. Luther-Davies, “Application of Penumbral Imaging to Thermonuclear Neutrons,” J. Appl. Phys. 58, 2508 (1985).
    [Crossref]
  7. K. A. Nugent, B. Luther-Davies, “Penumbral Neutron Imaging: Optimization and Simulation,” J. Appl. Phys. 60, 1289 (1986).
    [Crossref]
  8. J. Gunson, B. Polychronopulos, “Optimum Design of a Coded Mask X-Ray Telescope for Rocket Applications,” Mon. Not. R. Astron. Soc. 177, 485 (1976).
  9. See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
    [Crossref]
  10. K. T. Stalker, J. G. Kelly, “Coded Aperture Imaging System for Nuclear Fuel Motion Detection,” 1980 Int. Opt. Comput. Conf.SPIE, 231 (1980).
  11. K. A. Nugent, B. Luther-Davies, “Potential and Limitations of Penumbral Imaging,” Appl. Opt. 25, 1008 (1986).
    [Crossref] [PubMed]
  12. E. E. Fenimore, T. M. Cannon, “Coded Aperture Imaging with Uniformly Redundant Arrays,” Appl. Opt. 17, 337 (1978).
    [Crossref] [PubMed]
  13. E. E. Fenimore, “Coded Aperture Imaging: The Modulation Transfer Function for Uniformly Redundant Arrays,” Appl. Opt. 19, 2465 (1980).
    [Crossref] [PubMed]
  14. L. I. Yin, J. I. Trombka, S. M. Seltzer, M. J. Bielefeld, “X-Ray Imaging of Extended Objects Using Nonoverlapping Redundant Array,” Appl. Opt. 22, 2155 (1983).
    [Crossref] [PubMed]
  15. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, “Three-Dimensional Radiographic Imaging with a Restricted View Angle,” J. Opt. Soc. Am. 69, 1323 (1979).
    [Crossref]
  16. G. R. Gindi, R. G. Paxman, H. H. Barrett, “Reconstruction of an Object from Its Coded Image and Object Constraints,” Appl. Opt. 23, 851 (1984).
    [Crossref] [PubMed]
  17. W. E. Smith, R. G. Paxman, H. H. Barrett, “Image Reconstruction from Coded Data: I. Reconstruction Algorithms and Experimental Results,” J. Opt. Soc. Am. A 2, 491 (1985).
    [Crossref] [PubMed]
  18. T. S. Huang, J. L. C. Sanz, H. Fan, J. Shafii, B. M. Tsai, “Numerical Comparison of Several Algorithms for Band-Limited Signal Extrapolation,” Appl. Opt. 23, 307 (1984).
    [Crossref] [PubMed]
  19. N. N. Abdelmalek, “Restoration of Images with Missing High-Frequency Components Using Quadratic Programming,” Appl. Opt. 22, 2182 (1983).
    [Crossref] [PubMed]
  20. T. M. Cannon, E. E. Fenimore, “Tomographical Imaging Using Uniformly Redundant Arrays,” Appl. Opt. 18, 1052 (1979).
    [Crossref] [PubMed]
  21. N. M. Ceglio, D. W. Sweeney, “Zone Plate Coded Imaging: Theory and Applications,” Prog. Opt. 21, 289 (1984).

1986 (2)

K. A. Nugent, B. Luther-Davies, “Penumbral Neutron Imaging: Optimization and Simulation,” J. Appl. Phys. 60, 1289 (1986).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Potential and Limitations of Penumbral Imaging,” Appl. Opt. 25, 1008 (1986).
[Crossref] [PubMed]

1985 (2)

1984 (4)

T. S. Huang, J. L. C. Sanz, H. Fan, J. Shafii, B. M. Tsai, “Numerical Comparison of Several Algorithms for Band-Limited Signal Extrapolation,” Appl. Opt. 23, 307 (1984).
[Crossref] [PubMed]

K. A. Nugent, B. Luther-Davies, “Penumbral Imaging of High Energy X-Rays from Laser-Produced Plasmas,” Opt. Commun. 49, 393 (1984).
[Crossref]

G. R. Gindi, R. G. Paxman, H. H. Barrett, “Reconstruction of an Object from Its Coded Image and Object Constraints,” Appl. Opt. 23, 851 (1984).
[Crossref] [PubMed]

N. M. Ceglio, D. W. Sweeney, “Zone Plate Coded Imaging: Theory and Applications,” Prog. Opt. 21, 289 (1984).

1983 (2)

1980 (1)

1979 (3)

1978 (1)

1977 (2)

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
[Crossref]

N. M. Ceglio, L. W. Coleman, “Spatially Solved α Emission from Laser Fusion Targets,” Phys. Rev. Lett. 39, 20 (1977).
[Crossref]

1976 (1)

J. Gunson, B. Polychronopulos, “Optimum Design of a Coded Mask X-Ray Telescope for Rocket Applications,” Mon. Not. R. Astron. Soc. 177, 485 (1976).

1973 (1)

See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
[Crossref]

Abdelmalek, N. N.

Arendt, J. W.

Attwood, D. T.

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
[Crossref]

Barrett, H. H.

Bielefeld, M. J.

Cannon, T. M.

Ceglio, N. M.

N. M. Ceglio, D. W. Sweeney, “Zone Plate Coded Imaging: Theory and Applications,” Prog. Opt. 21, 289 (1984).

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
[Crossref]

N. M. Ceglio, L. W. Coleman, “Spatially Solved α Emission from Laser Fusion Targets,” Phys. Rev. Lett. 39, 20 (1977).
[Crossref]

Chiu, M. Y.

Chou, C.

Coleman, L. W.

N. M. Ceglio, L. W. Coleman, “Spatially Solved α Emission from Laser Fusion Targets,” Phys. Rev. Lett. 39, 20 (1977).
[Crossref]

DeMeester, G. D.

See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
[Crossref]

Fan, H.

Fenimore, E. E.

George, E. V.

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
[Crossref]

Gindi, G. R.

Gunson, J.

J. Gunson, B. Polychronopulos, “Optimum Design of a Coded Mask X-Ray Telescope for Rocket Applications,” Mon. Not. R. Astron. Soc. 177, 485 (1976).

Huang, T. S.

Kelly, J. G.

K. T. Stalker, J. G. Kelly, “Coded Aperture Imaging System for Nuclear Fuel Motion Detection,” 1980 Int. Opt. Comput. Conf.SPIE, 231 (1980).

Lee, P.

Luther-Davies, B.

K. A. Nugent, B. Luther-Davies, “Penumbral Neutron Imaging: Optimization and Simulation,” J. Appl. Phys. 60, 1289 (1986).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Potential and Limitations of Penumbral Imaging,” Appl. Opt. 25, 1008 (1986).
[Crossref] [PubMed]

K. A. Nugent, B. Luther-Davies, “Application of Penumbral Imaging to Thermonuclear Neutrons,” J. Appl. Phys. 58, 2508 (1985).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Penumbral Imaging of High Energy X-Rays from Laser-Produced Plasmas,” Opt. Commun. 49, 393 (1984).
[Crossref]

Nugent, K. A.

K. A. Nugent, B. Luther-Davies, “Penumbral Neutron Imaging: Optimization and Simulation,” J. Appl. Phys. 60, 1289 (1986).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Potential and Limitations of Penumbral Imaging,” Appl. Opt. 25, 1008 (1986).
[Crossref] [PubMed]

K. A. Nugent, B. Luther-Davies, “Application of Penumbral Imaging to Thermonuclear Neutrons,” J. Appl. Phys. 58, 2508 (1985).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Penumbral Imaging of High Energy X-Rays from Laser-Produced Plasmas,” Opt. Commun. 49, 393 (1984).
[Crossref]

Paxman, R. G.

Polychronopulos, B.

J. Gunson, B. Polychronopulos, “Optimum Design of a Coded Mask X-Ray Telescope for Rocket Applications,” Mon. Not. R. Astron. Soc. 177, 485 (1976).

Sanz, J. L. C.

Scharfman, H.

See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
[Crossref]

Seltzer, S. M.

Shafii, J.

Simpson, R. G.

Smith, W. E.

Stalker, K. T.

K. T. Stalker, J. G. Kelly, “Coded Aperture Imaging System for Nuclear Fuel Motion Detection,” 1980 Int. Opt. Comput. Conf.SPIE, 231 (1980).

Sweeney, D. W.

N. M. Ceglio, D. W. Sweeney, “Zone Plate Coded Imaging: Theory and Applications,” Prog. Opt. 21, 289 (1984).

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).

Trombka, J. I.

Tsai, B. M.

van Hulsteyn, D. B.

Wilson, D. T.

See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
[Crossref]

Yin, L. I.

Appl. Opt. (9)

J. Appl. Phys. (3)

N. M. Ceglio, D. T. Attwood, E. V. George, “Zone-Plate Coded Imaging of Laser-Produced Plasmas,” J. Appl. Phys. 48, 1566 (1977).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Application of Penumbral Imaging to Thermonuclear Neutrons,” J. Appl. Phys. 58, 2508 (1985).
[Crossref]

K. A. Nugent, B. Luther-Davies, “Penumbral Neutron Imaging: Optimization and Simulation,” J. Appl. Phys. 60, 1289 (1986).
[Crossref]

J. Opt. Soc. Am. (1)

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

Mon. Not. R. Astron. Soc. (1)

J. Gunson, B. Polychronopulos, “Optimum Design of a Coded Mask X-Ray Telescope for Rocket Applications,” Mon. Not. R. Astron. Soc. 177, 485 (1976).

Opt. Commun. (1)

K. A. Nugent, B. Luther-Davies, “Penumbral Imaging of High Energy X-Rays from Laser-Produced Plasmas,” Opt. Commun. 49, 393 (1984).
[Crossref]

Opt. Eng. (1)

See, e.g., H. H. Barrett, D. T. Wilson, G. D. DeMeester, H. Scharfman, “Fresnel Zone Plate Imaging in Radiology and Nuclear Medicine,” Opt. Eng. 12, 8 (1973).
[Crossref]

Phys. Rev. Lett. (1)

N. M. Ceglio, L. W. Coleman, “Spatially Solved α Emission from Laser Fusion Targets,” Phys. Rev. Lett. 39, 20 (1977).
[Crossref]

Prog. Opt. (1)

N. M. Ceglio, D. W. Sweeney, “Zone Plate Coded Imaging: Theory and Applications,” Prog. Opt. 21, 289 (1984).

Other (2)

H. H. Barrett, W. Swindell, Radiological Imaging (Academic, New York, 1981).

K. T. Stalker, J. G. Kelly, “Coded Aperture Imaging System for Nuclear Fuel Motion Detection,” 1980 Int. Opt. Comput. Conf.SPIE, 231 (1980).

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Figures (4)

Fig. 1
Fig. 1

Geometry of a coded aperture imaging system. The symbols are those used in the text.

Fig. 2
Fig. 2

Schematic of the Fourier transform of coded data obtained using an array of coding apertures. The lines of data become points for a 2-D object.

Fig. 3
Fig. 3

Schematic of an orthogonal view system showing its relationship to the flat detector arrangement discussed in the text.

Fig. 4
Fig. 4

Modulation transfer function of a zone plate coded imaging system. The interval Δρ is the minimum frequency interval that can be supported over the complete spatial frequency range.

Equations (47)

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d ( x , y ) = T 4 π ( l + z ) 2 o ( - z l x , - z l y ) * a ( z z + l x , z z + l y ) ,
cos 3 θ 1 ,
k 0 = T 4 π ( l + z ) 2 .
d ( x , y ) = k 0 - o ( - z l x , - z l y , z ) * a ( z z + l x , z z + l y ) d z ,
D ( ω , ν ) = k 0 - O ( ω , ν , z ) A ( ω , ν , z ) d z ,
O ( ω , ν , z ) = - o ( - z l x , - z l y , z ) exp [ 2 π i ( ω x + ν y ) ] d x d y ,
A ( ω , ν , z ) = - a ( z l + z x , z l + z y , z ) exp [ 2 π i ( ω x + ν y ) ] d x d y .
I z 1 ( ω , ν ) = F ( ω , ν , z 1 ) D ( ω , ν ) ,
I z 1 ( ω , ν ) = k 0 - O ( ω , ν , z ) A ( ω , ν , z ) F ( ω , ν , z 1 ) d z ,
A ( ω , ν , z ) F ( ω , ν , z 1 ) = C 0 δ ( z - z 1 ) ,
I z 1 ( ω , ν ) = C 0 k 0 - O ( ω , ν , z ) δ ( z - z 1 ) d z = C 0 k 0 O ( ω , ν , z 1 )
A ( ω , ν , z ) A * ( ω , ν , z 1 ) = C 0 δ ( z - z 1 ) ,
A ( ω , ν , z ) = C 0 δ ( z - z 1 ) .
Δ ν s = 1 / M d 0 ,
a 2 ( x , y ) = k 0 m = - n = - C m n exp [ 2 π i ( m x + n y ) ( 1 + l z ) d a p ] ,
C m n C m n * = constant .
A ( ω , ν , z . ) = k 0 m = n = - C m n sinc [ ( 1 + l z ) d a p ω - m ] × sinc [ ( 1 + l z ) d a p ν - n ] ,
sinc [ ( 1 + l z ) d a p ω - m ] = sinc [ ( l + l z ) d a p ν - n ] = 1 ,
ω n = n ( 1 + l z ) d a p ,             ν m = m ( 1 + l z ) d a p ,             n , m = - 2 , - 1 , 0 , 1 , 2 ,
l z d 0 ( 1 + l z ) d a p
A ( ω , ν , z ) = C , ω m = m Δ f ( z ) , ν n = n Δ f ( z ) , m , n = , - 1 , 0 , 1 , = 0 eleswhere ,
Δ f ( z ) = 1 ( 1 + l z ) d s ,
l z d 0 ( z ) ( 1 + l z ) d s
A ( ω , ν , z ) = A s [ ( 1 + l z ) ω , ( 1 + l z ) ν ] × m n δ [ ω - m ( 1 + l z ) d s , ν - n ( 1 + l z ) d s ] ,
a z ( x , y ) = a s ( z l + z x , z l + z y ) * m n δ [ x - m ( 1 + l z ) d s , y - n ( 1 + l z ) d s ]
d ( x , y ) = k 0 m n O m n C m n exp [ 2 π i ( m x + n y ) ( 1 + l z ) d s ] ,
D ( ω , ν ) = k 0 m n O m n C m n δ [ ω - m ( 1 + l z ) d s , ν - n ( 1 + l z ) d s ] .
D ( ω , ν ) = k 0 - m n O m n ( z ) C m n × δ [ ω - m ( 1 + l z ) d s , ν - n ( 1 + l z ) d s ] · d z
ω = m d s z z + l ,             ν = n d s z z + l ,
ν = n m ω ,
m ( 1 + l z 1 ) d s = m + 1 ( 1 + l z 2 ) d s ,
D ( ω , ν ) = k 0 m n O m n ( ω ) C m n ν = n m ω = 0 elsewhere . }
d det = N ( 1 + l z max ) d s ,
D N ( ω , ν ) = D ( ω , ν ) * B ( ω , ν ) ,
B ( ω , ν ) = d det 2 sinc ( 1 2 d det ω ) sinc ( 1 2 d det ν ) .
D N ( ω , ν ) - k 0 d det 2 m n C m n { O m n ( ω ) * [ sinc ( 1 2 d det ω ) × sinc ( 1 2 d det n m ω ) ] } ,
Δ ω = 1 / d det .
Δ z = z l ( z + l ) 2 ,
Z = z max N ( z max + l ) ,
Δ z z max N .
m z ( x , y ) = ( z + l ) 2 [ x 2 + y 2 + ( l + z ) 2 ] 3 / 2 .
d ( x , y ) = o ( - z l x , - z l y ) * { 1 + cos [ π r 2 / ( 1 + l z ) r 2 a 2 ] } ,
G ( ρ ) = δ ( ρ ) + ( 1 + l z ) 2 r a 2 sin [ π ( 1 + l z ) r 2 a 2 ρ 2 ] ,
π ( 1 + l z ) r 2 a 2 ρ 2 = 0 ,
π ( 1 + l z ) r 2 a 2 ρ 2 = π / 2 ,
Δ ρ min = 1 2 ( 1 + l z ) r a .
l z d o b 2 ( 1 + l z ) r a ,

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