Abstract

Coded aperture imaging uses many pinholes to increase the SNR for intrinsically weak sources when the radiation can be neither reflected nor refracted. Effectively, the signal is multiplexed onto an image and then decoded, often by computer, to form a reconstructed image. We derive the modulation transfer function (MTF) of such a system employing uniformly redundant arrays (URA). We show that the MTF of a URA system is virtually the same as the MTF of an individual pinhole regardless of the shape or size of the pinhole. Thus, only the location of the pinholes is important for optimum multiplexing and decoding. The shape and size of the pinholes can then be selected based on other criteria. For example, one can generate self-supporting patterns, useful for energies typically encountered in the imaging of laser-driven compressions or in soft x-ray astronomy. Such patterns contain holes that are all the same size, easing the etching or plating fabrication efforts for the apertures. A new reconstruction method is introduced called δ decoding. It improves the resolution capabilities of a coded aperture system by mitigating a blur often introduced during the reconstruction step.

© 1980 Optical Society of America

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References

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  1. L. Mertz, N. Young, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961), p. 305.
  2. H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2686 (1973).
    [CrossRef] [PubMed]
  3. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  4. C. M. Brown, “Multiplex Imaging and Random Arrays,” Ph.D. Thesis, U. Chicago (1972).
  5. R. G. Simpson, H. H. Barrett, Opt. Eng. 14, 490 (1975).
    [CrossRef]
  6. E. E. Fenimore, T. M. Cannon, Appl. Opt. 17, 337 (1978).
    [CrossRef] [PubMed]
  7. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  8. E. E. Fenimore, T. M. Cannon, D. B. Van Hulsteyn, P. Lee, Appl. Opt. 18, 945 (1979).
    [CrossRef] [PubMed]
  9. E. E. Fenimore, Appl. Opt. 17, 3562 (1978).
    [CrossRef] [PubMed]

1979

1978

1975

R. G. Simpson, H. H. Barrett, Opt. Eng. 14, 490 (1975).
[CrossRef]

1973

1968

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Barrett, H. H.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Brown, C. M.

C. M. Brown, “Multiplex Imaging and Random Arrays,” Ph.D. Thesis, U. Chicago (1972).

Cannon, T. M.

Dicke, R. H.

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Fenimore, E. E.

Horrigan, F. A.

Lee, P.

Mertz, L.

L. Mertz, N. Young, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961), p. 305.

Simpson, R. G.

R. G. Simpson, H. H. Barrett, Opt. Eng. 14, 490 (1975).
[CrossRef]

Van Hulsteyn, D. B.

Young, N.

L. Mertz, N. Young, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961), p. 305.

Appl. Opt.

Astrophys. J.

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Opt. Eng.

R. G. Simpson, H. H. Barrett, Opt. Eng. 14, 490 (1975).
[CrossRef]

Other

L. Mertz, N. Young, in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961), p. 305.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, N.J., 1974).

C. M. Brown, “Multiplex Imaging and Random Arrays,” Ph.D. Thesis, U. Chicago (1972).

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Figures (6)

Fig. 1
Fig. 1

Two cycles of an r × s URA pattern. Note it has periods rc and sc with square c × c pinholes.

Fig. 2
Fig. 2

Demonstration of how the aperture function can be decomposed into a function describing the locations of the pinholes and a function representing the pinhole shape.

Fig. 3
Fig. 3

Various functions used to derive the MTF of a URA aperture: (a) |U(μ,ν)|; (b) H(μ,ν); (c) |U(μ,ν) · H(μ,ν)|; (d) B(μ,ν); (e) |F(A)|.

Fig. 4
Fig. 4

(a) MTF of a URA whose pinholes are c/2 × c/2 squares; (b) MTF of a URA whose pinholes are round with a diameter of c/2.

Fig. 5
Fig. 5

(a) Fourier transform of u(x,y) for a random array; (b) MTF of a random array.

Fig. 6
Fig. 6

(a) SPSF for a correlation analysis; (b) MTF for balanced correlation; (c) SPSF for δ decoding; (d) MTF for δ decoding.

Equations (18)

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P ( x , y ) = S ( x , y ) * A ( x , y ) + N ( x , y ) ,
R ( x , y ) = P ( x , y ) * G ( x , y ) = S * [ A * G ] + N * G ,
MTF A * G = | F ( A * G ) | = | F ( A ) · F ( G ) | ,
A ( x , y ) = [ u ( x , y ) * h ( x , y ) ] b ( x , y ) .
F ( A ) = [ U ( μ , ν ) · H ( μ , ν ) ] * B ( μ , ν ) .
U ( μ , ν ) = U ( μ + 1 / c , ν + 1 / c ) , U ( μ , ν ) = l = 0 s 1 k = 0 r 1 D l k δ ( μ k / r c ) δ ( ν l / s c ) ,
u * u ( x , y ) = u * u ( x + r c , y + s c ) u * u ( x , y ) = d 0 δ ( x ) δ ( y ) + d 1 l = 1 s 1 k = 1 r 1 δ ( x k c ) δ ( y l c ) ,
u * u ( x , y ) = ( d 0 d 1 ) δ ( x ) δ ( y ) + d 1 l = 0 s 1 k = 0 r 1 δ ( x k c ) δ ( y l c ) ,
U 2 ( μ , ν ) d 1 δ ( μ ) δ ( ν ) + [ ( d 0 d 1 ) / r s ] × l = 0 s 1 k = 0 r 1 δ ( μ k / r c ) δ ( ν l / s c ) ,
U ( k / r c , l / s c ) = β if A ( k c , l c ) = 1 = β if A ( k c , l c ) = 0 ,
H ( μ , ν ) sin ( 2 π c μ ) sin ( 2 π c ν ) / μ ν .
B ( μ , ν ) sin ( π c r μ ) sin ( π c s ν ) / μ ν ,
G B ( x , y ) = 1 if A ( x , y ) = 1 G B ( x , y ) = M / ( M N ) if A ( x , y ) 1
G δ ( x , y ) = u ( x , y ) · b ( x , y ) ,
u ( x , y ) = 1 if u ( x , y ) = 1 = M / ( M N ) δ ( x k c ) δ ( y l c ) otherwise
U 2 ( μ , ν ) F ( μ * ν ) δ ( μ k / r c ) δ ( ν l / s c ) ,
MTF s = | F ( A * G δ ) | = | [ U ( μ , ν ) · H ( μ , ν ) * B ( μ , ν ) ] · [ U ( μ , ν ) · H δ ( μ , ν ) * B ( μ , ν ) ] | ,
MTF system = | F ( A * G δ * D * Q ) | .

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