Abstract

Uniformly redundant arrays (URA) have autocorrelation functions with perfectly flat sidelobes. The URA combines the high-transmission characteristics of the random array with the flat sidelobe advantage of the nonredundant pinhole arrays. A general expression for the signal-to-noise ratio (SNR) has been developed for the URA as a function of the type of object being imaged and the design parameters of the aperture. The SNR expression is used to obtain an expression for the optimum aperture transmission. Currently, the only 2-D URAs known have a transmission of ½. This, however, is not a severe limitation because the use of the nonoptimum transmission of ½ never causes a reduction in the SNR of more than 30%. The predicted performance of the URA system is compared to the image obtainable from a single pinhole camera. Because the reconstructed image of the URA contains virtually uniform noise regardless of the original object’s structure, the improvement over the single pinhole camera is much larger for the bright points than it is for the low intensity points. For a detector with high background noise, the URA will always give a much better image than the single pinhole camera regardless of the structure of the object. In the case of a detector with low background noise, the improvement of the URA relative to the single pinhole camera will have a lower limit of ~(2f)−1/2, where f is the fraction of the field of view that is uniformly filled by the object.

© 1978 Optical Society of America

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References

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  1. L. Mertz, N. Young, in Proceedings of the International Conference on Optical Instruments and Techniques (Chapman and Hall, London, 1961), p. 305.
  2. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  3. E. E. Fenimore, T. M. Cannon, Appl. Opt. 17, 337 (1978).
    [CrossRef] [PubMed]
  4. H. H. Barrett, G. D. DeMeester, Appl. Opt. 13, 1100 (1974).
    [CrossRef] [PubMed]
  5. C. M. Brown, Ph.D. thesis, “Multiplex Imaging and Random Arrays,” U. Chicago (1972).
  6. C. M. Brown, J. Appl. Phys. 45, 1806 (1973).
    [CrossRef]
  7. F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).
  8. M. Oda, Appl. Opt. 4, 143 (1965).
    [CrossRef]
  9. J. G. Underwood, J. E. Milligan, A. C. Deloach, R. B. Hoover, Appl. Opt. 16, 858 (1977).
    [PubMed]
  10. R. P. Godwin, Adv. X-Ray Anal. 19, 533 (1975).
  11. H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
    [PubMed]

1978 (1)

1977 (1)

1976 (1)

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

1975 (1)

R. P. Godwin, Adv. X-Ray Anal. 19, 533 (1975).

1974 (1)

1973 (1)

C. M. Brown, J. Appl. Phys. 45, 1806 (1973).
[CrossRef]

1972 (1)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

1968 (1)

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

1965 (1)

Barrett, H. H.

Brown, C. M.

C. M. Brown, J. Appl. Phys. 45, 1806 (1973).
[CrossRef]

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

Cannon, T. M.

Deloach, A. C.

DeMeester, G. D.

Dicke, R. H.

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

Fenimore, E. E.

Godwin, R. P.

R. P. Godwin, Adv. X-Ray Anal. 19, 533 (1975).

Gunson, F.

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Hoover, R. B.

Mertz, L.

L. Mertz, N. Young, in Proceedings of the International Conference on Optical Instruments and Techniques (Chapman and Hall, London, 1961), p. 305.

Milligan, J. E.

Oda, M.

Polychronopulos, B.

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Underwood, J. G.

Young, N.

L. Mertz, N. Young, in Proceedings of the International Conference on Optical Instruments and Techniques (Chapman and Hall, London, 1961), p. 305.

Adv. X-Ray Anal. (1)

R. P. Godwin, Adv. X-Ray Anal. 19, 533 (1975).

Appl. Opt. (4)

Astrophys. J. (1)

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

J. Appl. Phys. (1)

C. M. Brown, J. Appl. Phys. 45, 1806 (1973).
[CrossRef]

J. Nucl. Med. (1)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

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

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Other (2)

L. Mertz, N. Young, in Proceedings of the International Conference on Optical Instruments and Techniques (Chapman and Hall, London, 1961), p. 305.

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

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Figures (6)

Fig. 1
Fig. 1

(A) This coded aperture arrangement employs only the basic r by s pattern for the aperture and has the disadvantage that the detector must be large enough to contain the image from the full field of view. (B) This arrangement employs a 2r by 2s aperture composed of a mosaic of basic r by s patterns. Emitting points in the source produce shadows of cyclic permutations of the basic aperture pattern, and thus the detector needs to be only r by s in size.

Fig. 2
Fig. 2

A 1-D slice through a reconstructed object for a point source. The SNR is defined to be the peak height divided by the expected size of the fluctuations of the peak and the background.

Fig. 3
Fig. 3

(A) Orientation of the G array and the picture from the Sij source when calculating the Ŝij point in the reconstructed object. (B) Same as A except the picture is due to the Skl source.

Fig. 4
Fig. 4

Orientation of the G array and the picture from the Skl source when calculating the Ŝij point in the reconstructed object. The random array geometry of Fig. 1(A) is assumed.

Fig. 5
Fig. 5

(A) Optimum density of the aperture as a function of the relative intensity of a point in the source ψ for select values of the detector background noise ξ. (B) The ratio of the SNR with a density of ½ to the SNR with the optimum density as a function of ψ and ξ. Note that using a density of ½ never causes a large effect.

Fig. 6
Fig. 6

Predicted SNR of a URA system and a pinhole camera for a source described in the text. Note that the improvement for the brighter points in the source (ψ large) is much better than for the low-intensity points.

Equations (24)

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P = S * A + D ,
S ˆ P * G = S * ( A * G ) + D * G ,
G i j = 1 if A i j = 1 ; G i j = 1 if A i j = 0 ,
G i j = 1 if A i j = 1 ; G i j = M / ( M N ) if A i j = 0 ,
SNR u ( i , j ) = E ( S i j ) { VAR [ C ( S i j , S ˆ i j ) ] + k l VAR [ C ( S k l , S ˆ i j ) ] + VAR [ C ( D , S ˆ i j ) ] } 1 / 2 ,
k l VAR [ C ( S k l , S ˆ i j ) ]
C ( S i j , S ˆ i j ) = u υ P u υ i j G u υ i j .
E [ C ( S i j , S ˆ i j ) ] = S y ¨ u υ A u υ i j G u υ i j .
E [ C ( S i j , S ˆ i j ) ] = N S i j .
E [ C ( S k l , S ˆ i j ) ] = S k l u υ A u υ k l G u υ i j .
E [ C ( S k l , S ˆ i j ) ] = S k l [ M · 1 + ( N M ) · M / ( M N ) ] = 0 ,
E [ C ( D , S ˆ i j ) ] = u υ D u υ G u υ i j = B u υ G u υ = B · [ N · 1 + ( r s N ) M / ( M N ) ] ,
E ( S ˆ i j ) = N S i j + removable dc term .
VAR [ C ( S i j , S ˆ i j ) ] = N S i j .
VAR [ C ( S k l , S ˆ i j ) ] = S k l [ M · 1 2 + ( N M ) ( M / M N ) 2 ] , VAR [ C ( D , S ˆ i j ) ] λ B = [ N · 1 2 + ( r s M ) ( M / M N ) 2 ] B .
SNR u ( i , j ) = N S i j ( N S i j + M N N M I T + λ B ) 1 / 2 ,
k l S k l .
E [ C R ( S k l , S ˆ i j ) ] = [ γ · 1 + η ρ / ( ρ 1 ) ] S k l β S k l .
ρ N / ( r · s ) , ψ i j = S i j / I T , M ρ 2 ( r · s ) , ξ B / I T .
SNR u ( i , j ) = ( 1 ρ ) 1 / 2 ρ ψ i j ( r · s · I T ) 1 / 2 [ ρ 2 ( 1 ψ i j ) + ρ ( ψ i j + ξ ) ] 1 / 2 .
ρ T = [ ( ψ i j + ξ ) 2 + ( 1 ψ i j ) ( ψ i j + ξ ) ] 1 / 2 ( ψ i j + ξ ) 1 ψ i j .
SNR u ( i , j ) = N 1 / 2 S i j ( S i j + I T + 2 B ) 1 / 2 = N 1 / 2 ψ i j I T ( 1 + 2 ξ + ψ i j ) 1 / 2 .
SNR p ( i , j ) = S i j ( S i j + B ) 1 / 2 = ψ i j I T ( ψ i j + ξ ) 1 / 2 .
F i j SNR u SNR p = [ N ( ψ i j + ξ ) 2 ξ + ψ i j + 1 ] 1 / 2 .

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