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. This gives the URA the capability to image low-intensity, low-contrast sources. Furthermore, whereas the inherent noise in random array imaging puts a limit on the obtainable SNR, the URA has no such limit. Computer simulations show that the URA with significant shot and background noise is vastly superior to random array techniques without noise. Implementation permits a detector which is smaller than its random array counterpart.

© 1978 Optical Society of America

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  1. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  2. J. G. Abies, Proc. Astron. Soc. Aust. 4, 172 (1968).
  3. H. H. Barrett, F. A. Norrigan, Appl. Opt. 12, 2686 (1973).
    [CrossRef] [PubMed]
  4. J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
    [CrossRef]
  5. C. M. Brown, Ph.D. thesis, “Multiplex Imaging and Random Arrays,” U. Chicago (1972).
  6. R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
    [CrossRef]
  7. M. J. E. Golay, J. Opt. Soc. Am. 61, 272 (1970).
    [CrossRef]
  8. W. K. Klemperer, Astron. Astrophys. Suppl. 15, 449 (1974).
  9. D. Calabro, J. K. Wolf, Inform. Control 11, 537 (1968).
    [CrossRef]
  10. F. J. MacWilliams, N. J. A. Sloane, Proc. IEEE 64, 1715 (1976).
    [CrossRef]
  11. C. M. Brown, J. Appl. Phys. 45, 1806 (1973).
    [CrossRef]

1976

F. J. MacWilliams, N. J. A. Sloane, Proc. IEEE 64, 1715 (1976).
[CrossRef]

1975

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

1974

W. K. Klemperer, Astron. Astrophys. Suppl. 15, 449 (1974).

1973

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

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

H. H. Barrett, F. A. Norrigan, Appl. Opt. 12, 2686 (1973).
[CrossRef] [PubMed]

1970

1968

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

J. G. Abies, Proc. Astron. Soc. Aust. 4, 172 (1968).

D. Calabro, J. K. Wolf, Inform. Control 11, 537 (1968).
[CrossRef]

Abies, J. G.

J. G. Abies, Proc. Astron. Soc. Aust. 4, 172 (1968).

Barrett, H. H.

Blake, R. L.

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

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).

Burek, A. J.

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

Calabro, D.

D. Calabro, J. K. Wolf, Inform. Control 11, 537 (1968).
[CrossRef]

Dicke, R. H.

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

Ekstrom, M. P.

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

Fenimore, E. E.

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

Golay, M. J. E.

Klemperer, W. K.

W. K. Klemperer, Astron. Astrophys. Suppl. 15, 449 (1974).

MacWilliams, F. J.

F. J. MacWilliams, N. J. A. Sloane, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Norrigan, F. A.

Palmieri, T. M.

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

Puetter, R.

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

Sloane, N. J. A.

F. J. MacWilliams, N. J. A. Sloane, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Twogood, R. E.

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

Wolf, J. K.

D. Calabro, J. K. Wolf, Inform. Control 11, 537 (1968).
[CrossRef]

Woods, J. W.

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

Appl. Opt.

Astron. Astrophys. Suppl.

W. K. Klemperer, Astron. Astrophys. Suppl. 15, 449 (1974).

Astrophys. J.

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

IEEE Trans. Nucl. Sci.

J. W. Woods, M. P. Ekstrom, T. M. Palmieri, R. E. Twogood, IEEE Trans. Nucl. Sci. NS-22, 379 (1975).
[CrossRef]

Inform. Control

D. Calabro, J. K. Wolf, Inform. Control 11, 537 (1968).
[CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am.

Proc. Astron. Soc. Aust.

J. G. Abies, Proc. Astron. Soc. Aust. 4, 172 (1968).

Proc. IEEE

F. J. MacWilliams, N. J. A. Sloane, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Rev. Sci. Instrum.

R. L. Blake, A. J. Burek, E. E. Fenimore, R. Puetter, Rev. Sci. Instrum. 45, 513 (1973).
[CrossRef]

Other

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

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Figures (12)

Fig. 1
Fig. 1

The basic steps involved in coded aperture imaging are shown above. In an attempt to obtain a higher SNR, a multiple-pinhole aperture is used to form many overlapping images of the object. The resulting recorded picture must be decoded, using either a digital or optical method. The resulting reconstruction is of higher quality than that obtained by using a simple pinhole.

Fig. 2
Fig. 2

A coded aperture comprised of a 2r by 2s section of a mosaic of uniformly redundant arrays each of size r by s. In this particular case r = 43 and s = 41.

Fig. 3
Fig. 3

This arrangement for a coded aperture imaging system 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 versions of the basic aperture pattern upon the detector, which need be only r by s in size.

Fig. 4
Fig. 4

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.

Fig. 5
Fig. 5

A 40 × 40 random array. As shown in Fig. 6, the uniformly redundant array is superior to the random array because of its ideal system point-spread function.

Fig. 6
Fig. 6

The system point-spread function (SPSF) for the random pinhole aperture and the matched decoding procedure. This and Fig. 6(b) represent the same 40 × 40 aperture. (b) The SPSF for the random pinhole aperture and the balanced correlation decoding procedure. The sidelobes have an expected value of zero, although some trends are possible. (c) The SPSF for the nonredundant pinhole aperture used in conjunction with the matched decoding procedure. Note the difference in the vertical scale from the accompanying graphs. The height of the small plateaus is 1. (d) The SPSF of the uniformly redundant array in conjunction with balanced correlation. Note that the sidelobes are deterministically zero out to ±41.

Fig. 7
Fig. 7

Shown above are the two test objects used in the computer simulations of this paper. Each point in the man emits 210 photons/sec/pinhole. The disks are similar with the large background disk emitting 210 photons/sec/aperture opening and the smaller ones emitting 10% or 20% more or less than this is shown.

Fig. 8
Fig. 8

This figure represents the result of having imaged the man (Fig. 7) through a random pinhole aperture and then having decoded using the matched decoding process. The high background bias, which is signal-dependent, nearly obliterates the man. The simulation was noise-free, hence the bias stems entirely from the nature of the SPSF of Fig. 6(a). In some cases in which the distribution of the object is partially known, an attempt could be made to mitigate the bias effects.

Fig. 9
Fig. 9

This figure is the result of having imaged the man through a random aperture and then having decoded using the balanced correlation method. The geometry was that of Fig. 4. This was a noise-free simulation and hence represents an upper limit on the obtainable image quality.

Fig. 10
Fig. 10

This picture demonstrates the result of having used a uniformly redundant array and the geometry of Fig. 3. Quantum statistics on the source as well as background noise were included in the simulation. Even higher quality is obtainable through longer exposure time.

Fig. 11
Fig. 11

This image represents the results of having encoded and decoded the source of Fig. 7(b) using the uniformly redundant array and the geometry of Fig. 3. The simulation of this 1-sec exposure included quantum fluctuations in the source as well as background noise. Twenty percent variations in the source are easily discernible.

Fig. 12
Fig. 12

This image is identical to that in Fig. 11 except a 10-sec exposure was simulated. Ten percent variations are now discernible.

Equations (13)

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P = ( O * A ) + N ,
O ̂ = R 1 [ ( P ) / ( A ) ] = O + R 1 [ ( N ) / ( A ) ] ,
O ̂ = P * G = RO * ( A * G ) + N * G ,
p ( x , y ) = o ( ξ , η , b ) a ( f ξ / b + x m , f η / b + y m ) d ξ d η db ,
P ( k , l ) O * A + N i j O ( i , j ) A ( i + k , j + l ) + N ( k , l ) ,
O ̂ ( i , j ) = P * G k l P ( k , l ) G ( k + i , l + j ) ,
A ( I , J ) = 0 if I = 0 , 1 if J = 0 , I 0 , 1 if C r ( I ) C s ( J ) = 1 , 0 otherwise ,
C r ( I ) = 1 if there exist an integer x , 1 x < r such that I = mod r x 2 1 otherwise .
G ( i , j ) = 1 if A ( i , j ) = 1 = 1 if A ( i , j ) = 0 ,
i j A ( i , j ) G ( i + k , j + 1 ) ( rs 1 ) / 2 if mod r k = 0 and mod s l = 0 = 0 otherwise .
O ̂ ( k , l ) = i l P ( i , j ) G ( i + k , j + l ) ,
SPSF = A * G .
G ( i , j ) = 1 if A ( i , j ) = 1 , = ρ / 1 ρ if A ( i , j ) = 0 ,

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