Abstract

We introduce a new family of binary arrays for use in coded aperture imaging which are predicted to have properties and sensitivity (SNR) equal to that of the uniformly redundant array (URA). The new arrays, called MURAs (modified URAs), have decoding coefficients all of which are unimodular, resulting in a reconstructed image with noise terms completely independent of image–source structure. Although the new arrays are derived from quadratic residues, they do not belong to the cyclic difference set or set of pseudonoise sequences and consequently are constructible in configurations forbidden to those designs, thus providing the user with a wider selection of aperture patterns to match his particular needs. With the addition of MURAs to the family of binary arrays, all prime numbers can now be used for making optimal coded apertures, increasing the number of available square patterns by more than a factor of 3.

© 1989 Optical Society of America

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References

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  1. E. Caroli et al., “Coded Aperture Imaging in X- and Gamma-Ray Astronomy,” Space Sci. Rev. 45, 349–403 (1987).
    [CrossRef]
  2. L. Mertz, N. O. Young, “Fresnel Transformations of Images,” in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 305.
  3. R. H. Dicke, “Scatter-Hole Cameras for X-Rays and Gamma Rays,” Astrophys. J. 153, L101 (1968).
    [CrossRef]
  4. E. E. Fenimore, T. M. Cannon, “Coded Aperture Imaging with Uniformly Redundant Arrays,” Appl. Opt. 17, 337 (1978).
    [CrossRef] [PubMed]
  5. F. J. MacWilliams, N. J. A. Sloane, “Pseudo-Random Sequences and Arrays,” Proc. IEEE 64, 1715–1729 (1976).
    [CrossRef]
  6. M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).
  7. E. E. Fenimore, “Coded Aperture Imaging: The Modulation Transfer Function for Uniformly Redundant Arrays,” Appl. Opt. 19, 2465–2471 (1980).
    [CrossRef] [PubMed]
  8. E. E. Fenimore, “Coded Aperture Imaging: Predicted Performance of Uniformly Redundant Arrays,” Appl. Opt. 17, 3562 (1978).
    [CrossRef] [PubMed]
  9. S. Miyamoto, “Hadamard Transform X-Ray Telescope,” Space Sci. Instrum. 3, 473–481 (1977).
  10. E. E. Fenimore, G. S. Weston, “Fast Delta Hadamard Transform,” Appl. Opt. 20, 3058–3067 (1981).
    [CrossRef] [PubMed]
  11. S. R. Gottesman, E. J. Schneid, “PNP—A New Class of Coded Aperture Arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 (1986).
    [CrossRef]
  12. M. H. Finger, T. A. Prince, “Hexagonal Uniformly Redundant Arrays for Coded Aperture Imaging,” in Proceedings, Nineteenth International Cosmic Ray Conference, La Jolla, CA, 11–23 Aug.1985.
  13. D. Calabro, J. K. Wolf, “On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties,” Inf. Control 11, 537–560 (1968).
    [CrossRef]
  14. E. E. Fenimore, T. M. Cannon, “Uniformly Redundant Arrays: Digital Reconstruction Methods,” Appl. Opt. 20, 1858– 1864 (1981).
    [CrossRef] [PubMed]
  15. J. P. Roques, “Fast Decoding Algorithm for Uniformly Redundant Arrays,” Appl. Opt. 26, 3862–3865 (1987).
    [CrossRef] [PubMed]

1987 (2)

E. Caroli et al., “Coded Aperture Imaging in X- and Gamma-Ray Astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

J. P. Roques, “Fast Decoding Algorithm for Uniformly Redundant Arrays,” Appl. Opt. 26, 3862–3865 (1987).
[CrossRef] [PubMed]

1986 (1)

S. R. Gottesman, E. J. Schneid, “PNP—A New Class of Coded Aperture Arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 (1986).
[CrossRef]

1981 (2)

1980 (1)

1978 (2)

1977 (1)

S. Miyamoto, “Hadamard Transform X-Ray Telescope,” Space Sci. Instrum. 3, 473–481 (1977).

1976 (1)

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-Random Sequences and Arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

1968 (2)

D. Calabro, J. K. Wolf, “On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

R. H. Dicke, “Scatter-Hole Cameras for X-Rays and Gamma Rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

Calabro, D.

D. Calabro, J. K. Wolf, “On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Cannon, T. M.

Caroli, E.

E. Caroli et al., “Coded Aperture Imaging in X- and Gamma-Ray Astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Scatter-Hole Cameras for X-Rays and Gamma Rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

Fenimore, E. E.

Finger, M. H.

M. H. Finger, T. A. Prince, “Hexagonal Uniformly Redundant Arrays for Coded Aperture Imaging,” in Proceedings, Nineteenth International Cosmic Ray Conference, La Jolla, CA, 11–23 Aug.1985.

Gottesman, S. R.

S. R. Gottesman, E. J. Schneid, “PNP—A New Class of Coded Aperture Arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 (1986).
[CrossRef]

Harwit, M.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

MacWilliams, F. J.

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-Random Sequences and Arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

Mertz, L.

L. Mertz, N. O. Young, “Fresnel Transformations of Images,” in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 305.

Miyamoto, S.

S. Miyamoto, “Hadamard Transform X-Ray Telescope,” Space Sci. Instrum. 3, 473–481 (1977).

Prince, T. A.

M. H. Finger, T. A. Prince, “Hexagonal Uniformly Redundant Arrays for Coded Aperture Imaging,” in Proceedings, Nineteenth International Cosmic Ray Conference, La Jolla, CA, 11–23 Aug.1985.

Roques, J. P.

Schneid, E. J.

S. R. Gottesman, E. J. Schneid, “PNP—A New Class of Coded Aperture Arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 (1986).
[CrossRef]

Sloane, N. J. A.

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-Random Sequences and Arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

Weston, G. S.

Wolf, J. K.

D. Calabro, J. K. Wolf, “On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Young, N. O.

L. Mertz, N. O. Young, “Fresnel Transformations of Images,” in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 305.

Appl. Opt. (6)

Astrophys. J. (1)

R. H. Dicke, “Scatter-Hole Cameras for X-Rays and Gamma Rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

S. R. Gottesman, E. J. Schneid, “PNP—A New Class of Coded Aperture Arrays,” IEEE Trans. Nucl. Sci. NS-33, 745–749 (1986).
[CrossRef]

Inf. Control (1)

D. Calabro, J. K. Wolf, “On the Synthesis of Two-Dimensional Arrays with Desirable Correlation Properties,” Inf. Control 11, 537–560 (1968).
[CrossRef]

Proc. IEEE (1)

F. J. MacWilliams, N. J. A. Sloane, “Pseudo-Random Sequences and Arrays,” Proc. IEEE 64, 1715–1729 (1976).
[CrossRef]

Space Sci. Instrum. (1)

S. Miyamoto, “Hadamard Transform X-Ray Telescope,” Space Sci. Instrum. 3, 473–481 (1977).

Space Sci. Rev. (1)

E. Caroli et al., “Coded Aperture Imaging in X- and Gamma-Ray Astronomy,” Space Sci. Rev. 45, 349–403 (1987).
[CrossRef]

Other (3)

L. Mertz, N. O. Young, “Fresnel Transformations of Images,” in Proceedings, International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman & Hall, London, 1961), p. 305.

M. Harwit, N. J. A. Sloane, Hadamard Transform Optics (Academic, New York, 1979).

M. H. Finger, T. A. Prince, “Hexagonal Uniformly Redundant Arrays for Coded Aperture Imaging,” in Proceedings, Nineteenth International Cosmic Ray Conference, La Jolla, CA, 11–23 Aug.1985.

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Figures (5)

Fig. 1
Fig. 1

Schematic of coded aperture detector system. The aperture is a mosaic of a 5 × 7 URA.

Fig. 2
Fig. 2

Comparison of the throughputs for PN sequences and the new family discussed in the text. They both converge rapidly to 50%.

Fig. 3
Fig. 3

(a) Thirty-seven-element array mosaicked onto an infinite hexagonal grid. A physical aperture would be constructed from a section of the mosaic delineated by a thick line. This section is large enough to contain all possible cyclic permutations of the basic pattern but will have no ambiguities, (b) Black and white MURA pattern produced by mapping the thirty-seven-element array of Table I onto the hexagonal grid at left. Black represents opaque, and white represents transparent regions. Lattice lines are shown only for reference.

Fig. 4
Fig. 4

Hexagonal MURA apertures with (a) 61, (b) 397, (c) 1657, and (d) 1801 elements in the central or basic pattern. The four figures have been mosaicked as in Fig. 3(a).

Fig. 5
Fig. 5

Square MURA apertures whose basic patterns measure (a) 5 × 5, (b) 11 × 1l, (c) 29 × 20, and (d) 59 × 59 elements. All the patterns have been mosaicked to form a physical aperture. (a) The basic pattern is delineated by a thick line; grid marks are shown only for reference.

Tables (3)

Tables Icon

Table I Binary Patterns of the First Ten Linear MURA Sequences Treated in the Text

Tables Icon

Table II First Twenty-Five Hexagons Listed by Size

Tables Icon

Table III Spectrum of Square and Nearly Square Arrays

Equations (36)

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D = S * A + B ,
D k l = i j S i j A i + k , j + l + B k l .
S ̂ = D * G .
S ̂ = ( S * A ) * G + B * G .
A * G δ ,
S ̂ = S + B * G ,
N = i j A i j ,
A * G = N δ ,
S ̂ = N S + B * G .
σ ( S ̂ i j ) = υ a r ( S i j , S ̂ i j ) + k l υ a r ( S k l , S ̂ i j ) + υ a r ( B , S ̂ i j ) ,
var ( S i j , S ̂ i j ) = S i j k l A k l G k l 2 ;
k l var ( S k l , S ̂ i j ) = [ ( S * A ) * G 2 ] i j ;
var ( B , S ̂ i j ) = ( B * G 2 ) i j ,
k l var ( S k l , S ̂ i j ) = ( k l S k l ) ( m n A m n ) ( G i j ) 2 ,
L = 4 m + 1 , m = 1 , 2 , 3 , ,
A i = { 0 if i = 0 , 1 if i is a quadratic residue modulo L , i 0 , 0 otherwise .
G i = { + 1 if i = 0 , + 1 if A i = 1 , i 0 , 1 if A i = 0 , i 0 .
A = 0 1 0 1 1 0 0 0 0 1 1 0 1
A = 0 0 1 1 0 1 0 1 0 1 1 0 0 ,
A = 1 1 0 0 1 0 0 0 1 0 0 1 1 .
G = + + + + + + + ,
A = { A i } i = 0 L 1 ,
A = { A i j } i , j = 0 p 1
A i j = { 0 if i = 0 , 1 if j = 0 , i 0 , 1 if C i C j = + 1 , 0 otherwise ,
C i = { + 1 if i is aquadratic residue modulo p , 1 otherwise .
G i j = { + 1 if i + j = 0 ; + 1 if A i j = 1 , ( i + j 0 ) ; 1 if A i j = 0 , ( i + j 0 ) ,
A = | 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 1 0 0 1 | G = | + + + + + + + + + + + + + | .
S ̂ i j = N S i j + ( B * G ) i j ,
N = i j A i j
i j G i j = + 1 .
( B * G ) i j = B i j ,
var ( S i j , S ̂ i j ) = N S i j .
k l var ( S k l , S ̂ i j ) = N k l S k l .
var ( B , S ̂ i j ) = k l B k l ,
SNR i j = N S i j N S i j + N k l S k l + m n B m n ,
SNR i j = ( net source counts ) ( net source counts ) + ( total detector counts ) o

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