Abstract

Two-dimensional binary signals (arrays) with good autocorrelation properties are needed for coded-aperture imaging systems. In many astrophysical instruments, circular detectors with hexagonally packed detector elements are used, such that hexagonal coded apertures are often preferable to rectangular ones. A general method for folding a one-dimensional sequence into a hexagonal array is presented, by which the periodic or odd-periodic correlation properties of the original sequence are preserved. This method is applied to a known family of sequences with perfect odd-periodic correlation, yielding a new family of almost-binary and odd-perfect—or binary and almost odd-perfect—hexagonal arrays with optimum properties for coded-aperture imaging. The new odd-perfect arrays have near-uniform side lengths and exist for many more sizes than known families of even-periodic hexagonal arrays with good imaging properties.

© 2001 Optical Society of America

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References

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  1. G. K. Skinner, “Imaging with coded-aperture masks,” Nucl. Instrum. Methods 221, 33–40 (1984).
    [CrossRef]
  2. A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
    [CrossRef]
  3. E. E. Fenimore, T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef] [PubMed]
  4. A. Busboom, H. Elders-Boll, H. D. Schotten, “Uniformly redundant arrays,” Exp. Astron. 8, 97–123 (1998).
    [CrossRef]
  5. J. Jedwab, C. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
    [CrossRef]
  6. H. D. Lüke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
    [CrossRef]
  7. W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
    [CrossRef]
  8. A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).
  9. I. D. Jupp, A. R. Green, A. J. Dean, “Imaging with two angular scales using coded mask techniques,” in Imaging in High Energy Astronomy: Proceedings of the International Workshop Held in Anacapri (Capri, Italy) 26–30 September 1994, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 209–212.
    [CrossRef]
  10. M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1984), pp. 295–298.
  11. S. R. Gottesman, E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352 (1989).
    [CrossRef] [PubMed]
  12. H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
    [CrossRef]
  13. H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
    [CrossRef]
  14. See http://www.ient.rwth-aachen.de/oddperfect.html .
  15. H. D. Lüke, A. Busboom, “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. 34, 1345–1350 (1998).
    [CrossRef]
  16. H. D. Lüke, A. Busboom, “Mismatched filtering of periodic and odd-periodic binary arrays,” Appl. Opt. 37, 856–864 (1998).
    [CrossRef]

1998 (3)

A. Busboom, H. Elders-Boll, H. D. Schotten, “Uniformly redundant arrays,” Exp. Astron. 8, 97–123 (1998).
[CrossRef]

H. D. Lüke, A. Busboom, “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. 34, 1345–1350 (1998).
[CrossRef]

H. D. Lüke, A. Busboom, “Mismatched filtering of periodic and odd-periodic binary arrays,” Appl. Opt. 37, 856–864 (1998).
[CrossRef]

1997 (3)

1995 (1)

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

1990 (2)

J. Jedwab, C. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

1989 (1)

1984 (2)

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

G. K. Skinner, “Imaging with coded-aperture masks,” Nucl. Instrum. Methods 221, 33–40 (1984).
[CrossRef]

1978 (1)

Busboom, A.

Byard, K.

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

Cannon, T. M.

Cook, W. R.

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

Dean, A. J.

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

I. D. Jupp, A. R. Green, A. J. Dean, “Imaging with two angular scales using coded mask techniques,” in Imaging in High Energy Astronomy: Proceedings of the International Workshop Held in Anacapri (Capri, Italy) 26–30 September 1994, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 209–212.
[CrossRef]

Elders-Boll, H.

A. Busboom, H. Elders-Boll, H. D. Schotten, “Uniformly redundant arrays,” Exp. Astron. 8, 97–123 (1998).
[CrossRef]

A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
[CrossRef]

Fenimore, E. E.

Finger, M.

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

Finger, M. H.

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1984), pp. 295–298.

Goldwurm, A.

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

Gottesman, S. R.

Green, A. R.

I. D. Jupp, A. R. Green, A. J. Dean, “Imaging with two angular scales using coded mask techniques,” in Imaging in High Energy Astronomy: Proceedings of the International Workshop Held in Anacapri (Capri, Italy) 26–30 September 1994, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 209–212.
[CrossRef]

Hall, C. J.

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

Harding, J. S. J.

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

Jedwab, J.

J. Jedwab, C. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

Jupp, I. D.

I. D. Jupp, A. R. Green, A. J. Dean, “Imaging with two angular scales using coded mask techniques,” in Imaging in High Energy Astronomy: Proceedings of the International Workshop Held in Anacapri (Capri, Italy) 26–30 September 1994, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 209–212.
[CrossRef]

Lüke, H. D.

H. D. Lüke, A. Busboom, “Mismatched filtering of periodic and odd-periodic binary arrays,” Appl. Opt. 37, 856–864 (1998).
[CrossRef]

H. D. Lüke, A. Busboom, “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. 34, 1345–1350 (1998).
[CrossRef]

H. D. Lüke, A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612–6619 (1997).
[CrossRef]

H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
[CrossRef]

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

Mitchell, C.

J. Jedwab, C. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

Prince, T. A.

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1984), pp. 295–298.

Schotten, H. D.

A. Busboom, H. Elders-Boll, H. D. Schotten, “Uniformly redundant arrays,” Exp. Astron. 8, 97–123 (1998).
[CrossRef]

A. Busboom, H. D. Schotten, H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058–1065 (1997).
[CrossRef]

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

Skinner, G. K.

G. K. Skinner, “Imaging with coded-aperture masks,” Nucl. Instrum. Methods 221, 33–40 (1984).
[CrossRef]

Stone, E. C.

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

A. Goldwurm, K. Byard, A. J. Dean, C. J. Hall, J. S. J. Harding, “Laboratory images with HURA coded apertures,” Astron. Astrophys. 227, 640–648 (1990).

Electron. Lett. (1)

J. Jedwab, C. Mitchell, “Infinite families of quasiperfect and doubly quasiperfect binary arrays,” Electron. Lett. 26, 294–295 (1990).
[CrossRef]

Exp. Astron. (1)

A. Busboom, H. Elders-Boll, H. D. Schotten, “Uniformly redundant arrays,” Exp. Astron. 8, 97–123 (1998).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (2)

H. D. Lüke, H. D. Schotten, “Odd-perfect, almost binary correlation sequences,” IEEE Trans. Aerosp. Electron. Syst. 31, 495–498 (1995).
[CrossRef]

H. D. Lüke, A. Busboom, “Mismatched filtering of odd-periodic binary sequences,” IEEE Trans. Aerosp. Electron. Syst. 34, 1345–1350 (1998).
[CrossRef]

IEEE Trans. Inf. Theory (1)

H. D. Lüke, “Binary odd-periodic complementary sequences,” IEEE Trans. Inf. Theory 43, 365–367 (1997).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

W. R. Cook, M. Finger, T. A. Prince, E. C. Stone, “Gamma-ray imaging with a rotating hexagonal uniformly redundant array,” IEEE Trans. Nucl. Sci. NS-31, 771–775 (1984).
[CrossRef]

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

Nucl. Instrum. Methods (1)

G. K. Skinner, “Imaging with coded-aperture masks,” Nucl. Instrum. Methods 221, 33–40 (1984).
[CrossRef]

Other (3)

See http://www.ient.rwth-aachen.de/oddperfect.html .

I. D. Jupp, A. R. Green, A. J. Dean, “Imaging with two angular scales using coded mask techniques,” in Imaging in High Energy Astronomy: Proceedings of the International Workshop Held in Anacapri (Capri, Italy) 26–30 September 1994, L. Bassani, G. di Cocco, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1995), pp. 209–212.
[CrossRef]

M. H. Finger, T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the Nineteenth International Cosmic Ray Conference (Scientific and Technical Information Branch, NASA, Washington, D.C., 1984), pp. 295–298.

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Figures (5)

Fig. 1
Fig. 1

Hexagonal array on a hexagonal grid (top) and sheared visualization of the same array on a rectangular grid (bottom).

Fig. 2
Fig. 2

Three methods of tiling hexagonal arrays such as to cover the entire plane seamlessly.

Fig. 3
Fig. 3

Folding of a sequence of length N = 14 into a type B hexagonal array with a = 2, b = 3, c = 3, Δ = 0. Folding diagonal n x = -1, n y = 2.

Fig. 4
Fig. 4

Three methods of odd-periodically tiling hexagonal arrays with alternation of the signs along three different axes. Each hexagon depicts one basic array with (-) or without (+) alternated signs of its element with respect to the original array.

Fig. 5
Fig. 5

Example of an odd-perfect hexagonal array with N = 114 elements and a = 6, b = c = 7, Δ = 0, type A, n x = 1, n y = 0.

Tables (3)

Tables Icon

Table 1 Offsets for Tiling Hexagonal Arrays of Types A, B, and C

Tables Icon

Table 2 Necessary and Sufficient Conditions for Periodic Hexagonal Folding

Tables Icon

Table 3 New Odd-Perfect Hexagonal Arrays up to a = 5, b = 5, c = 5

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

a+b=d+e,
b+c=e+f,
c+d=a+f.
Δ=d-a=b-e=f-c.
N=b+c-1a+f-1-cc-12-ff-12.
N=b+c-1a+c+Δ-1-cc-12-c+Δc+Δ-12=ab+bc+ac-a-b-c+bΔ-ΔΔ+12+1.
N=ab+bc+ac-a-b-c+bΔ-Δ+1.
Δxij=iΔx1+jΔx2,
Δyij=iΔy1+jΔy2.
ax, y=sn,
xnxn mod N,ynyn mod N,
knx=iΔx1+jΔx2,
kny=iΔy1+jΔy2,
i=-mΔx2/gcdΔx1, Δx2;m=, -2, -1, 0, 1, 2,.
j=mΔx1/gcdΔx1, Δx2;m=, -2, -1, 0, 1, 2,.
kny=mΔx1Δy2-Δx2Δy1;m=, -2, -1, 0, 1, 2,.
Δx1Δy2-Δx2Δy1=N
k=mΔx1Δy2-Δx2Δy1=mN;m=, -2, -1, 0, 1, 2, ,
gcdΔy1, Δy2=1,
gcdnx, N=1.
0=inyΔx1-nxΔy1+jnyΔx2-nxΔy2.
i=-mnyΔx2-nxΔy2;m=, -2, -1, 0, 1, 2,,
j=mnyΔx1-nxΔy1.
knx=m-nyΔx1Δx2+nxΔx1Δy2+nyΔx1Δx2-nxΔy1Δx2,k=mΔx1Δy2-Δy1Δx2,
sˆn=-1n/Nsn mod N,
ϕˆssm=n=0N-1 snsˆn+m.
imin=nyΔx2-nxΔy2.
jmin=nxΔy1-nyΔx1.
N=pa+1,

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