Abstract

A class of flexible coded apertures, called index class apertures, is presented. The configurations are shown to possess similar properties to the geometric apertures of Gourlay and Stephen [Appl. Opt. 22, 4042 (1983) [CrossRef]  ], and it is demonstrated that the modified uniformly redundant arrays (MURAs) are a special case of the index class apertures. The apertures are shown to offer both a larger range of throughput values and better imaging capability than is available to the geometric apertures, while at the same time possessing more rigidity of structure than other designs, such as the MURAs and the uniformly redundant arrays.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Mertz and N. O. Young, “Fresnel transformations of images,” in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, 1961), pp. 305–310.
  2. R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. Lett. 153, L101–L106(1968).
    [CrossRef]
  3. J. Gunson and B. Polychronopulos, “Optimum design of a coded mask x-ray telescope for rocket applications,” Mon. Not. R. Astron. Soc. 177, 485–497 (1976).
  4. M. H. Finger and T. A. Prince, ”Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the 19th International Cosmic Ray Conference, F. C. Jones, ed. (Scientific and Technical Information Branch, NASA, 1985), Vol. 3, pp. 295–298.
  5. S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745–749(1986).
    [CrossRef]
  6. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352(1989).
    [CrossRef]
  7. K. Byard, “Square element antisymmetric coded apertures,” Exp. Astron. 2, 227–232 (1992).
    [CrossRef]
  8. K. Byard, “On self-supporting coded aperture arrays,” Nucl. Instrum. Methods Phys. Res. A 322, 97–100(1992).
  9. K. Byard, “Synthesis of binary arrays with perfect correlation properties-coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 336, 262–268 (1993).
  10. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef]
  11. A. R. Gourlay and J. B. Stephen, “Geometric coded aperture masks,” Appl. Opt. 22, 4042–4047 (1983).
    [CrossRef]
  12. A. R. Gourlay and N. G. Young, “Coded aperture imaging: a class of flexible mask designs,” Appl. Opt. 23, 4111–4117 (1984).
    [CrossRef]
  13. A. B. Giles, “Self-supporting perfect masks for 2D infrared and x-ray imaging,” Appl. Opt. 20, 3068–3072 (1981).
    [CrossRef]
  14. A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
    [CrossRef]
  15. W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
    [CrossRef]
  16. E. E. Fenimore, “Coded aperture imaging: predicted performance of uniformly redundant arrays,” Appl. Opt. 17, 3562–3570 (1978).
    [CrossRef]
  17. L. D. Baumert, Cyclic Difference Sets, Vol. 182 of Lecture Notes in Mathematics (Springer-Verlag, 1971).
  18. P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).
  19. G. K. Skinner and T. J. Ponman, “On the properties of images from coded-mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518–522 (1994).
  20. D. Gordon, La Jolla Difference Set Repository, http://www.ccrwest.org/diffsets/diff_sets/index.html .
  21. B. Schmidt, “Cyclotomic integers and finite geometry,” J. Am. Math. Soc. 12, 929–952 (1999).
    [CrossRef]
  22. H. D. Luke, “Sequences and arrays with perfect periodic correlation,” IEEE Trans. Aerosp. Electron. Syst. 24, 287–294(1988).
    [CrossRef]
  23. I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps,” IEEE Trans. Inf. Theory 8, 10–12 (1962).
    [CrossRef]
  24. W. K. Klemperer, “Very large array configurations for the observation of rapidly varying sources,” Astron. Astrophys. Suppl. 15, 449–451 (1974).

2012 (1)

P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).

2010 (1)

W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
[CrossRef]

1999 (1)

B. Schmidt, “Cyclotomic integers and finite geometry,” J. Am. Math. Soc. 12, 929–952 (1999).
[CrossRef]

1994 (1)

G. K. Skinner and T. J. Ponman, “On the properties of images from coded-mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518–522 (1994).

1993 (1)

K. Byard, “Synthesis of binary arrays with perfect correlation properties-coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 336, 262–268 (1993).

1992 (2)

K. Byard, “Square element antisymmetric coded apertures,” Exp. Astron. 2, 227–232 (1992).
[CrossRef]

K. Byard, “On self-supporting coded aperture arrays,” Nucl. Instrum. Methods Phys. Res. A 322, 97–100(1992).

1989 (1)

1988 (1)

H. D. Luke, “Sequences and arrays with perfect periodic correlation,” IEEE Trans. Aerosp. Electron. Syst. 24, 287–294(1988).
[CrossRef]

1986 (1)

S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745–749(1986).
[CrossRef]

1984 (1)

1983 (1)

1981 (1)

1980 (1)

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

1978 (2)

1976 (1)

J. Gunson and B. Polychronopulos, “Optimum design of a coded mask x-ray telescope for rocket applications,” Mon. Not. R. Astron. Soc. 177, 485–497 (1976).

1974 (1)

W. K. Klemperer, “Very large array configurations for the observation of rapidly varying sources,” Astron. Astrophys. Suppl. 15, 449–451 (1974).

1968 (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. Lett. 153, L101–L106(1968).
[CrossRef]

1962 (1)

I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps,” IEEE Trans. Inf. Theory 8, 10–12 (1962).
[CrossRef]

Baumert, L. D.

L. D. Baumert, Cyclic Difference Sets, Vol. 182 of Lecture Notes in Mathematics (Springer-Verlag, 1971).

Beron, B. L.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Byard, K.

K. Byard, “Synthesis of binary arrays with perfect correlation properties-coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 336, 262–268 (1993).

K. Byard, “Square element antisymmetric coded apertures,” Exp. Astron. 2, 227–232 (1992).
[CrossRef]

K. Byard, “On self-supporting coded aperture arrays,” Nucl. Instrum. Methods Phys. Res. A 322, 97–100(1992).

Campbell, L.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Cannon, T. M.

Dicke, R. H.

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. Lett. 153, L101–L106(1968).
[CrossRef]

Eichler, R.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Fenimore, E. E.

Finger, M. H.

M. H. Finger and T. A. Prince, ”Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the 19th International Cosmic Ray Conference, F. C. Jones, ed. (Scientific and Technical Information Branch, NASA, 1985), Vol. 3, pp. 295–298.

Giles, A. B.

Gorodetsky, P.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Gottesman, S. R.

S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344–4352(1989).
[CrossRef]

S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745–749(1986).
[CrossRef]

Gourlay, A. R.

Gunson, J.

J. Gunson and B. Polychronopulos, “Optimum design of a coded mask x-ray telescope for rocket applications,” Mon. Not. R. Astron. Soc. 177, 485–497 (1976).

Hofstadter, R.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Hughes, E. B.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Jo, A.

W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
[CrossRef]

Johansson, A.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Klemperer, W. K.

W. K. Klemperer, “Very large array configurations for the observation of rapidly varying sources,” Astron. Astrophys. Suppl. 15, 449–451 (1974).

Lee, W.

W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
[CrossRef]

Luke, H. D.

H. D. Luke, “Sequences and arrays with perfect periodic correlation,” IEEE Trans. Aerosp. Electron. Syst. 24, 287–294(1988).
[CrossRef]

Mertz, L.

L. Mertz and N. O. Young, “Fresnel transformations of images,” in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, 1961), pp. 305–310.

Polychronopulos, B.

J. Gunson and B. Polychronopulos, “Optimum design of a coded mask x-ray telescope for rocket applications,” Mon. Not. R. Astron. Soc. 177, 485–497 (1976).

Ponman, T. J.

G. K. Skinner and T. J. Ponman, “On the properties of images from coded-mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518–522 (1994).

Prince, T. A.

M. H. Finger and T. A. Prince, ”Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the 19th International Cosmic Ray Conference, F. C. Jones, ed. (Scientific and Technical Information Branch, NASA, 1985), Vol. 3, pp. 295–298.

Reed, I. S.

I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps,” IEEE Trans. Inf. Theory 8, 10–12 (1962).
[CrossRef]

Schmidt, B.

B. Schmidt, “Cyclotomic integers and finite geometry,” J. Am. Math. Soc. 12, 929–952 (1999).
[CrossRef]

Schneid, E. J.

S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745–749(1986).
[CrossRef]

Shutler, P. M. E.

P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).

Skinner, G. K.

G. K. Skinner and T. J. Ponman, “On the properties of images from coded-mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518–522 (1994).

Springham, S. V.

P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).

Stephen, J. B.

Stewart, R. M.

I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps,” IEEE Trans. Inf. Theory 8, 10–12 (1962).
[CrossRef]

Talebitaher, A.

P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).

Wilson, S.

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

Yoon, C.

W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
[CrossRef]

Young, N. G.

Young, N. O.

L. Mertz and N. O. Young, “Fresnel transformations of images,” in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, 1961), pp. 305–310.

Appl. Opt. (6)

Astron. Astrophys. Suppl. (1)

W. K. Klemperer, “Very large array configurations for the observation of rapidly varying sources,” Astron. Astrophys. Suppl. 15, 449–451 (1974).

Astrophys. J. Lett. (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. Lett. 153, L101–L106(1968).
[CrossRef]

Exp. Astron. (1)

K. Byard, “Square element antisymmetric coded apertures,” Exp. Astron. 2, 227–232 (1992).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

H. D. Luke, “Sequences and arrays with perfect periodic correlation,” IEEE Trans. Aerosp. Electron. Syst. 24, 287–294(1988).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. S. Reed and R. M. Stewart, “Note on the existence of perfect maps,” IEEE Trans. Inf. Theory 8, 10–12 (1962).
[CrossRef]

IEEE Trans. Nucl. Sci. (3)

S. R. Gottesman and E. J. Schneid, “PNP—a new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745–749(1986).
[CrossRef]

A. Johansson, B. L. Beron, L. Campbell, R. Eichler, P. Gorodetsky, R. Hofstadter, E. B. Hughes, and S. Wilson, “The use of and active coded aperture for improved directional measurements in high energy γ-ray astronomy,” IEEE Trans. Nucl. Sci. 27, 375–380 (1980).
[CrossRef]

W. Lee, A. Jo, and C. Yoon, “Multiple modality γ-ray imager using LaCl3(Ce) scintillators with active collimation method,” IEEE Trans. Nucl. Sci. 57, 1396–1403 (2010).
[CrossRef]

J. Am. Math. Soc. (1)

B. Schmidt, “Cyclotomic integers and finite geometry,” J. Am. Math. Soc. 12, 929–952 (1999).
[CrossRef]

Mon. Not. R. Astron. Soc. (2)

G. K. Skinner and T. J. Ponman, “On the properties of images from coded-mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518–522 (1994).

J. Gunson and B. Polychronopulos, “Optimum design of a coded mask x-ray telescope for rocket applications,” Mon. Not. R. Astron. Soc. 177, 485–497 (1976).

Nucl. Instrum. Methods Phys. Res. A (3)

K. Byard, “On self-supporting coded aperture arrays,” Nucl. Instrum. Methods Phys. Res. A 322, 97–100(1992).

K. Byard, “Synthesis of binary arrays with perfect correlation properties-coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 336, 262–268 (1993).

P. M. E. Shutler, A. Talebitaher, and S. V. Springham, “Signal-to-noise ratio in coded aperture imaging, ” Nucl. Instrum. Methods Phys. Res. A 669, 22–31 (2012).

Other (4)

M. H. Finger and T. A. Prince, ”Hexagonal uniformly redundant arrays for coded-aperture imaging,” in Proceedings of the 19th International Cosmic Ray Conference, F. C. Jones, ed. (Scientific and Technical Information Branch, NASA, 1985), Vol. 3, pp. 295–298.

L. Mertz and N. O. Young, “Fresnel transformations of images,” in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman and Hall, 1961), pp. 305–310.

D. Gordon, La Jolla Difference Set Repository, http://www.ccrwest.org/diffsets/diff_sets/index.html .

L. D. Baumert, Cyclic Difference Sets, Vol. 182 of Lecture Notes in Mathematics (Springer-Verlag, 1971).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Apertures with p=13. (a) n=4, t=28.4%; (b) n=3, t=35.5%.

Fig. 2.
Fig. 2.

Distribution of possible aperture throughput values t versus aperture size p using the geometric and the index class apertures. The dashed line indicates the MURA equivalent, n=2.

Fig. 3.
Fig. 3.

Possible index class aperture unit patterns for p=31.

Fig. 4.
Fig. 4.

Regions contributing to the integrated intensities of the source distribution at all pixels other then the (0,0)th to evaluate SNR00 for the aperture with p=13, n=4.

Fig. 5.
Fig. 5.

Graph of SNRij/SNRG3 versus B/Sij for low background for index class apertures with p=31 and selected values of n.

Fig. 6.
Fig. 6.

SNR versus fraction of extended source inside region T1 for a total source flux of 100 counts per pixel: (a) small proportion of source in (i,j) pixel, low background, (b) small proportion of source in (i,j) pixel, high background, (c) half of source in (i,j) pixel, low background, (d) half of source in (i,j) pixel, high background.

Tables (2)

Tables Icon

Table 1. Ratio of SNR Values for the Index Class versus G3 Apertures for p=31

Tables Icon

Table 2. Results of SNR for the Index Class Apertures (SNRij) versus the URAs Created from Cyclic Difference Sets (SNRijU)a

Equations (25)

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

R(x)={gun+x(modp):0uf1}={gx,gn+x,g2n+x,,g(f1)n+x}(modp).
A(i,j)={1ifj=0,i01ifiR(x)andjR(x)0otherwise,
G(i,j)={αif  i=0,j=01if  A(i,j)=1βotherwise,
α=(p2)(n2)+1n1,β=1n1.
A*G=C(k,l)=i=0p1j=0p1A(i,j)G(i+k,j+l)={N0ifk=l=00otherwise,
R(0)={24u+0(mod13):0u2}={20,24,28}(mod13)={1,16,256}(mod13)={1,3,9},
R(1)={24u+1(mod13):0u2}={21,25,29}(mod13)={2,5,6},
R(2)={24u+2(mod13):0u2}={22,26,210}(mod13)={4,10,12},
R(3)={24u+3(mod13):0u2}={23,27,211}(mod13)={7,8,11}.
A=[0000000000000110100000100010100110000001101000001000100010000010110100110000001010011000000100000011001010000001100101101000001000100010000010110000001100101000100000101],
G=[αββββββββββββ11β1βββββ1βββ1β1ββ11ββββββ11β1βββββ1βββ1βββ1βββββ1β11β1ββ11ββββββ1β1ββ11ββββββ1ββββββ11ββ1β1ββββββ11ββ1β11β1βββββ1βββ1βββ1βββββ1β11ββββββ11ββ1β1βββ1βββββ1β1],
t=(p1)(p1+n)p2n.
SNRij=N0Sij[N0Sij+(α2+N1+N2β2)IT1+(N3+N4β2)IT2+Bλ]1/2,
N0=(p1)(p1+n)n=tp2,
N1=p2+p(n+1)(n2)(2n+1)(n1)n2,
N2=(p1)[p(n1)+1]n2,
N3=(p1)(p+n1)n2,
N4=(p1)(n1)(p+n1)n2,
λ=α2+N0+(p2N01)β2,
SNRij=(N0Sij)1/2[1+(λ/N0)(B/Sij)]1/2.
SNRG3=[(4p4)Sij]1/2{1+[λG/(4p4)](B/Sij)}1/2,
SNRijSNRG3=p1+n4n(λGλ)1/2.
n=(p1)/3.
l=N0(N01)(p21).
SNRijU=N0Sij{N0Sij+[l+(N0l)g2]IT+[N0+(p2N0)g2]B}1/2,

Metrics