Abstract

Simple formulas are often used to estimate the sensitivity of coded mask x-ray or gamma-ray telescopes, but these are strictly applicable only if a number of basic assumptions are met. Complications arise, for example, if a grid structure is used to support the mask elements, if the detector spatial resolution is not good enough to completely resolve all the detail in the shadow of the mask, or if any of a number of other simplifying conditions are not fulfilled. We derive more general expressions for the Poisson-noise-limited sensitivity of astronomical telescopes using the coded mask technique, noting explicitly in what circumstances they are applicable. The emphasis is on using nomenclature and techniques that result in simple and revealing results. Where no convenient expression is available a procedure is given that allows the calculation of the sensitivity. We consider certain aspects of the optimization of the design of a coded mask telescope and show that when the detector spatial resolution and the mask to detector separation are fixed, the best source location accuracy is obtained when the mask elements are equal in size to the detector pixels.

© 2008 Optical Society of America

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  1. S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
    [CrossRef]
  2. C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
    [CrossRef]
  3. A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
    [CrossRef]
  4. F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
    [CrossRef]
  5. R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).
  6. R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).
  7. E. Caroli, J. B. Stephen, G. di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349-403 (1987).
    [CrossRef]
  8. G. K. Skinner, “Coding (and decoding) coded mask telescopes,” Exp. Astron. 6, 1-7 (1995).
    [CrossRef]
  9. J. G. Ables, “Fourier transform photography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172-173(1968).
  10. R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
    [CrossRef]
  11. L. Mertz, Transformations in Optics (Wiley, 1965).
  12. L. Mertz, “Ancestry of indirect techniques for x-ray imaging,” Proc. SPIE 1159, 14-17 (1989).
  13. 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).
  14. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337-347 (1978).
  15. R. J. Proctor, G. K. Skinner, and A. P. Willmore, “The design of optimum coded mask x-ray telescopes,” Mon. Not. R. Astron. Soc. 187, 633-643 (1979).
  16. A. B. Giles, “Self-supporting perfect masks for 2-D infrared and x-ray imaging,” Appl. Opt. 20, 3068-3072 (1981).
  17. M. Oda, “X-ray imaging techniques--modulation collimator and coded mask.” Adv. Space Res. 2, 207-216 (1982).
    [CrossRef]
  18. W. J. Wild, “Dilute uniformly redundant sequences for use in coded-aperture imaging,” Opt. Lett. 8, 247-249 (1983).
  19. A. R. Gourlay and N. G. S. Young, “Coded aperture imaging: a class of flexible mask designs,” Appl. Opt. 23, 4111-4117(1984).
  20. M. H. Finger and T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in “Proceedings of the International Cosmic Ray Conference,” F.C.Jones, ed. (NASA, 1985), pp. 295-298.
  21. S. R. Gottesman and E. J. Schneid, “PNP--A new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745-749(1986).
  22. K. Byard, “Square element antisymmetric coded apertures,” Exp. Astron. 2, 227-232 (1992).
    [CrossRef]
  23. K. Byard, “On self-supporting coded aperture arrays,” Nucl. Instrum. Methods Phys. Res. A 322, 97-100 (1992).
  24. L. E. Kopilovich and L. G. Sodin, “Synthesis of coded masks for gamma-ray and x-ray telescopes,” Mon. Not. R. Astron. Soc. 266, 357-359 (1994).
  25. L. G. Sodin, “Synthesis of coded masks for x-ray and gamma-ray telescopes,” Astron. Lett. 21, 423-427 (1995).
  26. H. D. Lüke and A. Busboom, “Binary arrays with perfect odd-periodic autocorrelation,” Appl. Opt. 36, 6612-6619 (1997).
  27. A. Busboom, H. D. Schotten, and H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. A 14, 1058-1065 (1997).
    [CrossRef]
  28. R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273-284 (2001).
  29. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344-4352 (1989).
  30. H. D. Lüke and A. Busboom, “Mismatched filtering of periodic and odd-periodic binary arrays,” Appl. Opt. 37, 856-864 (1998).
  31. M. R. Sims, M. J. L. Turner, and R. Willingale, “The influence of disturbing effects on the performance of a wide field coded mass x-ray camera,” Nucl. Instrum. Methods Phys. Res. A 228, 512-531 (1985).
  32. D. Vigneau and D. W. Robinson, “Large coded aperture mask for spaceflight hard x-ray images,” Proc. SPIE 4851, 1326-1335 (2003).
    [CrossRef]
  33. A. P. Hammersley and G. K. Skinner, “Data processing of imperfectly coded images.” Nucl. Instrum. Methods Phys. Res. A 221, 45-48 (1984).
  34. R. Willingale, “Use of the maximum entropy method in x-ray astronomy,” Mon. Not. R. Astron. Soc. 194, 359-364 (1981).
  35. R. Willingale, M. R. Sims, and M. J. L. Turner, “Advanced deconvolution techniques for coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 221, 60-66 (1984).
  36. M. L. McConnell, E. L. Chupp, P. P. Dunphy, D. J. Forrest, and A. Owens, “The problem of nonuniform background rates in a coded aperture gamma-ray telescope,” in Proceedings of the International Cosmic Ray Conference (Nauka, 1987), p. 309-311.
  37. A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).
  38. J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.
  39. R. M. Rideout and G. K. Skinner, “Minimum error image reconstruction for coded mask telescopes,” Astron. Astrophys. 120, 579-585 (1996).
  40. L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
    [CrossRef]
  41. G. Skinner and P. Connell, “The Spiros imaging software for the INTEGRAL SPI spectrometer,” Astron. Astrophys. 411, L123-L126 (2003).
    [CrossRef]
  42. A. W. Strong, “Maximum entropy imaging with INTEGRAL/SPI data,” Astron. Astrophys. 411, L127-L129 (2003).
    [CrossRef]
  43. B. M. Schäfer and N. Kawai, “A Fourier-based algorithm for modelling aberrations in HETE-2's imaging system,” Nucl. Instrum. Methods Phys. Res. A 500, 263-271 (2003).
  44. C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
    [CrossRef]
  45. J. E. Grindlay and J. Hong, “Optimizing wide-field coded aperture imaging: radial mask holes and scanning,” Proc. SPIE 5168, 402-410 (2004).
    [CrossRef]
  46. D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).
  47. P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
    [CrossRef]
  48. T. M. Palmieri, “An x-ray telescope sensitive at high energies,” Ap. Space Sci. 26, 431-445 (1974).
  49. E. E. Fenimore, “Coded aperture imaging--predicted performance of uniformly redundant arrays,” Appl. Opt. 17, 3562-3570 (1978).
  50. E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).
  51. J. J. M. in 't Zand, J. Heise, and R. Jager, “The optimum open fraction of coded apertures: with an application to the wide field x-ray cameras of SAX,” Astron. Astrophys. 288, 665-674 (1994).
  52. This illustrates an approximation in the approach used here. The shadow of a source at trial positions away from the peak cannot be totally independent of that expected for a source at the peak position . The process described here is equivalent to correlation with a mean-subtracted form of the expected count distribution, which must produce a function with a mean of zero. Thus, a positive peak implies a negative mean level elsewhere. The effect goes inversely with the number of resolution elements in the field of view and becomes negligible for sufficiently large N.
  53. G. K. Skinner and T. J. Ponman, “On the properties of images from coded mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518 (1994).
  54. The same symbols are used indiscriminately here for the observed and expected numbers of events as the one is the best estimate of the other.
  55. W. Cash, “Parameter estimation in astronomy through application of the likelihood ratio,” Astrophys. J. 228, 939-947 (1979).
    [CrossRef]
  56. R. N. Bracewell, “The Fourier Transform and Its Applications,” 2nd ed. (McGraw-Hill, 1986)
  57. Note that the Ci are counts per pixel where CS and CB were totals for all the pixels of a particular category.
  58. Sometimes S/(σS-1) is used for a better approximation but the differences are small in practical cases.

2008 (1)

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

2005 (2)

C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
[CrossRef]

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

2004 (1)

J. E. Grindlay and J. Hong, “Optimizing wide-field coded aperture imaging: radial mask holes and scanning,” Proc. SPIE 5168, 402-410 (2004).
[CrossRef]

2003 (6)

G. Skinner and P. Connell, “The Spiros imaging software for the INTEGRAL SPI spectrometer,” Astron. Astrophys. 411, L123-L126 (2003).
[CrossRef]

A. W. Strong, “Maximum entropy imaging with INTEGRAL/SPI data,” Astron. Astrophys. 411, L127-L129 (2003).
[CrossRef]

B. M. Schäfer and N. Kawai, “A Fourier-based algorithm for modelling aberrations in HETE-2's imaging system,” Nucl. Instrum. Methods Phys. Res. A 500, 263-271 (2003).

P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
[CrossRef]

D. Vigneau and D. W. Robinson, “Large coded aperture mask for spaceflight hard x-ray images,” Proc. SPIE 4851, 1326-1335 (2003).
[CrossRef]

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

2001 (2)

L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
[CrossRef]

R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273-284 (2001).

1999 (1)

R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

1998 (1)

1997 (3)

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

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

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

1996 (3)

A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
[CrossRef]

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

R. M. Rideout and G. K. Skinner, “Minimum error image reconstruction for coded mask telescopes,” Astron. Astrophys. 120, 579-585 (1996).

1995 (2)

L. G. Sodin, “Synthesis of coded masks for x-ray and gamma-ray telescopes,” Astron. Lett. 21, 423-427 (1995).

G. K. Skinner, “Coding (and decoding) coded mask telescopes,” Exp. Astron. 6, 1-7 (1995).
[CrossRef]

1994 (3)

L. E. Kopilovich and L. G. Sodin, “Synthesis of coded masks for gamma-ray and x-ray telescopes,” Mon. Not. R. Astron. Soc. 266, 357-359 (1994).

J. J. M. in 't Zand, J. Heise, and R. Jager, “The optimum open fraction of coded apertures: with an application to the wide field x-ray cameras of SAX,” Astron. Astrophys. 288, 665-674 (1994).

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

1992 (3)

A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

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 (2)

L. Mertz, “Ancestry of indirect techniques for x-ray imaging,” Proc. SPIE 1159, 14-17 (1989).

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

1987 (1)

E. Caroli, J. B. Stephen, G. di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349-403 (1987).
[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).

1985 (1)

M. R. Sims, M. J. L. Turner, and R. Willingale, “The influence of disturbing effects on the performance of a wide field coded mass x-ray camera,” Nucl. Instrum. Methods Phys. Res. A 228, 512-531 (1985).

1984 (4)

A. P. Hammersley and G. K. Skinner, “Data processing of imperfectly coded images.” Nucl. Instrum. Methods Phys. Res. A 221, 45-48 (1984).

R. Willingale, M. R. Sims, and M. J. L. Turner, “Advanced deconvolution techniques for coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 221, 60-66 (1984).

E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).

A. R. Gourlay and N. G. S. Young, “Coded aperture imaging: a class of flexible mask designs,” Appl. Opt. 23, 4111-4117(1984).

1983 (1)

1982 (1)

M. Oda, “X-ray imaging techniques--modulation collimator and coded mask.” Adv. Space Res. 2, 207-216 (1982).
[CrossRef]

1981 (2)

R. Willingale, “Use of the maximum entropy method in x-ray astronomy,” Mon. Not. R. Astron. Soc. 194, 359-364 (1981).

A. B. Giles, “Self-supporting perfect masks for 2-D infrared and x-ray imaging,” Appl. Opt. 20, 3068-3072 (1981).

1979 (2)

W. Cash, “Parameter estimation in astronomy through application of the likelihood ratio,” Astrophys. J. 228, 939-947 (1979).
[CrossRef]

R. J. Proctor, G. K. Skinner, and A. P. Willmore, “The design of optimum coded mask x-ray telescopes,” Mon. Not. R. Astron. Soc. 187, 633-643 (1979).

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)

T. M. Palmieri, “An x-ray telescope sensitive at high energies,” Ap. Space Sci. 26, 431-445 (1974).

1968 (2)

J. G. Ables, “Fourier transform photography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172-173(1968).

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

Ables, J. G.

J. G. Ables, “Fourier transform photography: a new method for x-ray astronomy,” Proc. Astron. Soc. Australia 1, 172-173(1968).

Accorsi, R.

R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273-284 (2001).

Ballet, J.

L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
[CrossRef]

Band, D. L.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Barbier, L. M.

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

Barret, D.

J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.

Barthelmy, S.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Barthelmy, S. D.

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
[CrossRef]

Bertram, D.

A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

Bond, I. A.

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

Bouchet, L.

L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, “The Fourier Transform and Its Applications,” 2nd ed. (McGraw-Hill, 1986)

Bradt, H.

A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
[CrossRef]

Brinkman, A. C.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

Busboom, A.

Butler, R. C.

E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).

Byard, K.

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P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
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M. L. McConnell, E. L. Chupp, P. P. Dunphy, D. J. Forrest, and A. Owens, “The problem of nonuniform background rates in a coded aperture gamma-ray telescope,” in Proceedings of the International Cosmic Ray Conference (Nauka, 1987), p. 309-311.

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A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

Clausen, K.

P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
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G. Skinner and P. Connell, “The Spiros imaging software for the INTEGRAL SPI spectrometer,” Astron. Astrophys. 411, L123-L126 (2003).
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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
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S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
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Dean, A. J.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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Di Cocco, G.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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[CrossRef]

E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).

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R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
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R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

Dunphy, P. P.

M. L. McConnell, E. L. Chupp, P. P. Dunphy, D. J. Forrest, and A. Owens, “The problem of nonuniform background rates in a coded aperture gamma-ray telescope,” in Proceedings of the International Cosmic Ray Conference (Nauka, 1987), p. 309-311.

Elders-Boll, H.

Eyles, C. J.

A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

Fenimore, E. E.

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
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F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
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S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. 28, 4344-4352 (1989).

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E. E. Fenimore, “Coded aperture imaging--predicted performance of uniformly redundant arrays,” Appl. Opt. 17, 3562-3570 (1978).

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M. H. Finger and T. A. Prince, “Hexagonal uniformly redundant arrays for coded-aperture imaging,” in “Proceedings of the International Cosmic Ray Conference,” F.C.Jones, ed. (NASA, 1985), pp. 295-298.

Fishman, G.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Forrest, D. J.

M. L. McConnell, E. L. Chupp, P. P. Dunphy, D. J. Forrest, and A. Owens, “The problem of nonuniform background rates in a coded aperture gamma-ray telescope,” in Proceedings of the International Cosmic Ray Conference (Nauka, 1987), p. 309-311.

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R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

Galassi, M.

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
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Garson, A.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Gasparini, F.

R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273-284 (2001).

Gehrels, N.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
[CrossRef]

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Giles, A. B.

Giménez, A.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Goldwurm, A.

L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
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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).

S. R. Gottesman and E. J. Schneid, “PNP--A new class of coded aperture arrays,” IEEE Trans. Nucl. Sci. 33, 745-749(1986).

Goulooze, H.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

Gourlay, A. R.

Grebenev, S.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Grindlay, J. E.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

J. E. Grindlay and J. Hong, “Optimizing wide-field coded aperture imaging: radial mask holes and scanning,” Proc. SPIE 5168, 402-410 (2004).
[CrossRef]

J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.

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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).

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A. P. Hammersley and G. K. Skinner, “Data processing of imperfectly coded images.” Nucl. Instrum. Methods Phys. Res. A 221, 45-48 (1984).

Hartmann, D. H.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Hatsukade, I.

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

Heise, J.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

J. J. M. in 't Zand, J. Heise, and R. Jager, “The optimum open fraction of coded apertures: with an application to the wide field x-ray cameras of SAX,” Astron. Astrophys. 288, 665-674 (1994).

Hermsen, W.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Herring, J. R. H.

A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

Hong, J.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

J. E. Grindlay and J. Hong, “Optimizing wide-field coded aperture imaging: radial mask holes and scanning,” Proc. SPIE 5168, 402-410 (2004).
[CrossRef]

Hullinger, D.

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

in 't Zand, J. J.

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

in 't Zand, J. J. M.

J. J. M. in 't Zand, J. Heise, and R. Jager, “The optimum open fraction of coded apertures: with an application to the wide field x-ray cameras of SAX,” Astron. Astrophys. 288, 665-674 (1994).

Jager, R.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

J. J. M. in 't Zand, J. Heise, and R. Jager, “The optimum open fraction of coded apertures: with an application to the wide field x-ray cameras of SAX,” Astron. Astrophys. 288, 665-674 (1994).

Janin, G.

P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
[CrossRef]

Jensen, P. L.

P. L. Jensen, K. Clausen, C. Cassi, F. Ravera, G. Janin, C. Winkler, and R. Much, “The INTEGRAL spacecraft--in-orbit performance,” Astron. Astrophys. 411, L7-L17 (2003).
[CrossRef]

Jernigan, J. G.

R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
[CrossRef]

Kawai, N.

B. M. Schäfer and N. Kawai, “A Fourier-based algorithm for modelling aberrations in HETE-2's imaging system,” Nucl. Instrum. Methods Phys. Res. A 500, 263-271 (2003).

F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

Kopilovich, L. E.

L. E. Kopilovich and L. G. Sodin, “Synthesis of coded masks for gamma-ray and x-ray telescopes,” Mon. Not. R. Astron. Soc. 266, 357-359 (1994).

Krawczynski, H.

D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

Krimm, H. A.

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

Lanza, R. C.

R. Accorsi, F. Gasparini, and R. C. Lanza, “Optimal coded aperture patterns for improved SNR in nuclear medicine imaging,” Nucl. Instrum. Methods Phys. Res. A 474, 273-284 (2001).

Lebrun, F.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Levine, A.

R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

Levine, A. M.

A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
[CrossRef]

Lowes, P.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

Lüke, H. D.

Lum, K. S. K.

J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.

Lund, N.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Maggioli, P. P.

E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).

Manandhar, R. P.

J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.

Markwardt, C. B.

C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
[CrossRef]

S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
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L. Bouchet, J. P. Roques, J. Ballet, A. Goldwurm, and J. Paul, “The SIGMA/Granat telescope: calibration and data reduction,” Astrophys. J. 548, 990-1009 (2001).
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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

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R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

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A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
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R. Willingale, M. R. Sims, and M. J. L. Turner, “Advanced deconvolution techniques for coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 221, 60-66 (1984).

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D. L. Band, J. E. Grindlay, J. Hong, G. Fishman, D. H. Hartmann, A. GarsonIII, H. Krawczynski, S. Barthelmy, N. Gehrels, and G. Skinner, “EXIST's gamma-ray burst sensitivity,” Astrophys. J. 673, 1225-1232 (2008).

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C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
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G. K. Skinner and T. J. Ponman, “On the properties of images from coded mask telescopes,” Mon. Not. R. Astron. Soc. 267, 518 (1994).

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R. J. Proctor, G. K. Skinner, and A. P. Willmore, “The design of optimum coded mask x-ray telescopes,” Mon. Not. R. Astron. Soc. 187, 633-643 (1979).

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A. M. Levine, H. Bradt, W. Cui, J. G. Jernigan, E. H. Morgan, R. Remillard, R. E. Shirey, and D. A. Smith, “First results from the all-sky monitor on the Rossi x-ray timing explorer,” Astrophys. J. 469, L33-L36 (1996).
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E. Caroli, J. B. Stephen, G. di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349-403 (1987).
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E. Caroli, R. C. Butler, G. di Cocco, P. P. Maggioli, L. Natalucci, and A. Spizzichino, “Coded masks in x- and gamma-ray astronomy--The problem of the signal-to-noise ratio evaluation,” Nuovo Cimento C 7, 786-804 (1984).

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E. Caroli, J. B. Stephen, G. di Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349-403 (1987).
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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
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F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
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S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

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S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
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F. Tokanai, M. Matsuoka, N. Kawai, A. Yoshida, M. Yamauchi, K. Takagishi, I. Hatsukade, E. E. Fenimore, M. Galassi, and J. J. in 't Zand, I. A. Bond, K. Morimoto, and H. Nakamura, “Performance of wide-field x-ray monitor on board HETE (High-Energy Transient Experiment),” Proc. SPIE 2808, 563-570(1996).
[CrossRef]

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S. D. Barthelmy, L. M. Barbier, J. R. Cummings, E. E. Fenimore, N. Gehrels, D. Hullinger, H. A. Krimm, C. B. Markwardt, D. M. Palmer, A. Parsons, G. Sato, M. Suzuki, T. Takahashi, M. Tashiro, and J. Tueller, “The Burst Alert Telescope (BAT) on the SWIFT Midex mission,” Space Sci. Rev. 120, 143-164 (2005).
[CrossRef]

C. B. Markwardt, J. Tueller, G. K. Skinner, N. Gehrels, S. D. Barthelmy, and R. F. Mushotzky, “The SWIFT/BAT high-latitude survey: first results,” Astrophys. J. 633, L77-L80 (2005).
[CrossRef]

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M. R. Sims, M. J. L. Turner, and R. Willingale, “The influence of disturbing effects on the performance of a wide field coded mass x-ray camera,” Nucl. Instrum. Methods Phys. Res. A 228, 512-531 (1985).

R. Willingale, M. R. Sims, and M. J. L. Turner, “Advanced deconvolution techniques for coded aperture imaging,” Nucl. Instrum. Methods Phys. Res. A 221, 60-66 (1984).

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C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

Vance, S.

J. E. Grindlay, D. Barret, K. S. K. Lum, R. P. Manandhar, B. Robbason, and S. Vance, “New wavelet methods for flatfielding coded aperture images,” in Imaging in High Energy Astronomy, L. Bassani and G. DiCocco, eds. (Kluwer Academic, 1995), p. 213-220.

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R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

Vedrenne, G.

C. Winkler, T. J.-L. Courvoisier, G. Di Cocco, N. Gehrels, A. Giménez, S. Grebenev, W. Hermsen, J. M. Mas-Hesse, F. Lebrun, N. Lund, G. G. C. Palumbo, J. Paul, J.-P. Roques, H. Schnopper, V. Schönfelder, R. Sunyaev, B. Teegarden, P. Ubertini, G. Vedrenne, and A. J. Dean, “The INTEGRAL mission,” Astron. Astrophys. 411, L1-L6 (2003).
[CrossRef]

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D. Vigneau and D. W. Robinson, “Large coded aperture mask for spaceflight hard x-ray images,” Proc. SPIE 4851, 1326-1335 (2003).
[CrossRef]

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R. Vanderspek, J. Villaseñor, J. Doty, J. G. Jernigan, A. Levine, G. Monnelly, and G. R. Ricker, “GRB observations with the HETE soft x-ray cameras,” Astron. Astrophys. 138, 565-566(1999).

Watt, M. P.

A. P. Willmore, D. Bertram, M. P. Watt, G. K. Skinner, T. J. Ponman, M. J. Church, J. R. H. Herring, and C. J. Eyles, “Image correction in a coded mask x-ray telescope,” Mon. Not. R. Astron. Soc. 258, 621-628 (1992).

Wiersma, G.

R. Jager, W. A. Mels, A. C. Brinkman, M. Y. Galama, H. Goulooze, J. Heise, P. Lowes, J. M. Muller, A. Naber, A. Rook, R. Schuurhof, J. J. Schuurmans, and G. Wiersma, “The wide field cameras onboard the BeppoSAX x-ray astronomy satellite,” Astron. Astrophys. Suppl. Ser. 125, 557-572 (1997).

Wild, W. J.

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The same symbols are used indiscriminately here for the observed and expected numbers of events as the one is the best estimate of the other.

This illustrates an approximation in the approach used here. The shadow of a source at trial positions away from the peak cannot be totally independent of that expected for a source at the peak position . The process described here is equivalent to correlation with a mean-subtracted form of the expected count distribution, which must produce a function with a mean of zero. Thus, a positive peak implies a negative mean level elsewhere. The effect goes inversely with the number of resolution elements in the field of view and becomes negligible for sufficiently large N.

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Note that the Ci are counts per pixel where CS and CB were totals for all the pixels of a particular category.

Sometimes S/(σS-1) is used for a better approximation but the differences are small in practical cases.

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Figures (5)

Fig. 1
Fig. 1

Part of a mask in which the opaque elements are supported by a grid with bar width g and pitch p, leaving holes of width m. The plot beneath shows the response of a square detector pixel of side d as it is moved across the mask shadow along the line shown. The widths of the transition regions are indicated in the case where g < d < m .

Fig. 2
Fig. 2

Top: nomogram for determining the optimum mask open fraction f opt given the parameters b and s, which are, respectively, the sky background and the strength of the observed source, relative to the intrinsic detector background. f opt can be read from the horizontal scale at the intersection of lines of constant s (continuous lines, logarithmically spaced) and of constant b (dashed lines, also logarithmically spaced). The arrows illustrate the effect of a finite detector resolution (see Subsection 6C). Bottom: signal-to-noise ratio when using f = f opt relative to that with f = 1 2 .

Fig. 3
Fig. 3

Numbers of events needed in the region of the detector plane that are exposed to the source ( C S ) and that shadowed by the mask ( C B ) needed to achieve a given level of detection significance. The dashed and continuous curves are calculated using the Cash statistic [Eq. (21)] which correctly handles Poisson statistics and are for f = 0.4 and f = 0.6 . respectively. Dotted curves are based on δ χ 2 [Eq. (22)] and so use the approximation of Gaussian statistics. The curves for each family are at levels of 17.3, 26.3, 37.4, and 50.4 (left to right), which for both the Cash statistic and χ 2 correspond to 4 σ , 5 σ , 6 σ , and 7 σ , respectively, for 1 degree of freedom (appropriate if the source position is known).

Fig. 4
Fig. 4

Loss in sensitivity if the detector spatial resolution is not perfect. The solid curve (a) is for a simple mask with square elements having open fraction f = 0.5 and a detector with square pixels [Eq. (24)]. The dotted curve (b) shows the corresponding curve with f = 0.4 . The dashed curve (c) is for a mask in which 50% of the elements are open but in which a supporting grid like that in Fig. 1 reduces the transparency to f = 0.4 . If the detector resolution is good (low d / p ) the grid provides coding and so the curve approaches (b). If the detector resolution is poor, the grid simply attenuates the flux and reduces the sensitivity (assumptions 3–5, 8, and 9).

Fig. 5
Fig. 5

Variation of relative values of signal-to-noise ratio, angular resolution, and source location accuracy with mask element size for a fixed detector element size. Assumptions 3, 5, 8, and 9 are made; in particular, noise is assumed to be background dominated ( B f , t S ).

Equations (46)

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C S = f A ( S + B f , t ) ,
C B = ( 1 f ) A B f , t .
S ^ = C S f A C B ( 1 f ) A ,
σ S 2 = C S ( f A ) 2 + C B ( 1 f ) 2 A 2 ,
= S + B f , t f A + B f , t ( 1 f ) A .
S ^ σ S = S f ( 1 f ) A ( 1 f ) S + B f , t
( assumptions   4 9 ) .
S ^ σ S = S f ( 1 f ) A B f , t
( assumptions   3 9 ) ,
S ^ σ S = ( S / 2 ) A [ ( S / 2 ) + B 1 / 2 ] A = ( S σ S ) ref
( assumptions   1 , 4 9 ) ,
S ^ σ S = S 2 A B 1 / 2
( assumptions   1 , 3 9 ) .
C S = f A ( f S + B f , t ) ,
C B = ( 1 f ) A ( f S + B f , t ) .
σ I 2 = 1 A ( 1 f ) ( S + B f , t f ) ,
S ^ σ I = S f ( 1 f ) A f S + B f , t
( assumptions   4 6 , 8 , 9 ) ,
f opt ( b , s ) = 1 1 + 1 + b 1 + s
( assumptions   4 9 ) .
g ( b , s ) 2 = 1 + 4 ( f opt ( b , s ) 1 2 ) 2 .
C S = f A ( t 1 S + B f , t ) ,
C B = ( 1 f ) A ( t 0 S + B f , t ) .
S ^ σ S = S ( t 1 t 0 ) f ( 1 f ) A [ ( 1 f ) t 1 + f t 0 ] S + B f , t
( assumptions   5 9 ) ,
S ^ σ I = S ( t 1 t 0 ) f ( 1 f ) A [ ( 1 f ) t 0 + f t 1 ] S + B f , t
( assumptions   5 , 6 , 8 , 9 ) .
s = t 1 s + t 0 b , b = t 0 s + t 1 b
δ C 2 = C S ln ( C S f ) + C B ln ( C B 1 f ) ( C S + C B ) ln ( C S + C B )
( assumptions   4 , 6 , 8 , 9 ) .
δ χ 2 = ( C S f ( C S + C B ) ) 2 f ( C S + C B ) + ( C B ( 1 f ) ( C S + C B ) ) 2 ( 1 f ) ( C S + C B ) = ( C S f ( C S + C B ) ) 2 f ( 1 f ) ( C S + C B )
( assumptions   4 6 , 8 , 9 ) .
( S σ S ) = Δ ( S σ S ) ref = Δ S 2 A B f , t ,
Δ 2 4 = 1 n i P i 2 - ( 1 n i P i ) 2
( assumptions   3 , 5 , 8 , 9 ) ,
Δ = ( 1 - d 3 m ) ( t 1 - t 0 ) 4 f ( 1 - f ) if     m d = m d ( 1 - m 3 d ) ( t 1 - t 0 ) 4 f ( 1 - f ) if     m d
( assumptions   3 , 5 , 8 , 9 ) .
C i = a ( B f , t + P i S ) ± C i ,
σ S 2 = 2 Δ w ( 1 C i ) - 1 ,
Δ w 2 4 = ( P i 2 C i ) / ( 1 C i ) - [ ( P i C i ) / ( 1 C i ) ] 2
( assumptions   5 , 7 9 ) .
C 2 = - C i ln ( P i S + B f , t ) ,
δ C 2 = - C i ln ( P i S + B f , t ) + 2 n C i ln ( C i )
( only assumptions   7 9   necessary ) ,
δ θ 2 = ( m / l ) 2 + ( d / l ) 2 ,
δ α = ( σ S S ) k [ ( m / l ) 2 + ( d / l ) ] 1 2 = Δ ( σ S S ) d = 0 k [ ( m / l ) 2 + ( d / l ) ] 1 2 ,

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