Abstract

Expressions for the minimum number of photons required to measure the position of a light pattern on a noncoherent detector array with a desired accuracy are obtained from the Cramér–Rao lower bounds. The dependence of the Cramér–Rao bound on the shape, on the statistical properties of the intensity, on spatial sampling by the detector array, and on the array area is examined quantitatively. Examples of linear and nonlinear algorithms for measuring the pattern position with accuracies approaching the Cramér–Rao bound are presented. The analysis is carried out for photon statistics modeled as conditional Poisson processes.

© 1983 Optical Society of America

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  1. J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.
  2. C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).
  3. M. Elbaum, P. Diament, M. King, and B. Edelson, "Maximum angular accuracy of pulsed laser radar in photocounting limit," Appl. Opt. 16, 1982–1992 (1977).
  4. D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.
  5. M. Elbaum and M. Greenebaum, "Angular apertures for angular tracking," Appl. Opt. 16, 2438–2440 (1977).
  6. J. F. Walkup and J. W. Goodman, "Limitations of fringe parameter estimation at low light levels," J. Opt. Soc. Am. 63, 399–407 (1973).
  7. M. Elbaum and P. Diament, "Estimation of image centroid size, and orientation with laser radar," Appl. Opt. 16, 2433–2437 (1977).
  8. M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).
  9. P. M. Solomon and T. A. Glavich, "Image signal processing in sub-pixel accuracy star trackers," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).
  10. L. Mandel, "Photoelectric counting measurements as a test for the existence of photons," J. Opt. Soc. Am. 67, 1101–1104 (1977).
  11. L. Mandel, "Fluctuations of photon beams: distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233–243 (1959).
  12. H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).
  13. J. Nowakowski and M. Elbaum, "Fundamental limits in estimating light pattern position," Quarterly Tech. Rep. T-3/481- 3-00, Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February 1982).
  14. J. Nowakowski, "Barankin bound and Cramér–Rao bound: an interpretation," Rep. No. N-3/481-3-00 (Riverside Research Institute, New York, December 1981).
  15. J. Nowakowski, "Barankin type bounds and estimators," Rep. No. N-2/481-3-00 (Riverside Research Institute, New York, December 1981).

1980 (2)

M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

P. M. Solomon and T. A. Glavich, "Image signal processing in sub-pixel accuracy star trackers," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

1978 (1)

J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.

1977 (4)

1973 (1)

1970 (1)

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

1959 (1)

L. Mandel, "Fluctuations of photon beams: distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233–243 (1959).

Chi, C.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

Diament, P.

Edelson, B.

Elbaum, M.

J. Nowakowski and M. Elbaum, "Fundamental limits in estimating light pattern position," Quarterly Tech. Rep. T-3/481- 3-00, Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February 1982).

M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

M. Elbaum and P. Diament, "Estimation of image centroid size, and orientation with laser radar," Appl. Opt. 16, 2433–2437 (1977).

M. Elbaum and M. Greenebaum, "Angular apertures for angular tracking," Appl. Opt. 16, 2438–2440 (1977).

M. Elbaum, P. Diament, M. King, and B. Edelson, "Maximum angular accuracy of pulsed laser radar in photocounting limit," Appl. Opt. 16, 1982–1992 (1977).

Glavich, T. A.

P. M. Solomon and T. A. Glavich, "Image signal processing in sub-pixel accuracy star trackers," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Goodman, J. W.

Greenebaum, M.

King, M.

MacEachin, J.

M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

Mandel, L.

L. Mandel, "Photoelectric counting measurements as a test for the existence of photons," J. Opt. Soc. Am. 67, 1101–1104 (1977).

L. Mandel, "Fluctuations of photon beams: distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233–243 (1959).

McIntyre, C.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

Nowakowski, J.

J. Nowakowski and M. Elbaum, "Fundamental limits in estimating light pattern position," Quarterly Tech. Rep. T-3/481- 3-00, Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February 1982).

J. Nowakowski, "Barankin bound and Cramér–Rao bound: an interpretation," Rep. No. N-3/481-3-00 (Riverside Research Institute, New York, December 1981).

J. Nowakowski, "Barankin type bounds and estimators," Rep. No. N-2/481-3-00 (Riverside Research Institute, New York, December 1981).

Orenstein, N.

M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

Peters, W. N.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

Shapiro, J. H.

J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.

Solomon, P. M.

P. M. Solomon and T. A. Glavich, "Image signal processing in sub-pixel accuracy star trackers," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Walkup, J. F.

Wischnia, H. F.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

Appl. Opt. (3)

J. Opt. Soc. Am. (2)

Proc. Phys. Soc. London (1)

L. Mandel, "Fluctuations of photon beams: distribution of the photo-electrons," Proc. Phys. Soc. London 74, 233–243 (1959).

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

M. Elbaum, N. Orenstein, and J. MacEachin, "Algorithms for estimating image position," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 98–104 (1980).

P. M. Solomon and T. A. Glavich, "Image signal processing in sub-pixel accuracy star trackers," Proc. Soc. Photo-Opt. Instrum. Eng. 252, 64–74 (1980).

Other (7)

D. L. Snyder, Random Point Processes (Wiley, New York, 1975), p. 450.

J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), pp. 183–193.

C. McIntyre, W. N. Peters, C. Chi, and H. F. Wischnia, "Optical components and technology in laser space communications systems," Proc. IEEE 58, 1491–1503 (1970).

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, Princeton, N.J., 1946).

J. Nowakowski and M. Elbaum, "Fundamental limits in estimating light pattern position," Quarterly Tech. Rep. T-3/481- 3-00, Vol. II of Studies of Electro-Optical Sensors for Laser Applications (Riverside Research Institute, New York, February 1982).

J. Nowakowski, "Barankin bound and Cramér–Rao bound: an interpretation," Rep. No. N-3/481-3-00 (Riverside Research Institute, New York, December 1981).

J. Nowakowski, "Barankin type bounds and estimators," Rep. No. N-2/481-3-00 (Riverside Research Institute, New York, December 1981).

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