Abstract

We study the maximum likelihood estimation of the location of an incoherent object, the light from which is distorted by an optical system and detected by photoelectric detectors with quantum noise. The noise is treated as a set of uncorrelated Gaussian variables with variances proportional to the signal plus background at their corresponding detectors. By the simulation of the noise, a large number of cases are tested and curves for the probability distribution of the error distance are obtained.

© 1974 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. J. Kishner, T. W. Barnard, J. Opt. Soc. Am. 62, 1366A (1972).
  2. C. W. Helstrom, Y. M. Hong, J. Opt. Soc. Am. 63, 480A (1973).
  3. C. W. Helstrom, J. Opt. Soc. Am. 62, 416 (1972).
    [CrossRef]
  4. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin Co., London, 1973), Vol. 2.
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

1973 (1)

C. W. Helstrom, Y. M. Hong, J. Opt. Soc. Am. 63, 480A (1973).

1972 (2)

C. W. Helstrom, J. Opt. Soc. Am. 62, 416 (1972).
[CrossRef]

S. J. Kishner, T. W. Barnard, J. Opt. Soc. Am. 62, 1366A (1972).

Barnard, T. W.

S. J. Kishner, T. W. Barnard, J. Opt. Soc. Am. 62, 1366A (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Helstrom, C. W.

C. W. Helstrom, Y. M. Hong, J. Opt. Soc. Am. 63, 480A (1973).

C. W. Helstrom, J. Opt. Soc. Am. 62, 416 (1972).
[CrossRef]

Hong, Y. M.

C. W. Helstrom, Y. M. Hong, J. Opt. Soc. Am. 63, 480A (1973).

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin Co., London, 1973), Vol. 2.

Kishner, S. J.

S. J. Kishner, T. W. Barnard, J. Opt. Soc. Am. 62, 1366A (1972).

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin Co., London, 1973), Vol. 2.

J. Opt. Soc. Am. (3)

S. J. Kishner, T. W. Barnard, J. Opt. Soc. Am. 62, 1366A (1972).

C. W. Helstrom, Y. M. Hong, J. Opt. Soc. Am. 63, 480A (1973).

C. W. Helstrom, J. Opt. Soc. Am. 62, 416 (1972).
[CrossRef]

Other (2)

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics (Charles Griffin Co., London, 1973), Vol. 2.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

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 (2)

Fig. 1
Fig. 1

The frequency of occurrence distribution of the estimated location based on a thousand simulated samples for several values of the noise factor F. In each graph is indicated the value of P, the probability that the estimation error is less than or equal to a resolution distance R/2.

Fig. 2
Fig. 2

The band-limited image, an arbitrary sample of the simulated noisy image, and the likelihood function whose minimum corresponds to the estimated location.

Equations (20)

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

I k = S ( x k u ) B ( u ) d u + N k , k = 1,2 , .. , M ,
Λ k 2 = S ( x k u ) B ( u ) d u + N B k .
B ( u ) = B 0 b ( u ) ,
s ( x ) = S ( x ) / S ( O ) , i k = F I k , n k = F N k , n B k = F N B k ,
F 1 = B 0 S ( O ) Δ u 2 .
i k = l s ( x k u l ) b ( u l ) + n k , k = 1 , .. , M ,
σ k 2 = F [ l s ( x k u l ) b ( u l ) + n B k ] .
L ( u 0 ) = prob ( { i k } ; u 0 ) .
L ( u 0 ) ~ k ( 1 / σ k ) exp [ 1 2 ( i k i k 0 ) 2 / σ k 2 ] ,
i k 0 = s ( x k u ) b ( u u 0 ) d u + n B k .
L * ( u 0 ) = k ln [ i k 0 ( u 0 ) ] + [ ( 1 / F ) [ i k i k 0 ] 2 / i k 0 .
b ( u l ) = δ u l , u 0 ,
S ( x u ) = β ( λ 2 / 4 π ) [ ( π / 4 R 2 ) sinc ( π | x u | / R ) ] 2
β = η ρ A T / ћ Ω ,
s ( x u ) = sinc 2 ( π | x u | / R )
F 1 = β ( λ 2 / 4 π ) ( π / 4 R 2 ) 2 B 0 = ( π / 64 ) η ρ ( λ Δ u / R 2 ) 2 ( B 0 A T / h Ω ) .
i k 0 = s ( x k u 0 ) + n B ,
L * ( u 0 ) = k ln [ s ( x k u 0 ) + n B ] + ( 1 / F ) [ i k s ( x k u 0 ) n B ] 2 / [ s ( x k u 0 ) + n B ] .
σ k 2 = F [ s ( x k u 0 ) + n B ] .
i k = s ( x k u 0 ) + n k ,

Metrics