Abstract

We present a method based on maximum-likelihood estimation to calculate the threshold of the number of converted photoelectrons according to Poisson and Gaussian statistics. The calculations for detection and false-alarm probabilities are given including a proposal of error probability. An efficient approach of setting threshold according to laser flight time is discussed as the laser power decreases in propagation. Comparison of the fixed threshold and the varied threshold in Poisson and Gaussian statistics is provided. The varied threshold approach tends to keep the error probability at the lowest level. This varied threshold approach is found to have a similar effect on ranging accuracy as a threshold-fixed approach.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. M. Baker, S. S. Duncan, and J. W. Copley, “A low noise, laser-gated imaging system for long-range target identification,” Proc. SPIE 5406, 133–144 (2004).
    [CrossRef]
  2. O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
    [CrossRef]
  3. J. G. Verly and R. L. Delanoy, “Model-based automatic target recognition (ATR) system for forward looking ground based and airborne imaging laser radars (LADAR),” Proc. IEEE 84, 126–163 (1996).
    [CrossRef]
  4. Z. Li, Q. Li, Z. Tian, Y. Wang, J. Sun, W. Du, and Q. Wang. “Processing for laser radar range images,” Chin. Opt. Lett. 2, 210–212 (2004).
  5. Y. Zhang, Y. Zhao, L. Liu, J. He, C. Jin, G. Mi, and X. Sun, “Improvement of range accuracy of range-gating laser radar using the centroid method,” Appl. Opt. 49, 267–271 (2010).
    [CrossRef] [PubMed]
  6. M. Henriksson, “Detection probabilities for photon-counting avalanche photodiodes applied to a laser radar system,” Appl. Opt. 44, 5140–5147 (2005).
    [CrossRef] [PubMed]
  7. J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700(1965).
    [CrossRef]
  8. J. W. Goodman, “Comparative performance of optical-radar detection techniques,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 526–535 (1966).
    [CrossRef]
  9. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed., 2nd ed. (Springer-Verlag, 1984).
  10. B. E. A. Saleh, Photoelectron Statistics: With Applications to Spectroscopy and Optical Communications (Springer, 1977).
  11. J. W. Goodman, Statistical Optics (Wiley, 1985).
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).
  13. R. M. Gagliardi and S. Karp, Optical Communications, Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, 1995).
  14. C. Jin, Y. Zhao, Y. Zhang, L. Liu, and X. Sun, “Scannerless three-dimensional imaging using a pulsed laser and an intensified charge-coupled device with linearly modulated gain,” Appl. Opt. 48, 3823–3829 (2009).
    [CrossRef] [PubMed]
  15. J. Busck and H. Heiselberg, “Gated viewing and high-accuracy three-dimensional laser radar,” Appl. Opt. 43, 4705–4710(2004).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

2005 (1)

2004 (3)

1999 (1)

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

1996 (1)

J. G. Verly and R. L. Delanoy, “Model-based automatic target recognition (ATR) system for forward looking ground based and airborne imaging laser radars (LADAR),” Proc. IEEE 84, 126–163 (1996).
[CrossRef]

1966 (1)

J. W. Goodman, “Comparative performance of optical-radar detection techniques,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 526–535 (1966).
[CrossRef]

1965 (1)

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700(1965).
[CrossRef]

Baker, I. M.

I. M. Baker, S. S. Duncan, and J. W. Copley, “A low noise, laser-gated imaging system for long-range target identification,” Proc. SPIE 5406, 133–144 (2004).
[CrossRef]

Bolander, G.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

Busck, J.

Copley, J. W.

I. M. Baker, S. S. Duncan, and J. W. Copley, “A low noise, laser-gated imaging system for long-range target identification,” Proc. SPIE 5406, 133–144 (2004).
[CrossRef]

Delanoy, R. L.

J. G. Verly and R. L. Delanoy, “Model-based automatic target recognition (ATR) system for forward looking ground based and airborne imaging laser radars (LADAR),” Proc. IEEE 84, 126–163 (1996).
[CrossRef]

Du, W.

Duncan, S. S.

I. M. Baker, S. S. Duncan, and J. W. Copley, “A low noise, laser-gated imaging system for long-range target identification,” Proc. SPIE 5406, 133–144 (2004).
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi and S. Karp, Optical Communications, Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, 1995).

Goodman, J. W.

J. W. Goodman, “Comparative performance of optical-radar detection techniques,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 526–535 (1966).
[CrossRef]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700(1965).
[CrossRef]

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed., 2nd ed. (Springer-Verlag, 1984).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Groenwall, C. A.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

He, J.

Heiselberg, H.

Henriksson, M.

Jin, C.

Karp, S.

R. M. Gagliardi and S. Karp, Optical Communications, Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, 1995).

Letalick, D.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

Li, Q.

Li, Z.

Liu, L.

Mi, G.

Olsson, H.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics: With Applications to Spectroscopy and Optical Communications (Springer, 1977).

Steinvall, O. K.

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

Sun, J.

Sun, X.

Tian, Z.

Verly, J. G.

J. G. Verly and R. L. Delanoy, “Model-based automatic target recognition (ATR) system for forward looking ground based and airborne imaging laser radars (LADAR),” Proc. IEEE 84, 126–163 (1996).
[CrossRef]

Wang, Q.

Wang, Y.

Zhang, Y.

Zhao, Y.

Appl. Opt. (4)

Chin. Opt. Lett. (1)

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

J. W. Goodman, “Comparative performance of optical-radar detection techniques,” IEEE Trans. Aerosp. Electron. Syst. AES-2, 526–535 (1966).
[CrossRef]

Proc. IEEE (2)

J. G. Verly and R. L. Delanoy, “Model-based automatic target recognition (ATR) system for forward looking ground based and airborne imaging laser radars (LADAR),” Proc. IEEE 84, 126–163 (1996).
[CrossRef]

J. W. Goodman, “Some effects of target-induced scintillation on optical radar performance,” Proc. IEEE 53, 1688–1700(1965).
[CrossRef]

Proc. SPIE (2)

I. M. Baker, S. S. Duncan, and J. W. Copley, “A low noise, laser-gated imaging system for long-range target identification,” Proc. SPIE 5406, 133–144 (2004).
[CrossRef]

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, “Gated viewing for target detection and target recognition,” Proc. SPIE 3707, 432–448 (1999).
[CrossRef]

Other (5)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J.C.Dainty, ed., 2nd ed. (Springer-Verlag, 1984).

B. E. A. Saleh, Photoelectron Statistics: With Applications to Spectroscopy and Optical Communications (Springer, 1977).

J. W. Goodman, Statistical Optics (Wiley, 1985).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1965).

R. M. Gagliardi and S. Karp, Optical Communications, Wiley Series in Telecommunications and Signal Processing (Wiley-Interscience, 1995).

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 (14)

Fig. 1
Fig. 1

Equivalent direct detection receiver model.

Fig. 2
Fig. 2

Detection probability. FT, fixed threshold; VT, varied threshold.

Fig. 3
Fig. 3

False-alarm probability.

Fig. 4
Fig. 4

Error probability.

Fig. 5
Fig. 5

Detection probability for fixed gain of Gaussian statistics.

Fig. 6
Fig. 6

False-alarm probability for fixed gain of Gaussian statistics.

Fig. 7
Fig. 7

Error probability for fixed gain of Gaussian statistics.

Fig. 8
Fig. 8

Detection probability for varied gain of Gaussian statistics.

Fig. 9
Fig. 9

False-alarm probability for varied gain of Gaussian statistics.

Fig. 10
Fig. 10

Error probability for varied gain of Gaussian statistics.

Fig. 11
Fig. 11

Absolute error for fixed gain.

Fig. 12
Fig. 12

Relative error for fixed gain.

Fig. 13
Fig. 13

Absolute error for varied gain.

Fig. 14
Fig. 14

Relative error for varied gain.

Equations (28)

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

p ( k ) = Γ ( k + M ) Γ ( k + 1 ) Γ ( M ) ( 1 + k ¯ M ) M ( 1 + M k ¯ ) k ,
p ( k ) = ( k ¯ s + k ¯ b ) k k ! e ( k ¯ s + k ¯ b ) ,
V ( t ) = i g i h ( t t i ) ,
P ( V ) = n = 0 P ( n ) g n ( V ) .
p ( V ) = 1 2 π σ exp { ( V μ ) 2 2 σ 2 } , μ = V ( t ) ¯ = ( g e τ R L ) k ¯ ( t τ , t ) , σ 2 = ( g e τ R L ) 2 F k ¯ ( t τ , t ) ,
p ( V ) = 1 2 π σ exp { ( V μ ) 2 2 σ 2 } , V ¯ = μ = g ¯ e τ R L ( k ¯ s + k ¯ b + k ¯ d ) , σ 2 = σ V 2 + σ V th 2 = ( g ¯ e τ R L ) 2 F k ¯ + 4 K T τ R L = ( g ¯ e τ ) 2 F ( k ¯ s + k ¯ b + k ¯ d ) + σ th 2 .
P f = k > k T k = P [ k | H ( 0 ) ] + 1 2 P [ k T | H ( 0 ) ] .
P d = k > k T k = P [ k | H ( 1 ) ] + 1 2 P [ k T | H ( 1 ) ] .
p ( k | H ( j ) ) = ( k ¯ s ( j ) + k ¯ b ) k k ! e ( k ¯ s ( j ) + k ¯ b ) .
φ ( j ) = ln ( p ( k | H ( j ) ) ) = ln ( ( k ¯ s ( j ) + k ¯ b ) k k ! e ( k ¯ s ( j ) + k ¯ b ) ) .
k T = k ¯ s ln ( 1 + k ¯ s k ¯ b ) ,
p ( V ) = 1 2 π σ exp [ ( V μ ) 2 2 σ 2 ] .
φ = ln [ p ( V ) ] = ln { 1 2 π σ exp [ ( V μ ) 2 2 σ 2 ] } = [ ln 1 2 π σ ( V μ ) 2 2 σ 2 ] .
ln 1 2 π σ 0 2 ( V μ 0 ) 2 2 σ 0 2 = ln 1 2 π σ 1 2 ( V μ 1 ) 2 2 σ 1 2 , μ 0 = μ ( 0 ) = g ¯ e τ R L ( k ¯ b + k ¯ d ) , μ 1 = μ ( 1 ) = g ¯ e τ R L ( k ¯ s + k ¯ b + k ¯ d ) , σ 0 2 = σ i 0 2 ( 0 ) = ( g ¯ e τ R L ) 2 F ( k ¯ b + k ¯ d ) + σ th 2 , σ 1 2 = σ i 1 2 ( 1 ) = ( g ¯ e τ R L ) 2 F ( k ¯ s + k ¯ b + k ¯ b ) + σ th 2 .
a = σ 1 2 σ 0 2 , b = 2 ( σ 0 2 μ 1 σ 1 2 μ 0 ) , c = σ 1 2 μ 0 2 σ 0 2 μ 1 2 + σ 1 2 σ 0 2 ln σ 0 2 σ 1 2 .
V T = b ± b 2 4 a c 2 a .
V T = b b 2 4 a c 2 a
V T = b + b 2 4 a c 2 a .
PE = 1 2 ( P f + P dr ) = 1 2 [ P f + ( 1 P d ) ] ,
P ( t ) = P 0 2 π τ p exp ( t 2 / 2 τ p 2 ) ,
t = τ p [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 .
d t = d τ p { [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 1 [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 } + d P 0 { τ p P 0 [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 } .
Δ t = [ ( t τ p ) 2 ( Δ τ p ) 2 + ( t P 0 ) 2 ( Δ P 0 ) 2 ] 1 / 2 .
Δ t = { ( [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 1 [ 2 ln ( P 0 2 π τ p P t ) ] 1 / 2 ) 2 ( Δ τ p ) 2 + τ p 2 P 0 2 [ 2 ln ( P 0 2 π τ p P t ) ] ( Δ P 0 ) 2 } 1 / 2 .
Δ t t = { 1 t 2 [ ( t τ p ) 2 ( Δ τ p ) 2 + ( t P 0 ) 2 ( Δ P 0 ) 2 ] } 1 / 2 .
Δ t t = [ ( 1 1 2 ln P 0 2 π τ p P t ) 2 ( Δ τ p τ p ) 2 + ( 1 2 ln P 0 2 π τ p P t ) 2 ( Δ P 0 P 0 ) 2 ] 1 / 2 .
g ¯ e τ p R L ( k ¯ s + k ¯ b + k ¯ d ) ,
( ( g ¯ e τ ) 2 F ( k ¯ s + k ¯ b + k ¯ d ) + σ th 2 ) 1 2 .

Metrics