Abstract

An optimal algorithm for detecting a target using a ladar system employing Geiger-mode avalanche photodiodes (GAPDs) is presented. The algorithm applies to any scenario where a ranging direct detection ladar is used to determine the presence of a target against a sky background within a specified range window. A complete statistical model of the detectionprocess for GAPDs is presented, including GAPDs that are inactive for a fixed period of time each time they fire. The model is used to develop a constant false alarm rate detection algorithm that minimizes acquisition time. Numerical performance predictions, simulation results, and experimental results are presented.

© 2008 Optical Society of America

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References

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  1. R. M. Marino and W. R. Davis, Jr., "Jigsaw: a foliage-penetrating 3D imaging laser radar system," Lincoln Lab. J. 15, 23-36 (2005).
  2. M. A. Albota, R. M. Heinrichs, D. G. Kocher, D. G. Fouche, B. E. Player, M. E. O'Brien, B. F. Aull, J. J. Zayhowski, J. Mooney, B. C. Willard, and R. R. Carlson, "Three-dimensional imaging laser radar with a photon-counting avalanche photodiode array and microchip laser," Appl. Opt. 41, 7671-7678 (2002).
    [CrossRef]
  3. S. Verghese, D. M. Cohen, E. A. Dauler, J. P. Donnelly, E. K. Duerr, S. H. Groves, P. I. Hopman, K. E. Jensen, Z.-L. Liau, L. J. Mahoney, K. A. McIntosh, D. C. Oakley, and G. M. Smith, "Geiger-mode avalanche photodiodes for photon-counting communications," in LEOS Summer Topical Meetings (IEEE, 2005), pp. 15-16.
    [CrossRef]
  4. The "SPCM-AQR" single photon counting module from Perkin-Elmer is one example, and it was used in this report's experimental investigation.
  5. D. G. Fouche, "Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors," Appl. Opt. 42, 5388-5398 (2003).
    [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. S. Johnson, P. Gatt, and T. Nichols, "Analysis of Geiger-mode APD laser radars," in Laser Radar Technology and Applications VIII, Proc. SPIE 5086, 359-368 (2003).
    [CrossRef]
  8. P. Gatt, S. Johnson, and T. Nichols, "Dead-time effects on Geiger-mode APD performance," in Laser Radar Technology and Applications XII, Proc. SPIE 6550, 65500I (2007).
    [CrossRef]
  9. A. V. Jelalian, Laser Radar System (Artech House, 1992).
  10. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).
  11. B. J. Klein and J. J. Degnan, "Optical antenna gain 1: transmitting antennas," Appl. Opt. 13, 2134-2141 (1974).
    [CrossRef] [PubMed]
  12. R. W. Engstrom, ed., RCA Electro-Optics Handbook (RCA/Commercial Engineering, 1974).
  13. B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, "Geiger-mode avalanche photodiodes for three-dimensional imaging," Lincoln Lab. J. 13, 335-48 (2002).
  14. J. W. Goodman, "Some effects of target-induced scintillation on optical radar performance," Proc. IEEE 53, 1688-1700 (1965).
    [CrossRef]
  15. D. G. Youmans, "Avalanche photodiode detection statistics for direct detection laser radar," in Laser Radar VII, Proc. SPIE 1633, 41-52 (1992).
    [CrossRef]
  16. G. Roussas, An Introduction to Probability and Statistical Inference (Academic, 2003).
  17. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  18. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 2001), Vol. 1.
  19. M. I. Skolnik, Radar Handbook (McGraw-Hill, 1990).
  20. J. T. Rickard and G. M. Dillard, "Adaptive detection algorithms for multiple target situations," IEEE Trans. Aerosp. Electron. Syst. AES-13, 338-343 (1977).
    [CrossRef]
  21. L. A. Jiang, E. A. Dauler, and J. T. Chang, "Photon-number resolving detector with 10-bits of resolution," Phys. Rev. A 75, 062325 (2007).
    [CrossRef]
  22. L. A. Jiang, D. R. Schue, D. C. Harrison, A. G. Hayes, E. L. Hines, J. M. Richardson, and K. I. Schultz, "Active range of the Optical Systems Test Facility at MIT Lincoln Laboratory," in Laser Radar Technology and Applications XI, Proc. SPIE 6214, 62140Q (2006).
    [CrossRef]
  23. B. Levin, "A representation for multinomial cumulative distribution functions," Ann. Stat. 9, 1123-1126 (1981).
    [CrossRef]

2007 (2)

L. A. Jiang, E. A. Dauler, and J. T. Chang, "Photon-number resolving detector with 10-bits of resolution," Phys. Rev. A 75, 062325 (2007).
[CrossRef]

P. Gatt, S. Johnson, and T. Nichols, "Dead-time effects on Geiger-mode APD performance," in Laser Radar Technology and Applications XII, Proc. SPIE 6550, 65500I (2007).
[CrossRef]

2006 (1)

L. A. Jiang, D. R. Schue, D. C. Harrison, A. G. Hayes, E. L. Hines, J. M. Richardson, and K. I. Schultz, "Active range of the Optical Systems Test Facility at MIT Lincoln Laboratory," in Laser Radar Technology and Applications XI, Proc. SPIE 6214, 62140Q (2006).
[CrossRef]

2005 (2)

R. M. Marino and W. R. Davis, Jr., "Jigsaw: a foliage-penetrating 3D imaging laser radar system," Lincoln Lab. J. 15, 23-36 (2005).

M. Henriksson, "Detection probabilities for photon-counting avalanche photodiodes applied to a laser radar system," Appl. Opt. 44, 5140-5147 (2005).
[CrossRef] [PubMed]

2003 (2)

D. G. Fouche, "Detection and false-alarm probabilities for laser radars that use Geiger-mode detectors," Appl. Opt. 42, 5388-5398 (2003).
[CrossRef] [PubMed]

S. Johnson, P. Gatt, and T. Nichols, "Analysis of Geiger-mode APD laser radars," in Laser Radar Technology and Applications VIII, Proc. SPIE 5086, 359-368 (2003).
[CrossRef]

2002 (2)

1992 (1)

D. G. Youmans, "Avalanche photodiode detection statistics for direct detection laser radar," in Laser Radar VII, Proc. SPIE 1633, 41-52 (1992).
[CrossRef]

1981 (1)

B. Levin, "A representation for multinomial cumulative distribution functions," Ann. Stat. 9, 1123-1126 (1981).
[CrossRef]

1977 (1)

J. T. Rickard and G. M. Dillard, "Adaptive detection algorithms for multiple target situations," IEEE Trans. Aerosp. Electron. Syst. AES-13, 338-343 (1977).
[CrossRef]

1974 (1)

1965 (1)

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

Ann. Stat. (1)

B. Levin, "A representation for multinomial cumulative distribution functions," Ann. Stat. 9, 1123-1126 (1981).
[CrossRef]

Appl. Opt. (4)

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

J. T. Rickard and G. M. Dillard, "Adaptive detection algorithms for multiple target situations," IEEE Trans. Aerosp. Electron. Syst. AES-13, 338-343 (1977).
[CrossRef]

Lincoln Lab. J. (2)

B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, "Geiger-mode avalanche photodiodes for three-dimensional imaging," Lincoln Lab. J. 13, 335-48 (2002).

R. M. Marino and W. R. Davis, Jr., "Jigsaw: a foliage-penetrating 3D imaging laser radar system," Lincoln Lab. J. 15, 23-36 (2005).

Phys. Rev. A (1)

L. A. Jiang, E. A. Dauler, and J. T. Chang, "Photon-number resolving detector with 10-bits of resolution," Phys. Rev. A 75, 062325 (2007).
[CrossRef]

Proc. IEEE (1)

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

Proc. SPIE (4)

D. G. Youmans, "Avalanche photodiode detection statistics for direct detection laser radar," in Laser Radar VII, Proc. SPIE 1633, 41-52 (1992).
[CrossRef]

S. Johnson, P. Gatt, and T. Nichols, "Analysis of Geiger-mode APD laser radars," in Laser Radar Technology and Applications VIII, Proc. SPIE 5086, 359-368 (2003).
[CrossRef]

P. Gatt, S. Johnson, and T. Nichols, "Dead-time effects on Geiger-mode APD performance," in Laser Radar Technology and Applications XII, Proc. SPIE 6550, 65500I (2007).
[CrossRef]

L. A. Jiang, D. R. Schue, D. C. Harrison, A. G. Hayes, E. L. Hines, J. M. Richardson, and K. I. Schultz, "Active range of the Optical Systems Test Facility at MIT Lincoln Laboratory," in Laser Radar Technology and Applications XI, Proc. SPIE 6214, 62140Q (2006).
[CrossRef]

Other (9)

S. Verghese, D. M. Cohen, E. A. Dauler, J. P. Donnelly, E. K. Duerr, S. H. Groves, P. I. Hopman, K. E. Jensen, Z.-L. Liau, L. J. Mahoney, K. A. McIntosh, D. C. Oakley, and G. M. Smith, "Geiger-mode avalanche photodiodes for photon-counting communications," in LEOS Summer Topical Meetings (IEEE, 2005), pp. 15-16.
[CrossRef]

The "SPCM-AQR" single photon counting module from Perkin-Elmer is one example, and it was used in this report's experimental investigation.

A. V. Jelalian, Laser Radar System (Artech House, 1992).

G. R. Osche, Optical Detection Theory for Laser Applications (Wiley, 2002).

R. W. Engstrom, ed., RCA Electro-Optics Handbook (RCA/Commercial Engineering, 1974).

G. Roussas, An Introduction to Probability and Statistical Inference (Academic, 2003).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 2001), Vol. 1.

M. I. Skolnik, Radar Handbook (McGraw-Hill, 1990).

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Figures (11)

Fig. 1
Fig. 1

Expected number of counts per pulse in each range bin, for ideal linear mode detector, rearm incapable Geiger mode detector, and rearm capable Geiger mode detector, assuming parameters given in Subsection 3D. Note that the vertical axes are truncated to highlight effects in nontarget bins.

Fig. 2
Fig. 2

Expected number of counts per pulse in each range bin for GAPD detectors and PRF varied from 10 to 100   kHz , assuming other parameters given in Subsection 3D. Note that the vertical axes are truncated to highlight effects in nontarget bins.

Fig. 3
Fig. 3

ROC curve illustrations: (a) a family of ROC curves is computed by varying the number of pulses U until one of them is above the specified point ( P F A ,beam , P D ,beam ) . (b) The randomized decision rule extends a discrete ROC to a continuous one via linear interpolation, allowing P D to be computed for P F A = 0.03 in the example shown.

Fig. 4
Fig. 4

Strategy for grouping finely quantized time-of-flight returns into coarse range bins that include the entire target. Appropriate selection of the bin edges maximizes the signal peak.

Fig. 5
Fig. 5

Predicted acquisition performance versus range for “dim target” scenario: (a) expected signal returns; (b) expected background returns and dark noise; (c) acquisition time for three detector types; (d) total signal returns required to acquire for three detector types.

Fig. 6
Fig. 6

Predicted acquisition performance versus range for “strong target” scenario: (a) expected signal returns; (b) expected background returns and dark noise; (c) acquisition time for three detector types; (d) total signal returns required to acquire for three detector types.

Fig. 7
Fig. 7

Predicted acquisition performance versus range for “strong target” scenario at 30 km range, where the PRF is varied from 5 to 100   kHz . (a) Expected signal returns–pulse versus PRF, and (b) acquisition times versus PRF for three detector types.

Fig. 8
Fig. 8

Monte Carlo numerical simulation results of CFAR detection algorithm for “weak target” scenario with both GAPD types. The histograms are shown in black, while the detection thresholds are shown in gray.

Fig. 9
Fig. 9

Monte Carlo numerical simulation results of CFAR detection algorithm for “strong target” scenario with both GAPD types. The histograms are shown in black, while the detection thresholds are shown in gray.

Fig. 10
Fig. 10

Illustration of the Optical Systems Test Facility Active Range, where the experiment was performed. Only 13.6   W of average power was used in the experiment.

Fig. 11
Fig. 11

Experimental results, showing threshold test for three different combinations of target and background noise strength. The histograms are shown in black, while the detection thresholds are shown in gray.

Equations (29)

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P R = I T ( x , z ) × λ h c × σ 4 π R 2 × A rcv × T atm 2 × T trans × T rcv × η ,
d σ = 4 ρ target d A cos 2 Θ ,
I T ( x , z ) = 2 P laser π ω 2 ( z ) e 2 x 2 / ω 2 ( z ) ,
ω 2 ( z ) = ω 0 2 [ 1 + ( λ z π ω 0 2 ) 2 ] ,
ω 0 = 2 π λ ϑ ,
P Sun = I Sun ( λ ) × λ h c × β × f ( ϕ ) × σ 4 π R 2 × A rcv × T atm × T atm × T rcv × η ,
Pr { m   noise   primary   PEs } = n m exp ( n ) m ! ,
Pr { g   signal   primary   PEs } = Γ ( g + M ) g ! Γ ( M ) ( 1 + M S ) g × ( 1 + S M ) M ,
Pr { w   total   primary   electrons } = g = 0 w [ n g exp ( n ) g ! × Γ ( w g + M ) ( w g ) ! Γ ( M ) ( 1 + M S ) g w ( 1 + S M ) M ] .  
a j Pr { at   least   1   primary   electron   in   bin   j } , = 1 Pr { 0   primary   electrons   in   bin   j } , = 1 exp ( n ) .
a j = 1 Pr { 0   primary   electrons   in   bin   j } , = 1 exp ( n ) ( 1 + S M ) M , = 1 exp [ n M log ( 1 + S M ) ] , 1 exp ( n S e f f ) .
θ j = i = 1 j 1 ( 1 a i ) × a j .
Pr { X = x } = U ! i = 1 B x i ! ( U i = 1 B x i ) ! i = 1 B θ i x i ( 1 i = 1 B θ i ) U i = 1 B x j .
Pr { X j = x j } = U ! x j ! ( U x j ) ! θ j x j ( 1 θ j ) U x j .
θ j = i = 1 j 1 ( 1 a i ) × a j ,     j r ,
= i = j r j 1 ( 1 a i ) × a j ,   j > r .
Pr { X = x } = Pr { X 1 , r + 1 = x 1 , r + 1 } × j = r + 2 B Pr { X j = x j | X 1 , j 1 = x 1 , j 1 } ,
Pr { X = x } = Pr { X 1 , r + 1 = x 1 , r + 1 } × j = r + 2 B Pr { X j = x j | X j r , j 1 = x j r , j 1 } .
Pr { X 1 , r + 1 = x 1 , r + 1 } = U ! i = 1 r + 1 x i ! ( U i = 1 r + 1 x i ) ! × i = 1 r + 1 θ i x i ( 1 i = 1 r + 1 θ i ) U i = 1 r + 1 x j .
Pr { X j = x j | X j r , j 1 = x j r , j 1 } = V j ! x j ! ( V j x j ) ! ϕ j x j ( 1 ϕ j ) V j x j ,
V j = U i = j r j 1 x i , ϕ j = θ j 1 i = j r j 1 θ i .
  P F A , j ( U , T j ) = 1 k = 0 T j 1 U ! k ! ( U k ) ! θ j , 0 k ( 1 θ j , 0 ) U k ,
P D , j ( U , T j ) = 1 k = 0 T j 1 U ! k ! ( U k ) ! θ j , 1 k ( 1 θ j ,1 ) U k .
P F A ,gate = 1 ( 1 P F A ,spec ) ( number   of   bins ) .
n ^ M L = arg max n n min log Pr { X = z ; n } ,
n ^ M L = arg max n n min { i = 1 B z i log θ i ( n ) + ( U i = 1 B z i ) × log ( 1 i = 1 B θ i ( n ) ) } .
n ^ M L = arg max n n min , S 0 log Pr { X = z ; n , S } .
n ^ M L = arg max n n min , S 0 { i = 1 B z i log θ i ( n , S ) + ( U i = 1 B z i ) log ( 1 i = 1 B θ i ( n , S ) ) } .
Pr { X x } = Pr { X 1 , r + 1 x 1 , r + 1 } × j = r + 2 B Pr { X j x j | X j r , j 1 x j r , j 1 } , = Pr { X 1 , r + 1 x 1, r + 1 } × j = r + 2 B Pr { X j r , j x j r , j } Pr { X j r , j 1 x j r , j 1 } .

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