Abstract

A Cramer–Rao lower bound on the range accuracy obtainable by a Flash light detection and ranging (LADAR) system receiving a return from a single surface in the instantaneous field of view of each detector is developed and verified with experimental data. The bound is compared to the performance of a new algorithm and that of a matched filter receiver by using both simulated and measured LADAR data. The simulated data are used to show that the estimator is nearly unbiased and efficient for systems that match the negative paraboloid model used in its derivation. It is found that the achievable range accuracy for the LADAR system and for the target geometry used to collect the measured data is of the order of 2.5 in. while the bound predicts a range accuracy limit of approximately 0.6 in.

© 2006 Optical Society of America

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  1. M. J. Halmos, "Eyesafe 3D FLASH LADAR for targets under obscuration," Proc. SPIE 5086, 70-83 (2003).
    [CrossRef]
  2. J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
    [CrossRef]
  3. A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
    [CrossRef]
  4. A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
    [CrossRef]
  5. J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
    [CrossRef]
  6. J. W. Goodman, Statistical Optics (Wiley, 1985).
  7. H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, 1968).
  8. B. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am 62, 55-59 (1972).

2005

J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
[CrossRef]

2003

M. J. Halmos, "Eyesafe 3D FLASH LADAR for targets under obscuration," Proc. SPIE 5086, 70-83 (2003).
[CrossRef]

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

1972

B. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am 62, 55-59 (1972).

Austin, W.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Bridges, R.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Bybee-Driscoll, S.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

Fetzer, G.

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Freeman, J.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

Gelbart, A.

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Goodman, J. W.

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

Halmos, M. J.

M. J. Halmos, "Eyesafe 3D FLASH LADAR for targets under obscuration," Proc. SPIE 5086, 70-83 (2003).
[CrossRef]

Hanna, K.

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

Khoury, J.

J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
[CrossRef]

Lorenzo, J.

J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
[CrossRef]

McCarley, K.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Moran, S. E.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Murray, J. T.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Richardson, B. H.

B. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am 62, 55-59 (1972).

Roddier, N.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Seales, T.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, 1968).

Vercillo, R.

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

Wasson, D.

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

Weber, C.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Woods, C. L.

J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
[CrossRef]

Wright, J.

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

Zhao, W.

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

J.

B. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am 62, 55-59 (1972).

Proc. SPIE

M. J. Halmos, "Eyesafe 3D FLASH LADAR for targets under obscuration," Proc. SPIE 5086, 70-83 (2003).
[CrossRef]

J. T. Murray, S. E. Moran, N. Roddier, R. Vercillo, R. Bridges, and W. Austin, "Advanced 3D polarimetric flash ladar imaging through foliage," Proc. SPIE 5086, 84-95 (2003).
[CrossRef]

A. Gelbart, S. Bybee-Driscoll, J. Freeman, G. Fetzer, D. Wasson, K. Hanna, and W. Zhao, "Signal processing, image registration, and visualization of FLASH lidar data," Proc. SPIE 5086, 197-208 (2003).
[CrossRef]

A. Gelbart, C. Weber, S. Bybee-Driscoll, J. Freeman, G. Fetzer, T. Seales, K. McCarley, and J. Wright, "FLASH lidar data collections in terrestrial and ocean environments," Proc. SPIE 5086, 27-38 (2003).
[CrossRef]

J. Khoury, C. L. Woods, and J. Lorenzo, "Resolution limits for time-of-flight laser radar," Proc. SPIE 5816, 270-276 (2005).
[CrossRef]

Other

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

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, 1968).

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Figures (11)

Fig. 1
Fig. 1

3D description of the resolution target used in the LADAR imaging experiments.

Fig. 2
Fig. 2

Four images gathered by the system in the region of the bar pattern at 4.0, 8.0, 12.0, and 6.0   ns (A, B, C, D, respectively) from the beginning of the range gate while the laser pulse propagates through the target.

Fig. 3
Fig. 3

Plot of typical return pulses as a function of time from the beginning of the range gate for the three surfaces present in the target.

Fig. 4
Fig. 4

Cramer–Rao bound for the range estimates as a function of range itself. It is assumed that the gain is 900 and the bias is 1650 photons. The times on the x axis are with respect to the beginning of the range gate. This shows that a pulse returning from the middle of the range gate should give a more accurate range estimate than one returning from the edge or even outside the gate.

Fig. 5
Fig. 5

Flow chart showing the steps of the reconstruction algorithm. LB stands for the computed lower bound on range accuracy.

Fig. 6
Fig. 6

Plot of the average range error as a function of the number of trials used in the average. This demonstrates that the algorithm appears to have a small range bias depending on the position of the surface in the range gate. For this set of experiments the target is on the outer edge of the range gate (dashed curve) with a bias of 1∕5 in. and in the middle of the range gate (solid curve) with no bias.

Fig. 7
Fig. 7

Plot showing the standard deviation of the RGB estimator versus the square root of the Cramer–Rao bound as a function of where the surface is in the range gate. This demonstrates that the algorithm is nearly efficient if the measured return pulse obeys the negative paraboloid single surface model under which the algorithms was derived.

Fig. 8
Fig. 8

Plot of the recovered range profile of the target as a function of position along its surface for the three different range estimation algorithms as well as a rendition of the target surface range profile. This plot demonstrates that the RGB estimator and the peak estimator can track the surface features of the target, but the matched filter receiver cannot.

Fig. 9
Fig. 9

Plot of a typical truncated return pulse with a maximum-likelihood fit to it obtained from the RGB estimator, demonstrating the ability of the algorithms to deal with the pulse truncation problem.

Fig. 10
Fig. 10

Plot of the recovered range profile of the target as a function of position along its surface for the three different range estimation algorithms as well as a rendition of the target surface range profile in a flat area of the target. It demonstrates that the matched filter receiver has a low variance, while the RGB estimator has a slightly higher variance. The peak estimator appears to have the highest variance due to its sensitivity to noise in the return pulse.

Fig. 11
Fig. 11

Gray-scale images of the estimated range to the target of the three algorithms and a rendition of the true target range, depicting the relative performance of the algorithms in different regions of the target. (A) The image of the true target; (B) the image obtained from the RGB estimator; (C) the image obtained from the matched filter receiver; and (D) the estimate obtained from the peak estimator.

Tables (1)

Tables Icon

Table 1 Root-Mean-Squared Error (RMSE) of the Estimated Range as Compared to the True Target for the Three Different Estimators a

Equations (53)

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1.876   ns
11 .82   m
7   ns
5   ns
1 .55   μm
10   cm
100   μm
100%
I ( t k , x n , y m ) = G ( x n , y m ) ( 1 [ R ( x n , y m ) t k c ] 2 ( c p w ) 2 ) × rect ( [ R ( x n , y m ) t k c ] 2 c p w ) + B ( x n , y m ) + q ( t k , x n , y m ) ,
p w
t k
1.876   ns
230   m
t 1 = 1.533 × 10 6 s
t k = t 1 + ( k 1 ) × 1.876 × 10 9
( x n , y m )
1 ( R t k c ) 2
t k
D ( t k , x n , y m )
D ( t k , x n , y m )
I ( t k , x n , y m )
1.55   μm
3   μm
D ( t k )
P [ ( D ( t k ) = d ( t k ) k ϵ ( 1 , 2 ,   , K ) ]
P [ D ( t k ) = d ( t k ) k ϵ ( 1 , 2 ,   , K ) ]
= k = 1 K I ( t k ) d ( t k ) e I ( t k ) d ( t k ) !
J a b = E [ δ 2 L δ a δ b ] ,
p w
3 × 3
L ( R , G , B ) = k = 1 K d ( t k ) ln [ I ( t k ) ] I ( t k ) .
3 × 3
δ L ( R , G , B ) δ R k = 1 K 2 [ d ( t k ) I ( t k ) 1 ] G ( R t k c ) ( c p w ) 2 × rect [ ( R t k c ) 2 c p w ] .
δ L ( R , G , B ) δ G = k = 1 K [ d ( t k ) I ( t k ) 1 ] ( 1 ( R t k c ) 2 ( c p w ) 2 ) × rect ( R t k c 2 c p w ) .
δ L ( R , G , B ) δ B = k = 1 K [ d ( t k ) I ( t k ) 1 ] .
E [ δ 2 L ( R , G , B ) δ 2 R ] k = 1 K 4 G 2 [ ( R t k c ) ( c p w ) 2 ] 2 rect [ ( R t k c ) 2 c p w ] I ( t k ) ,
E [ δ 2 L ( R , G , B ) δ R δ G ] k = 1 K 2 G ( R t k c ) ( c p w ) 2 [ 1 ( R t k c ) ( c p w ) 2 ] rect [ ( R t k c ) 2 c p w ] I ( t k ) ,
E [ δ 2 L ( R , G , B ) δ R δ B ] k = 1 K 2 G ( R t k c ) ( c p w ) 2   rect [ ( R t k c ) 2 c p w ] I ( t k ) ,
E [ δ 2 L ( R , G , B ) δ 2 G ] = k = 1 K [ 1 ( R t k c ) ( c p w ) 2 ] 2   rect [ ( R t k c ) 2 c p w ] I ( t k ) ,
E [ δ 2 L ( R , G , B ) δ G δ B ] = k = 1 K [ 1 ( R t k c ) ( c p w ) 2 ] rect [ ( R t k c ) 2 c p w ] I ( t k ) ,
E [ δ 2 L ( R , G , B ) δ 2 B ] = k = 1 K I ( t k ) I 2 ( t k ) .
[ E [ L ( R , G , B ) δ 2 R ] E [ L ( R , G , B ) δ R δ G ] E [ L ( R , G , B ) δ R δ B ] E [ L ( R , G , B ) δ G δ R ] E [ L ( R , G , B ) δ 2 G ] E [ L ( R , G , B ) δ G δ B ] E [ L ( R , G , B ) δ B δ R ] E [ L ( R , G , B ) δ R δ B ] E [ L ( R , G , B ) δ 2 B ] ] .
L B ( B ) = k = 1 K { 1 rect [ ( R t k c ) 2 c p w ] } [ d ( t k ) ln ( B ) B ] .
B = k = 1 K { 1 rect [ ( R t k c ) 2 c p w ] } d ( t k ) k = 1 K { 1 rect [ ( R t k c ) 2 c p w ] } .
0 = k = 1 K { d ( t k ) [ 1 ( R t k c ) 2 ( c p w ) 2 ] I ( t k ) ( 1 ( R t k c ) 2 ( c p w ) 2 ) } × rect [ ( R t k c ) 2 c p w ] .
G new = G old k = 1 K d ( t k ) [ 1 ( R t k c ) 2 ( c p w ) 2 ] rect [ ( R t k c ) 2 c p w ] I old ( t k ) k = 1 K [ 1 ( R t k c ) 2 ( c p w ) 2 ] rect [ ( R t k c ) 2 c p w ] .
G old
G new
G old
I old
G new G old / G old < γ
γ = 0.01
6.0   ns

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