Abstract

We examine the performance of amplitude-based height-estimation techniques for use with airborne synthetic aperture ladar (SAL) sensors in generating three-dimensional reconstructions of ground targets. Such techniques lend themselves to implementation more readily than phase-based techniques and are also more tolerant to phase instabilities that might be associated with SAL systems. For pairwise amplitude-comparison monopulse processing, we present analyses of the expected height sensitivity and bias of SAL systems in terms of the system parameters. We verify this analysis with simulations, and we also provide an overview of other SAL phenomena that affect height-estimation accuracy. We then propose an array-based joint-processing approach that can be applied instead of pairwise monopulse processing. We show that the joint-processing approach represents the maximum-likelihood estimator for obtaining the target height, and we demonstrate that the proposed approach significantly reduces bias-induced errors.

© 2005 Optical Society of America

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References

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  1. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27, 1983–1985 (2002).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. T. J. Karr, “Resolution of synthetic-aperture imaging through turbulence,” J. Opt. Soc. Am. A 20, 1067–1083 (2003).
    [CrossRef]
  4. J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).
  5. W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms (Artech House, Boston, Mass., 1995).
  6. S. M. Sherman, Monopulse Principles and Techniques (Artech House, Dedham, Mass., 1984).
  7. I. Kanter, “The probability density function of the monopulse ratio for N looks at a combination of constant and Rayleigh targets,” IEEE Trans. Inf. Theory IT-23, 643–648 (1977).
    [CrossRef]

2003 (1)

2002 (1)

1994 (1)

1977 (1)

I. Kanter, “The probability density function of the monopulse ratio for N looks at a combination of constant and Rayleigh targets,” IEEE Trans. Inf. Theory IT-23, 643–648 (1977).
[CrossRef]

Bashkansky, M.

Carrara, W. G.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms (Artech House, Boston, Mass., 1995).

Colella, B. D.

Curlander, J. C.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

Funk, E.

Goodman, R. S.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms (Artech House, Boston, Mass., 1995).

Green, T. J.

Kanter, I.

I. Kanter, “The probability density function of the monopulse ratio for N looks at a combination of constant and Rayleigh targets,” IEEE Trans. Inf. Theory IT-23, 643–648 (1977).
[CrossRef]

Karr, T. J.

Lucke, R. L.

Majewski, R. M.

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms (Artech House, Boston, Mass., 1995).

Marcus, S.

McDonough, R. N.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

Reintjes, J.

Rickard, L. J.

Sherman, S. M.

S. M. Sherman, Monopulse Principles and Techniques (Artech House, Dedham, Mass., 1984).

Appl. Opt. (1)

IEEE Trans. Inf. Theory (1)

I. Kanter, “The probability density function of the monopulse ratio for N looks at a combination of constant and Rayleigh targets,” IEEE Trans. Inf. Theory IT-23, 643–648 (1977).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (3)

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

W. G. Carrara, R. S. Goodman, R. M. Majewski, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms (Artech House, Boston, Mass., 1995).

S. M. Sherman, Monopulse Principles and Techniques (Artech House, Dedham, Mass., 1984).

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Figures (9)

Fig. 1
Fig. 1

Schematic comparison of microwave and optical systems for phase-comparison (top) and amplitude-comparison (bottom) monopulse. (Phased-array antennas are assumed for the rf cases.)

Fig. 2
Fig. 2

Diagram of the 1-D SAL system model. The focal-plane detector array allows for the use of amplitude-based height-estimation techniques.

Fig. 3
Fig. 3

Sum and difference patterns (left) and the monopulse discriminant (right) as functions of the normalized elevation angle for various beam patterns. The solid, dashed, and dotted curves correspond to jinc, Gaussian, and sinc beam patterns.

Fig. 4
Fig. 4

Sum and difference patterns (left) and the monopulse discriminant (right) as functions of the normalized elevation angle for the point-detector approximation (dashed curves) and for finite-sized detectors (solid curves). The solid and dashed curves are nearly indistinguishable, indicating that the point-detector approximation is excellent for the assumed detector size of 0.6 λ / D in elevation by 1.2 λ / D in azimuth.

Fig. 5
Fig. 5

Results from a Monte Carlo simulation of monopulse performance for a nonfluctuating point target in the presence of noise. Curves for the sample mean and standard deviation of monopulse ratio μ ˆ computed in the presence of noise are shown as functions of the true (noiseless) monopulse ratio Δ/Σ for CNR values of 1, 3, 10, 30, and 100 (linear units).

Fig. 6
Fig. 6

Results from a Monte Carlo simulation of monopulse performance for a target with Gaussian statistics in the presence of noise. Curves for the sample mean and standard deviation of monopulse ratio μ ˆ computed in the presence of noise are shown as functions of the true (noiseless) monopulse ratio Δ/Σ for CNR values of 1, 3, 10, 30, and 100 (linear units). Four looks are assumed.

Fig. 7
Fig. 7

The angular extent in elevation of the part of the target surface contained in a single resolution cell is limited by the system resolution.

Fig. 8
Fig. 8

Results from a Monte Carlo simulation of the performance of the ML elevation-angle estimator for a target with Gaussian statistics. Curves for the sample mean and standard deviation of the estimated normalized elevation angle θ ˆ 0 ( λ / D ) are shown as indicated as functions of the true elevation angle θ 0 ( λ / D ) for CNRs of 1, 3, and 10 (linear units). The curves marked with circles, diamonds, and triangles correspond to cases of one, four, and nine looks.

Fig. 9
Fig. 9

Example of a histogram over 10,000 trials of the ML elevation-angle estimator for fixed target location, CNR, and number of looks.

Equations (51)

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S f ( x f ) = - L / 2 L / 2 A ( x p ) exp - j   2 π λ   x p sin θ b exp j   2 π λ   x p sin tan - 1 x f F d x p
- L / 2 L / 2 A ( x p ) exp - j   2 π λ   x p θ b - x f F d x p ,
Σ d ( θ b ) = 0 d S f ( x f ) d x f + - d 0 S f ( x f ) d x f ,
Δ d ( θ b ) = 0 d S f ( x f ) d x f - - d 0 S f ( x f ) d x f .
Σ d ( θ b ) = - d d sinc π L λ   θ b - x f F d x f ,
Δ d ( θ b ) = 0 d sinc π L λ   θ b - x f F - sinc π L λ   θ b + x f F d x f ,
μ = Re Δ d Σ d ,
μ = - λ F π L   Si π L λ   θ b + d F + Si π L λ   θ b - d F - 2   Si π L θ b λ λ F π L   Si π L λ   θ b - d F - Si π L λ   θ b + d F ,
Si ( x ) = 0 x sinc ( t ) d t .
k θ = θ b   Δ d Σ d θ b = 0
= Σ d   Δ d θ b - Δ d   Σ d θ b Σ d 2 θ b = 0 .
k θ = 1 Σ d Δ d θ b θ b = 0 .
Δ d θ b = - F sinc π L λ   θ b + d F + sinc π L λ   θ b - d F - 2   sinc π L λ   θ b .
Si ( x ) x ,
sinc ( x ) 1 - x 2 6 ,
Σ d ( 0 ) = 2 λ F π L   Si π Ld λ F
2 λ F π L   π Ld λ F = 2 d ,
Δ d θ b θ b = 0 = - 2 F   sinc π Ld λ F + 2 F
- 2 F 1 - 1 6   π Ld λ F 2 + 2 F
= 2 F   1 6   π Ld λ F 2 .
k θ = 1 Σ d Δ d θ b θ b = 0
1 6   d F   π L λ 2 .
μ ( θ b ) π 2 6   d / F λ / L θ b λ / L .
μ 0 = Re Δ Σ ,
μ ˆ = Re Δ + n Δ Σ + n Σ ,
E [ μ ˆ | χ ] = [ 1 - exp ( - χ ) ] μ 0 ,
E [ μ ˆ ] = - E [ μ ˆ | χ ] f χ ( χ ) d χ ,
E [ μ ˆ ] = μ 0 0 [ 1 - exp ( - χ ) ] exp ( - χ / CNR ) CNR d χ ,
E [ μ ˆ ] = CNR CNR + 1   μ 0 .
σ μ ˆ = 1 + μ 0 2 ( N looks ) ( 2 CNR ) 1 / 2
1 [ ( N looks ) ( 2 CNR ) ] 1 / 2 ,
σ θ el λ D   σ μ ˆ .
σ z = ρ sin θ inc   σ θ el
ρ sin θ inc   λ D   σ μ ˆ ,
σ z λ D   ρ sin θ inc [ ( N looks ) ( 2 CNR ) ] 1 / 2 ,
g km = A k h ( θ m - θ tgt ) exp ( j ϕ k ) + n km ,
θ ˆ 0 = arg max [ f ( g | θ 0 ,   A ,   ϕ ) ]
= arg max k m f km ( g km | θ 0 ,   A k ,   ϕ k ) ,
f km ( g km | θ 0 ,   A k ,   ϕ k ) = 1 2 π σ n 2 exp 1 2 σ n 2   | g km - A k h ( θ m - θ 0 ) exp ( j ϕ k ) | 2 ,
θ ˆ 0 = arg max k m log [ f km ( g km | θ 0 ,   A k ,   ϕ k ) ] ,
θ ˆ 0 = arg max - k m | g km exp ( - j ϕ k ) - A k h ( θ m - θ 0 ) | 2 .
θ ˆ 0 = arg max - k m [ | g km | 2 - 2   Re [ g km exp ( - j ϕ k ) ] × A k h ( θ m - θ 0 ) + [ A k h ( θ m - θ 0 ) ] 2 ] .
θ ˆ 0 = arg max - k - 2 m   Re [ g km exp ( - j ϕ k ) ] × A k h ( θ m - θ 0 ) + A k 2 m [ h ( θ m - θ 0 ) ] 2 .
m [ h ( θ m - θ 0 ) ] 2 = 1
θ ˆ 0 = arg max k 2 A k   Re exp ( - j ϕ k ) × m g km h ( θ m - θ 0 ) - A k 2 ,
θ ˆ 0 = arg max k 2 A k m g km h ( θ m - θ 0 ) - A k 2 .
A k   2 A k m g km h ( θ m - θ 0 ) - A k 2
= 2 m g km h ( θ m - θ 0 ) - 2 A k ,
A k   ML = m g km h ( θ m - θ 0 ) .
θ ˆ 0 = arg max k m g km h ( θ m - θ 0 ) 2 .
ϕ = 4 π λ v tgt v plat   B x ,

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