Abstract

A methodology for analyzing an imaging sensor’s ability to assess target properties is developed. By the application of a Cramér–Rao covariance analysis to a statistical model relating the sensor measurements to the target, a lower bound can be calculated on the accuracy with which any unbiased algorithm can form estimates of target properties. Such calculations are important in understanding how a sensor’s design influences its performance for a given assessment task and in performing feasibility studies or system architecture design studies between sensor designs and sensing modalities. A novel numerical model relating a sensor’s measurements to a target’s three-dimensional geometry is developed in order to overcome difficulties in accurately performing the required numerical computations. The accuracy of the computations is verified against simple test cases that can be solved in closed form. Examples are presented in which the approach is used to investigate the influence of viewing perspective on orientation accuracy limits. These examples are also used to examine the potential accuracy improvement that could be gained by fusing multiperspective data.

© 2003 Optical Society of America

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  1. S. M. Hannon, J. H. Shapiro, “Laser radar target detection with a multipixel joint range-intensity processor,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 162–175 (1988).
    [CrossRef]
  2. S. M. Hannon, J. H. Shapiro, “Active-passive detection of multipixel targets,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 2–23 (1990).
    [CrossRef]
  3. T. J. Green, J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
    [CrossRef]
  4. J. Zhao, S. Ahalt, C. B. Stribling, “3-D orientation vector estimation from satellite imagery,” in Signal Processing, Sensor Fusion, and Target Recognition V, I. Kadar, V. Libby, eds., Proc. SPIE2755, 472–483 (1996).
    [CrossRef]
  5. L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
    [CrossRef]
  6. X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
    [CrossRef]
  7. B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
    [CrossRef]
  8. A. E. Koksal, J. H. Shapiro, W. M. Wells, “Model-based object recognition using laser radar range imagery,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 256–266 (1999).
    [CrossRef]
  9. R. Li, “Model-based target recognition using laser radar,” Opt. Eng. 31, 322–327 (1992).
    [CrossRef]
  10. T. J. Green, J. H. Shapiro, “Detecting objects in three-dimensional laser radar range images,” Opt. Eng. 33, 865–874 (1994).
    [CrossRef]
  11. M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
    [CrossRef]
  12. M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
    [CrossRef] [PubMed]
  13. A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
    [CrossRef]
  14. M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
    [CrossRef]
  15. J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
    [CrossRef]
  16. J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
    [CrossRef]
  17. A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
    [CrossRef]
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    [CrossRef]
  21. A. Srivastava, U. Grenander, “Metrics for target recognition,” in Applications of Artificial Neural Networks in Image Processing III, N. M. Nasrabadi, A. K. Katsaggelos, eds., Proc. SPIE3307, 29–36 (1998).
    [CrossRef]
  22. U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
    [CrossRef]
  23. U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
    [CrossRef]
  24. M. L. Cooper, M. Miller, “Information measures for object recognition accommodating signature variability,” IEEE Trans. Inf. Theory 46, 1896–1907 (2000).
    [CrossRef]
  25. H. Hendriks, “A Cramér–Rao type lower bound for estimators with values in a manifold,” J. Multivar. Analy. 38, 245–261 (1991).
    [CrossRef]
  26. As long as the noise of each measurement and sensor is statistically independent, which is true in a large number of situations, the joint pdf is simply the product of the pdfs corresponding to the individual measurements.
  27. J. V. D. R. Gerwe, P. S. Idell, “Cramer–Rao bound analysis of target characterization accuracy limits for imaging systems,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, A. R. Pirich, P. L. Repak, P. S. Idell, S. R. Czyzak, eds., Proc. SPIE4490, 245–255 (2001).
    [CrossRef]
  28. A. D. Lanterman, M. I. Miller, D. L. Snyder, “Representations of thermodynamic variability in the automated understanding of FLIR scenes,” in Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 26–37 (1996).
    [CrossRef]
  29. The bound given in relation (4) corresponds to the lowest MMSE achievable by an optimal estimator for the specific value of ξ used in the calculations. Another, more global bound can be calculated by averaging Eq. (3) over all values of ξ and weighted by the a prioridistribution p(ξ).This Bayesian bound gives the minimum-mean-square accuracy achievable by any estimator including those that are biased. See pp. 72–73 and 84–85 of Van Trees.18
  30. D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
    [CrossRef]
  31. D. Snyder, C. Helstrom, A. Lanterman, M. Faisal, R. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
    [CrossRef]
  32. D. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  33. R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).
  34. J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).
  35. Such a comparison was performed after the compilation of this paper. The details of the calculations are too long to include here but were similar to those presented in Section 4 and Appendix B. It was found that the numerical CRLB calculations were generally 5%–20% larger than that indicated by the closed-form expression. This difference is too small relative to the potential inaccuracies in the numerical calculations to infer much about how the obscuration effects influence the Fisher information. It does, however, provide an indication that the overall effect is fairly small and that the relative location of target edges dominates the Fisher information.
  36. The described modifications to the rendering algorithm were implemented subsequent to the compilation of this paper. Tests indicate that as a complex 3-D target was rotated, the pixel values in the vicinity of obscuration edges changed smoothly and continuously. A rigorous evaluation of the accuracy of the approach for computing CRLBs has not been performed, but preliminary results are promising.
  37. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

2002

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

2000

U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

M. L. Cooper, M. Miller, “Information measures for object recognition accommodating signature variability,” IEEE Trans. Inf. Theory 46, 1896–1907 (2000).
[CrossRef]

1998

U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
[CrossRef]

X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
[CrossRef]

1997

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
[CrossRef]

1995

M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
[CrossRef]

D. Snyder, C. Helstrom, A. Lanterman, M. Faisal, R. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
[CrossRef]

1994

T. J. Green, J. H. Shapiro, “Detecting objects in three-dimensional laser radar range images,” Opt. Eng. 33, 865–874 (1994).
[CrossRef]

1993

1992

T. J. Green, J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

R. Li, “Model-based target recognition using laser radar,” Opt. Eng. 31, 322–327 (1992).
[CrossRef]

1991

H. Hendriks, “A Cramér–Rao type lower bound for estimators with values in a manifold,” J. Multivar. Analy. 38, 245–261 (1991).
[CrossRef]

1988

E. Weinstein, A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inf. Theory 34, 338–342 (1988).
[CrossRef]

Ahalt, S.

X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
[CrossRef]

J. Zhao, S. Ahalt, C. B. Stribling, “3-D orientation vector estimation from satellite imagery,” in Signal Processing, Sensor Fusion, and Target Recognition V, I. Kadar, V. Libby, eds., Proc. SPIE2755, 472–483 (1996).
[CrossRef]

Angelisanti, D.

D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
[CrossRef]

Baer, C.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Blahut, R. E.

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).

Briscoe, D.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Butts, R.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Chatterjee, J.

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

Chellappa, R.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

Chhan, L. A.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

Clark, D.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Cochran, G.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Cooper, M.

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

Cooper, M. L.

M. L. Cooper, M. Miller, “Information measures for object recognition accommodating signature variability,” IEEE Trans. Inf. Theory 46, 1896–1907 (2000).
[CrossRef]

Crockett, G.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Dai, G.-M.

D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
[CrossRef]

Der, S.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

Du, X.

X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
[CrossRef]

Faisal, M.

Gerwe, J. V. D. R.

J. V. D. R. Gerwe, P. S. Idell, “Cramer–Rao bound analysis of target characterization accuracy limits for imaging systems,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, A. R. Pirich, P. L. Repak, P. S. Idell, S. R. Czyzak, eds., Proc. SPIE4490, 245–255 (2001).
[CrossRef]

Green, T.

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

Green, T. J.

T. J. Green, J. H. Shapiro, “Detecting objects in three-dimensional laser radar range images,” Opt. Eng. 33, 865–874 (1994).
[CrossRef]

T. J. Green, J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

Grenander, U.

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
[CrossRef]

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
[CrossRef]

A. Srivastava, U. Grenander, “Metrics for target recognition,” in Applications of Artificial Neural Networks in Image Processing III, N. M. Nasrabadi, A. K. Katsaggelos, eds., Proc. SPIE3307, 29–36 (1998).
[CrossRef]

M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
[CrossRef]

Hammoud, A. M.

Hannon, S. M.

S. M. Hannon, J. H. Shapiro, “Active-passive detection of multipixel targets,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 2–23 (1990).
[CrossRef]

S. M. Hannon, J. H. Shapiro, “Laser radar target detection with a multipixel joint range-intensity processor,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 162–175 (1988).
[CrossRef]

Hassebrook, L.

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

Helstrom, C.

Hendriks, H.

H. Hendriks, “A Cramér–Rao type lower bound for estimators with values in a manifold,” J. Multivar. Analy. 38, 245–261 (1991).
[CrossRef]

Idell, P. S.

J. V. D. R. Gerwe, P. S. Idell, “Cramer–Rao bound analysis of target characterization accuracy limits for imaging systems,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, A. R. Pirich, P. L. Repak, P. S. Idell, S. R. Czyzak, eds., Proc. SPIE4490, 245–255 (2001).
[CrossRef]

Jensen, G. R.

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

Kay, S.

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Koksal, A. E.

A. E. Koksal, J. H. Shapiro, W. M. Wells, “Model-based object recognition using laser radar range imagery,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 256–266 (1999).
[CrossRef]

Kostakis, J.

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

Kroncke, G.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Lanterman, A.

Lanterman, A. D.

A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
[CrossRef]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “Representations of thermodynamic variability in the automated understanding of FLIR scenes,” in Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 26–37 (1996).
[CrossRef]

Lhamon, M.

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

Li, B.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

Li, R.

R. Li, “Model-based target recognition using laser radar,” Opt. Eng. 31, 322–327 (1992).
[CrossRef]

Miller, M.

M. L. Cooper, M. Miller, “Information measures for object recognition accommodating signature variability,” IEEE Trans. Inf. Theory 46, 1896–1907 (2000).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

Miller, M. I.

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
[CrossRef]

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
[CrossRef]

M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
[CrossRef]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “Representations of thermodynamic variability in the automated understanding of FLIR scenes,” in Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 26–37 (1996).
[CrossRef]

M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
[CrossRef]

Nasrabadi, N. M.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

O’Sullivan, J.

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

O’Sullivan, J. A.

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

Riker, J.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Shapiro, J.

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

Shapiro, J. H.

T. J. Green, J. H. Shapiro, “Detecting objects in three-dimensional laser radar range images,” Opt. Eng. 33, 865–874 (1994).
[CrossRef]

T. J. Green, J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

S. M. Hannon, J. H. Shapiro, “Active-passive detection of multipixel targets,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 2–23 (1990).
[CrossRef]

S. M. Hannon, J. H. Shapiro, “Laser radar target detection with a multipixel joint range-intensity processor,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 162–175 (1988).
[CrossRef]

A. E. Koksal, J. H. Shapiro, W. M. Wells, “Model-based object recognition using laser radar range imagery,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 256–266 (1999).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Smith, W.

D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
[CrossRef]

Snyder, D.

D. Snyder, C. Helstrom, A. Lanterman, M. Faisal, R. White, “Compensation for readout noise in CCD images,” J. Opt. Soc. Am. A 12, 272–283 (1995).
[CrossRef]

D. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
[CrossRef]

Snyder, D. L.

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
[CrossRef]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “Representations of thermodynamic variability in the automated understanding of FLIR scenes,” in Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 26–37 (1996).
[CrossRef]

Srivastava, A.

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
[CrossRef]

M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
[CrossRef]

A. Srivastava, U. Grenander, “Metrics for target recognition,” in Applications of Artificial Neural Networks in Image Processing III, N. M. Nasrabadi, A. K. Katsaggelos, eds., Proc. SPIE3307, 29–36 (1998).
[CrossRef]

M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
[CrossRef]

Stephens, M.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Stribling, B.

X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
[CrossRef]

Stribling, C. B.

J. Zhao, S. Ahalt, C. B. Stribling, “3-D orientation vector estimation from satellite imagery,” in Signal Processing, Sensor Fusion, and Target Recognition V, I. Kadar, V. Libby, eds., Proc. SPIE2755, 472–483 (1996).
[CrossRef]

Suizu, R.

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory: Part 1 (Wiley, New York, 1968).

Wang, L.-C.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

Wang, M.

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

Weinstein, E.

E. Weinstein, A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inf. Theory 34, 338–342 (1988).
[CrossRef]

Weiss, A. J.

E. Weinstein, A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inf. Theory 34, 338–342 (1988).
[CrossRef]

Wells, W. M.

A. E. Koksal, J. H. Shapiro, W. M. Wells, “Model-based object recognition using laser radar range imagery,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 256–266 (1999).
[CrossRef]

White, R.

White, R. L.

Zhao, J.

J. Zhao, S. Ahalt, C. B. Stribling, “3-D orientation vector estimation from satellite imagery,” in Signal Processing, Sensor Fusion, and Target Recognition V, I. Kadar, V. Libby, eds., Proc. SPIE2755, 472–483 (1996).
[CrossRef]

Zheng, Q.

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

IEEE Trans. Image Process.

M. I. Miller, U. Grenander, J. A. O’Sullivan, D. L. Snyder, “Automatic target recognition organized via jump-diffusion algorithms,” IEEE Trans. Image Process. 6, 157–174 (1997).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory

U. Grenander, A. Srivastava, M. I. Miller, “Asymptotic performance analysis on Bayesian target recognition,” IEEE Trans. Inf. Theory 46, 1658–1665 (2000).
[CrossRef]

M. L. Cooper, M. Miller, “Information measures for object recognition accommodating signature variability,” IEEE Trans. Inf. Theory 46, 1896–1907 (2000).
[CrossRef]

E. Weinstein, A. J. Weiss, “A general class of lower bounds in parameter estimation,” IEEE Trans. Inf. Theory 34, 338–342 (1988).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

U. Grenander, M. I. Miller, A. Srivastava, “Hilbert–Schmidt lower bounds for estimators on matrix Lie groups,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 790–801 (1998).
[CrossRef]

IEEE Trans. Signal Process.

M. I. Miller, A. Srivastava, U. Grenander, “Conditional-mean estimation via jump-diffusion processes in multiple target tracking/recognition,” IEEE Trans. Signal Process. 43, 2678–2690 (1995).
[CrossRef]

J. Multivar. Analy.

H. Hendriks, “A Cramér–Rao type lower bound for estimators with values in a manifold,” J. Multivar. Analy. 38, 245–261 (1991).
[CrossRef]

J. Opt. Soc. Am. A

J. Stat. Plann. Infer.

A. Srivastava, U. Grenander, G. R. Jensen, M. I. Miller, “Jump-diffusion Markov processes on orthogonal groups for object pose estimation,” J. Stat. Plann. Infer. 103, 15–37 (2002).
[CrossRef]

Opt. Eng.

R. Li, “Model-based target recognition using laser radar,” Opt. Eng. 31, 322–327 (1992).
[CrossRef]

T. J. Green, J. H. Shapiro, “Detecting objects in three-dimensional laser radar range images,” Opt. Eng. 33, 865–874 (1994).
[CrossRef]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “General Metropolis–Hastings jump diffusions for automatic target recognition in infrared scenes,” Opt. Eng. 36, 1123–1137 (1997).
[CrossRef]

T. J. Green, J. H. Shapiro, “Maximum-likelihood laser radar range profiling with the expectation-maximization algorithm,” Opt. Eng. 31, 2343–2354 (1992).
[CrossRef]

L. Hassebrook, M. Lhamon, M. Wang, J. Chatterjee, “Postprocessing of correlation for orientation estimation,” Opt. Eng. 36, 2710–2718 (1997).
[CrossRef]

X. Du, S. Ahalt, B. Stribling, “Three-dimensional vector estimation for subcomponents of space object imagery,” Opt. Eng. 37, 798–807 (1998).
[CrossRef]

Other

B. Li, Q. Zheng, S. Der, R. Chellappa, N. M. Nasrabadi, L. A. Chhan, L.-C. Wang, “Experimental evaluation of neural, statistical and model-based approaches to FLIR ATR,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 388–397 (1998).
[CrossRef]

A. E. Koksal, J. H. Shapiro, W. M. Wells, “Model-based object recognition using laser radar range imagery,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 256–266 (1999).
[CrossRef]

J. Zhao, S. Ahalt, C. B. Stribling, “3-D orientation vector estimation from satellite imagery,” in Signal Processing, Sensor Fusion, and Target Recognition V, I. Kadar, V. Libby, eds., Proc. SPIE2755, 472–483 (1996).
[CrossRef]

S. M. Hannon, J. H. Shapiro, “Laser radar target detection with a multipixel joint range-intensity processor,” in Laser Radar III, R. J. Becherer, ed., Proc. SPIE999, 162–175 (1988).
[CrossRef]

S. M. Hannon, J. H. Shapiro, “Active-passive detection of multipixel targets,” in Laser Radar V, R. J. Becherer, ed., Proc. SPIE1222, 2–23 (1990).
[CrossRef]

M. Cooper, U. Grenander, M. I. Miller, A. Srivastava, “Accommodating geometric and thermodynamic variability for forward-looking infrared sensors,” in Algorithms for Synthetic Aperture Radar Imagery IV, E. G. Zelnio, ed., Proc. SPIE3070, 162–172 (1997).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral active-passive sensor fusion for ground-based target orientation estimation,” in Automatic Target Recognition VIII, F. A. Sadjadi, ed., Proc. SPIE3371, 500–507 (1998).
[CrossRef]

J. Kostakis, M. Cooper, T. Green, M. Miller, J. O’Sullivan, J. Shapiro, D. Snyder, “Multispectral sensor fusion for ground-based target orientation estimation: FLIR, LADAR, HRR,” in Automatic Target Recognition IX, F. A. Sadjadi, ed., Proc. SPIE3718, 14–24 (1999).
[CrossRef]

H. L. Van Trees, Detection, Estimation, and Modulation Theory: Part 1 (Wiley, New York, 1968).

S. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1993).

A. Srivastava, U. Grenander, “Metrics for target recognition,” in Applications of Artificial Neural Networks in Image Processing III, N. M. Nasrabadi, A. K. Katsaggelos, eds., Proc. SPIE3307, 29–36 (1998).
[CrossRef]

As long as the noise of each measurement and sensor is statistically independent, which is true in a large number of situations, the joint pdf is simply the product of the pdfs corresponding to the individual measurements.

J. V. D. R. Gerwe, P. S. Idell, “Cramer–Rao bound analysis of target characterization accuracy limits for imaging systems,” in Multifrequency Electronic/Photonic Devices and Systems for Dual-Use Applications, A. R. Pirich, P. L. Repak, P. S. Idell, S. R. Czyzak, eds., Proc. SPIE4490, 245–255 (2001).
[CrossRef]

A. D. Lanterman, M. I. Miller, D. L. Snyder, “Representations of thermodynamic variability in the automated understanding of FLIR scenes,” in Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 26–37 (1996).
[CrossRef]

The bound given in relation (4) corresponds to the lowest MMSE achievable by an optimal estimator for the specific value of ξ used in the calculations. Another, more global bound can be calculated by averaging Eq. (3) over all values of ξ and weighted by the a prioridistribution p(ξ).This Bayesian bound gives the minimum-mean-square accuracy achievable by any estimator including those that are biased. See pp. 72–73 and 84–85 of Van Trees.18

D. Snyder, D. Angelisanti, W. Smith, G.-M. Dai, “Correction for nonuniform flat-field response in focal-plane arrays,” in Digital Image Recovery and Synthesis III, P. S. Idell, T. J. Schulz, eds., Proc. SPIE2827, 60–67 (1996).
[CrossRef]

R. E. Blahut, Principles and Practice of Information Theory (Addison-Wesley, Reading, Mass., 1987).

J. Riker, R. Butts, G. Crockett, C. Baer, G. Kroncke, G. Cochran, D. Briscoe, M. Stephens, R. Suizu, D. Clark, “Tracking for anti-satellite (TASAT) systems simulation. Vol. II—Physical models,” (Research and Development Associates Logicon, 105 East Vermijo, Suite 450, Colorado Springs, CO, 1989).

Such a comparison was performed after the compilation of this paper. The details of the calculations are too long to include here but were similar to those presented in Section 4 and Appendix B. It was found that the numerical CRLB calculations were generally 5%–20% larger than that indicated by the closed-form expression. This difference is too small relative to the potential inaccuracies in the numerical calculations to infer much about how the obscuration effects influence the Fisher information. It does, however, provide an indication that the overall effect is fairly small and that the relative location of target edges dominates the Fisher information.

The described modifications to the rendering algorithm were implemented subsequent to the compilation of this paper. Tests indicate that as a complex 3-D target was rotated, the pixel values in the vicinity of obscuration edges changed smoothly and continuously. A rigorous evaluation of the accuracy of the approach for computing CRLBs has not been performed, but preliminary results are promising.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

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Figures (18)

Fig. 1
Fig. 1

3-D plot of the centers of the surface elements used to populate a target’s exterior. For purpose of illustration, the number of surface elements in the model shown here has been significantly reduced below that of a typical model.

Fig. 2
Fig. 2

Example of a depth map and corresponding radiometric rendering using the surface element target/sensor modeling approach.

Fig. 3
Fig. 3

(a) FPA image of the flat rectangular plate at a 20° out-of-plane rotation from the nominal position (face perpendicular to the LOS), (b) derivative of the image measurement with respect to out-of-plane rotational motion δ θ y about the vertical axis, (c) relative magnitude of the Fisher information at each pixel.

Fig. 4
Fig. 4

Plots of the pixel values corresponding to the image slices indicated in Fig. 3. Calculations performed by using the exact closed-form expression and by using the numerical model are plotted with solid curves and a series of plus signs, respectively. Except for (c), the difference in the results is too small to be visible. The dashed vertical lines indicate the position of the edge of the target.

Fig. 5
Fig. 5

(a) FPA image of the flat rectangular plate at its nominal position (face perpendicular to the LOS), (b) derivative of the image measurement with respect to in-plane rotational motion δ θ z about the LOS, (c) relative magnitude of the Fisher information at each pixel.

Fig. 6
Fig. 6

Plots of the pixel values corresponding to the image slices indicated in Fig. 5. Calculations performed by using the exact closed-form expression and by using the numerical model are plotted with solid curves and a series of plus signs, respectively. As seen from the plots, the results agree almost perfectly.

Fig. 7
Fig. 7

Dependency of the target orientation CRLB calculation on the size of the difference used in the derivative approximations and on the method of rendering (analytic or grid based).

Fig. 8
Fig. 8

Dependency of the FOV offset CRLB calculation on the size of the difference used in the derivative approximations and on the method of rendering.

Fig. 9
Fig. 9

Dependency of the PSF width CRLB calculation on the size of the difference used in the derivative approximations and on the method of rendering.

Fig. 10
Fig. 10

Imaging geometry of sensors A, B, and C with respect to the target and diagram of the yaw, pitch, and roll body-centered orientation axes { δ ϕ Y ,   δ ϕ P ,   δ ϕ R } . A series of orientations is considered as the HST is rotated around sensor B’s LOS with its pitch axis coaligned with the Y ˆ axis.

Fig. 11
Fig. 11

Mean image measurements obtained by sensors A, B, and C at a series of orientations of the HST as it is rotated about sensor B’s LOS or, equivalently, the HST’s P ˆ axis (pitch). To assist interpretation of the imaging geometry, a corresponding set of depth maps is also displayed. The x ˆ-, y ˆ-, and z ˆ-axis directions (see Fig. 10) are overlayed for convenience. The three labels P Y , P P , and P R denote particular viewing perspectives referred to later in Section 6.

Fig. 12
Fig. 12

CRLB on estimating the yaw aspect ( δ ϕ Y ) of the HST’s orientation with the use of image measurements from sensors A, B, and C, both individually and in combination. The bound is calculated for a sequence of orientations as the HST is rotated 360° about sensor B’s LOS or, equivalently, the HST’s P ˆ axis. As expected, the bounds vary as a function of the target/sensor geometry. Combining measurements from all three sensors always reduces the lower bound below that obtained for any single sensor.

Fig. 13
Fig. 13

CRLB on estimating the pitch aspect ( δ ϕ P ) of the HST’s orientation.

Fig. 14
Fig. 14

CRLB on estimating the roll aspect ( δ ϕ R ) of the HST’s orientation.

Fig. 15
Fig. 15

These images portray the relative strength of the Fisher information provided by each pixel of a FPA regarding the yaw, pitch, and roll orientation aspects of the HST. The top, middle, and bottom rows correspond to the viewing perspectives seen by sensors A, B, and C, respectively, with the HST positioned in its θ = 180 ° pose. These perspectives are referred to as P Y , P P , and P R and are identical to those seen in the bottom row of Fig. 11.  

Fig. 16
Fig. 16

Pristine image projection of the upper right corner of a flat plate as it is rotated by δθ, causing the edges to shift approximately 1 FPA pixel to the right. The fine lines correspond to the pristine grid, with the FPA element edges highlighted by the thicker grid lines. The illustration also contains an overlay of the PSF spot and a highlighted row of FPA elements.

Fig. 17
Fig. 17

Change in the simulated intensity distribution along a row of FPA elements as the edge of a target structure in Fig. 16 shifts approximately 1 FPA element to the right. The numbers on the ordinate axis denote FPA pixel edges, and the small steps indicate edges of the convolved image grid elements.

Fig. 18
Fig. 18

Dependency of the target orientation CRLB calculation on the size of the difference used in the derivative approximations when using TASAT to render the images.

Tables (3)

Tables Icon

Table 1 Sampling Density and Purpose of the Series of Grids Used by the Analytic Rendering Algorithm

Tables Icon

Table 2 Target/Sensor Parameters for Comparing the Numerical and Closed Expression Results for Imaging of a Flat Plate

Tables Icon

Table 3 Target/Sensor Parameters Used in Testing the Sensitivity of the CRLB Calculations to the Magnitude of the Perturbations Used in Approximating Derivatives by Finite Difference

Equations (59)

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

( ξ ^ k - ξ k ) 2 CRLB { ξ ^ k | ξ } [ F D - 1 ( ξ ) - 1 ] kk ,
[ F D ( ξ ) ] kl = - 2 ln   p ( d | ξ ) ξ k ξ l =   ln   p ( d | ξ ) ξ k   ln   p ( d | ξ ) ξ l .
[ F P ( ξ ) ] kl = - 2 ln   p ( ξ ) ξ k ξ l =   ln   p ( ξ ) ξ k   ln   p ( ξ ) ξ l .
( ξ ^ k - ξ k ) 2 [ { F D ( ξ ) + F P ( ξ ) } - 1 ] kk .
p ( ξ ) = [ ( 2 π ) N / 2 | Λ ξ | 1 / 2 ] - 1 × exp [ - 1 2 ( ξ - ξ ) T Λ ξ - 1 ( ξ - ξ ) ] ,
d ( x ¯ ) = Poisson { g ( x ¯ ,   ξ ) } + n σ x ¯ .
p ( d | ξ ) = x ¯ F exp [ - g ( x ¯ | ξ ) ] g ( x ¯ | ξ ) d ( x ¯ ) d ( x ¯ ) ! .
[ F D ( ξ ) ] kl = x ¯ F g ( x ¯ | ξ ) ξ k g ( x ¯ | ξ ) ξ l g ( x ¯ | ξ ) .
d ( x ¯ ) = g ( x ¯ ,   ξ ) + n σ x ¯ ,
p ( d | ξ ) = x ¯ F 1 2 π σ x ¯ 2 exp [ - | d ( x ¯ ) - g ( x ¯ | ξ ) | 2 / ( 2 σ x ¯ 2 ) ] .
[ F D ( ξ ) ] kl = x ¯ F g ( x ¯ | ξ ) ξ k g ( x ¯ | ξ ) ξ l σ x ¯ 2 .
d ( x ¯ ) = Poisson { g ( x ¯ ,   ξ ) + σ x ¯ 2 } - σ x ¯ 2 ,
[ F D ( ξ ) ] kl = x ¯ F g ( x ¯ | ξ ) ξ k g ( x ¯ | ξ ) ξ l g ( x ¯ | ξ ) + σ x ¯ 2 .
p ( d + σ ¯ 2 | ξ )
= q x ¯ F q × exp [ - g q ( x ¯ | ξ ) + σ x ¯ , q 2 ] [ g q ( x ¯ | ξ ) + σ x ¯ , q 2 ] d q ( x ¯ ) + σ x ¯ , q 2 [ d q ( x ¯ ) + σ x ¯ , q 2 ] ! ,
 
[ F D ( ξ ) ] kl q x ¯ F q g q ( x ¯ | ξ ) ξ k g q ( x ¯ | ξ ) ξ l g q ( x ¯ | ξ ) + σ x ¯ , q 2 ,
image intensity distribution ( x ,   y ) | ( surf . elem . α )
= K × ( radiant intensity ) α
× psf ( x - P x ( z ¯ α ) ,   y - P y ( z ¯ α ) ) .
[ g ( x ,   y | θ y ) / θ y ] 2 g ( x ,   y | θ y ) + σ x , y 2
g ( x | θ z ) θ z g ( x | θ z + δ θ z / 2 ) - g ( x | θ z - δ θ z / 2 ) δ θ z .
plate intensity ( x ,   y | θ y ) rect x W o ( θ y ) ,
W o ( θ y ) = W R cos   θ y ,
rect ( x ) = 0 , | x |   > 1 / 2 1 , | x |   1 / 2 .
psf ( x ,   y ) = tri x W p tri y W p ,
tri ( x ) = 1 + x , - 1 x 0 1 - x , 0 x 1 0 , otherwise .
g ( x ,   y ) ( ξ , η ) A x , y d ξ d η d ξ d η
× psf ( ξ - ξ ,   η - η ) plate intensity ( ξ ,   η ) .
g o ( x ,   y ) d ξ   rect ( x - ξ ) d ξ × tri ξ - ξ W p rect ξ W o ( θ y ) d η   rect ( y - η ) d η tri η - η W p .
g o ( x ,   y ) d ξ   rect ( x - ξ ) d ξ × tri ξ - ξ W p rect ξ W o ( θ y ) .
g o ( x ,   y ) 0 , W p + 0.5 χ I 1 ( χ ) , W p - 0.5 χ W p + 0.5 I 2 ( χ ) , 0.5 χ W p - 0.5 I 3 ( χ ) , - 0.5 χ 0.5 I 4 ( χ ) , - W p + 0.5 χ - 0.5 I 5 ( χ ) , - W p - 0.5 χ - W p + 0.5 I 6 ( χ ) , χ - W p - 0.5 χ = χ 0 | x | - W o ( θ y ) / 2 ,
I 1 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( χ ) ,
I 2 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( W p - 0.5 ) + T 2 ( W p - 0.5 ) - T 2 ( χ ) ,
I 3 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( W p - 0.5 ) + T 2 ( W p - 0.5 ) - T 2 ( 0.5 ) + T 3 ( 0.5 ) - T 3 ( χ ) ,
I 4 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( W p - 0.5 ) + T 2 ( W p - 0.5 ) - T 2 ( 0.5 ) + T 3 ( 0.5 ) - T 3 ( - 0.5 ) + T 4 ( - 0.5 ) - T 4 ( χ ) ,
I 5 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( W p - 0.5 ) + T 2 ( W p - 0.5 ) - T 2 ( 0.5 ) + T 3 ( 0.5 ) - T 3 ( - 0.5 ) + T 4 ( - 0.5 ) - T 4 ( - W p + 0.5 ) + T 5 ( - W p + 0.5 ) - T 5 ( χ ) ,
I 6 ( χ ) = T 1 ( W p + 0.5 ) - T 1 ( W p - 0.5 ) + T 2 ( W p - 0.5 ) - T 2 ( 0.5 ) + T 3 ( 0.5 ) - T 3 ( - 0.5 ) + T 4 ( - 0.5 ) - T 4 ( - W p + 0.5 ) + T 5 ( - W p + 0.5 ) - T 5 ( - W p - 0.5 ) ,
 
T 1 ( ζ ) = ( W p + 0.5 ) 2 ζ - ( W p + 0.5 ) ζ 2 + ζ 3 / 3 2 W p 2 ,
T 2 ( ζ ) = W p ζ - 0.5 ζ 2 W p 2 ,
T 3 ( ζ ) = ζ W p - ζ 4 W p 2 - ζ 3 3 W p 2 ,
T 4 ( ζ ) = W p ζ + 0.5 ζ 2 W p 2 ,
T 5 ( ζ ) = ( W p + 0.5 ) 2 ζ + ( W p + 0.5 ) ζ 2 + ζ 3 / 3 2 W p 2 .
g o ( x ,   y ) W o 0 , W p + 0.5 χ D 1 ( χ ) , W p - 0.5 χ W p + 0.5 D 2 ( χ ) , 0.5 χ W p - 0.5 D 3 ( χ ) , - 0.5 χ 0.5 D 2 ( χ ) , - W p + 0.5 χ - 0.5 D 1 ( χ ) , - W p - 0.5 χ - W p + 0.5 0 , χ - W p - 0.5 χ = χ o | x | - W o ( θ y ) / 2 ,
D 1 ( χ ) = ( W p + 0.5 - χ ) 2 4 W p 2 ,
D 2 ( χ ) = W p - χ 2 W p 2 ,
D 3 ( χ ) = 2 W p - 0.5 - 2 χ 2 4 W p 2 .
g o ( x ,   y | θ y ) θ y = g o ( x ,   y | θ y ) W o ( θ y ) W o ( θ y ) θ y = - g o ( x ,   y | θ y ) W o ( θ y )   W R sin ( θ y ) .
x y = cos   θ z - sin   θ z sin   θ z cos   θ z   x o y o .
x = ± W R / 2 - y   sin   θ z cos   θ z
plate intensity ( x ,   y | θ z ) = rect | x | W i ( x ,   y | θ z )
for θ z :   H   sin   θ z < W R ,
W i ( x ,   y | θ z ) = W R - sign ( x ) 2 y   sin   θ z cos   θ z
sign ( x ) = - 1 x < 0 1 x > 0 ,
g i ( x ,   y | θ z ) d ξ   rect ( x - ξ ) d ξ tri ξ - ξ W p × d η   rect ( y - η ) d η tri η - η W p × rect | ξ | W i ( x ,   y | θ z ) ,
g i ( x ,   y | θ z ) d ξ   rect ( x - ξ ) d ξ × tri ξ - ξ W p rect | ξ | W i ( x ,   y | θ z ) .
ξ = χ i | x | - W i ( x ,   y ,   θ z ) / 2 .
g i ( x ,   y | θ z ) θ z = g i ( x ,   y | θ z ) W i ( x ,   y | θ z ) W i ( x ,   y | θ z ) θ z = g i ( x ,   y | θ z ) W i ( x ,   y | θ z ) × - sign ( x ) y + [ W R / 2 - sign ( x ) y   cos   θ z ] sin   θ z cos 2   θ z .

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