Abstract

We present a regularized nonlinear least-squares algorithm for tracking the position and the orientation of a known object by using an active camera and an optical correlator situated between digital preprocessing and postprocessing operations. The numerical minimization required by the regularized least-squares solution is implemented by use of a rapid look-up table method. Performance of the algorithm is evaluated through a Monte Carlo sensitivity analysis that incorporates models for lens blur, image noise, illumination variation, and partial occlusion. This analysis shows robust performance with respect to image noise, partial occlusion of the object, and errors in the camera pan and tilt used to follow the moving object. The limiting factors in the algorithm’s performance are errors in the preprocessing step used to scale and rotate the input video images. These errors should be maintained within 6% and 3°, respectively.

© 1998 Optical Society of America

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References

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  1. S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. SPIE1704, 248–256 (1992).
  2. S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).
  3. D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).
  4. R. Talluri, J. K. Aggarwal, “Mobile robot self-location using model-image feature correspondence,” IEEE Trans. Robotics Automat. 12, 63–77 (1996).
  5. B. Sabata, J. K. Aggarwal, “Estimation of motion from a pair of range images: a review,” Comput. Vis. Graphics Image Process. Image Understand. 54, 309–324 (1991).
  6. H. Shariat, K. E. Price, “Motion estimation with more than two frames,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 417–434 (1990).
  7. O. Faugeras, S. Maybank, “Motion from point matches: multiplicity of solutions,” Int. J. Comput. Vis. 4, 225–246 (1990).
  8. T. J. Hebert, X. Yang, “A sequential algorithm for motion estimation from point correspondences with intermittent occlusions,” in Proceedings of the 1995 IEEE International Conference on Image Processing (ICIP) (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 2, pp. 221–224.
  9. R. Kumar, A. R. Hanson, “Robust methods for estimating pose and a sensitivity analysis,” Comput. Vis. Graphics Image Process. Image Understand. 60, 313–342 (1994).
  10. D. G. Lowe, “Fitting parameterized three-dimensional models to images,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 441–450 (1991).
  11. T. P. Wallace, O. R. Mitchell, “Analysis of three-dimensional movement using Fourier descriptors,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 583–588 (1980).
  12. P. Brou, “Using the Gaussian image to find the orientation of objects,” Int. J. Robotics Res. 3, 89–125 (1984).
  13. K. H. Fielding, J. L. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).
  14. V. Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
  15. B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).
  16. J. Figue, P. Réfrégier, “Angle determination of airplanes by multicorrelation technique with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).
  17. J. B. Bailey, “Stability simulations of a visual tracking algorithm for automated docking of space vehicles,” M.S. thesis (University of New Hampshire, Durham, N.H., December1989).
  18. D. Murray, A. Basu, “Motion tracking with an active camera,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 449–459 (1994).
  19. D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).
  20. T. J. Hebert, R. M. Leahy, “Statistic-based MAP image restoration from Poisson data using Gibbs priors,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
  21. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
  22. D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).
  23. J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications and Control, Vol. 2/e of Signal Processing Series (Prentice-Hall, Englewood Cliffs, N.J., 1995).

1996 (2)

D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).

R. Talluri, J. K. Aggarwal, “Mobile robot self-location using model-image feature correspondence,” IEEE Trans. Robotics Automat. 12, 63–77 (1996).

1994 (3)

R. Kumar, A. R. Hanson, “Robust methods for estimating pose and a sensitivity analysis,” Comput. Vis. Graphics Image Process. Image Understand. 60, 313–342 (1994).

J. Figue, P. Réfrégier, “Angle determination of airplanes by multicorrelation technique with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).

D. Murray, A. Basu, “Motion tracking with an active camera,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 449–459 (1994).

1992 (2)

T. J. Hebert, R. M. Leahy, “Statistic-based MAP image restoration from Poisson data using Gibbs priors,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).

V. Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).

1991 (2)

D. G. Lowe, “Fitting parameterized three-dimensional models to images,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 441–450 (1991).

B. Sabata, J. K. Aggarwal, “Estimation of motion from a pair of range images: a review,” Comput. Vis. Graphics Image Process. Image Understand. 54, 309–324 (1991).

1990 (3)

H. Shariat, K. E. Price, “Motion estimation with more than two frames,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 417–434 (1990).

O. Faugeras, S. Maybank, “Motion from point matches: multiplicity of solutions,” Int. J. Comput. Vis. 4, 225–246 (1990).

K. H. Fielding, J. L. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).

1989 (1)

B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).

1985 (1)

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

1984 (2)

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).

P. Brou, “Using the Gaussian image to find the orientation of objects,” Int. J. Robotics Res. 3, 89–125 (1984).

1980 (1)

T. P. Wallace, O. R. Mitchell, “Analysis of three-dimensional movement using Fourier descriptors,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 583–588 (1980).

Aggarwal, J. K.

R. Talluri, J. K. Aggarwal, “Mobile robot self-location using model-image feature correspondence,” IEEE Trans. Robotics Automat. 12, 63–77 (1996).

B. Sabata, J. K. Aggarwal, “Estimation of motion from a pair of range images: a review,” Comput. Vis. Graphics Image Process. Image Understand. 54, 309–324 (1991).

Bailey, J. B.

J. B. Bailey, “Stability simulations of a visual tracking algorithm for automated docking of space vehicles,” M.S. thesis (University of New Hampshire, Durham, N.H., December1989).

Ballard, D. H.

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

Basu, A.

D. Murray, A. Basu, “Motion tracking with an active camera,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 449–459 (1994).

Brou, P.

P. Brou, “Using the Gaussian image to find the orientation of objects,” Int. J. Robotics Res. 3, 89–125 (1984).

Brown, C. M.

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

Connelly, J. M.

B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).

Cotariu, S. S.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. SPIE1704, 248–256 (1992).

Davis, L.

D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).

DeMenthon, D.

D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).

Faugeras, O.

O. Faugeras, S. Maybank, “Motion from point matches: multiplicity of solutions,” Int. J. Comput. Vis. 4, 225–246 (1990).

Fielding, K. H.

K. H. Fielding, J. L. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).

Figue, J.

J. Figue, P. Réfrégier, “Angle determination of airplanes by multicorrelation technique with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).

Gianino, P. D.

Hanson, A. R.

R. Kumar, A. R. Hanson, “Robust methods for estimating pose and a sensitivity analysis,” Comput. Vis. Graphics Image Process. Image Understand. 60, 313–342 (1994).

Hebert, T. J.

T. J. Hebert, R. M. Leahy, “Statistic-based MAP image restoration from Poisson data using Gibbs priors,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).

T. J. Hebert, X. Yang, “A sequential algorithm for motion estimation from point correspondences with intermittent occlusions,” in Proceedings of the 1995 IEEE International Conference on Image Processing (ICIP) (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 2, pp. 221–224.

Horner, J. L.

K. H. Fielding, J. L. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).

Knopp, J.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. SPIE1704, 248–256 (1992).

Kumar, B. V. K.

B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).

Kumar, R.

R. Kumar, A. R. Hanson, “Robust methods for estimating pose and a sensitivity analysis,” Comput. Vis. Graphics Image Process. Image Understand. 60, 313–342 (1994).

Kumar, V.

Leahy, R. M.

T. J. Hebert, R. M. Leahy, “Statistic-based MAP image restoration from Poisson data using Gibbs priors,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).

Lee, A. J.

B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).

Lowe, D. G.

D. G. Lowe, “Fitting parameterized three-dimensional models to images,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 441–450 (1991).

Maybank, S.

O. Faugeras, S. Maybank, “Motion from point matches: multiplicity of solutions,” Int. J. Comput. Vis. 4, 225–246 (1990).

Mcknight, D. J.

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

Mendel, J. M.

J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications and Control, Vol. 2/e of Signal Processing Series (Prentice-Hall, Englewood Cliffs, N.J., 1995).

Mitchell, O. R.

T. P. Wallace, O. R. Mitchell, “Analysis of three-dimensional movement using Fourier descriptors,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 583–588 (1980).

Monroe, S. E.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. SPIE1704, 248–256 (1992).

Murray, D.

D. Murray, A. Basu, “Motion tracking with an active camera,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 449–459 (1994).

Oberkampf, D.

D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).

Price, K. E.

H. Shariat, K. E. Price, “Motion estimation with more than two frames,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 417–434 (1990).

Réfrégier, P.

J. Figue, P. Réfrégier, “Angle determination of airplanes by multicorrelation technique with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).

Sabata, B.

B. Sabata, J. K. Aggarwal, “Estimation of motion from a pair of range images: a review,” Comput. Vis. Graphics Image Process. Image Understand. 54, 309–324 (1991).

Serati, R. A.

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

Serati, S. A.

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

Shariat, H.

H. Shariat, K. E. Price, “Motion estimation with more than two frames,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 417–434 (1990).

Sharp, G. D.

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

Stockley, J. E.

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

Talluri, R.

R. Talluri, J. K. Aggarwal, “Mobile robot self-location using model-image feature correspondence,” IEEE Trans. Robotics Automat. 12, 63–77 (1996).

Titterington, D. M.

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

Wallace, T. P.

T. P. Wallace, O. R. Mitchell, “Analysis of three-dimensional movement using Fourier descriptors,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 583–588 (1980).

Yang, X.

T. J. Hebert, X. Yang, “A sequential algorithm for motion estimation from point correspondences with intermittent occlusions,” in Proceedings of the 1995 IEEE International Conference on Image Processing (ICIP) (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 2, pp. 221–224.

Appl. Opt. (2)

Astron. Astrophys. (1)

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

Comput. Vis. Graphics Image Process. Image Understand. (2)

B. Sabata, J. K. Aggarwal, “Estimation of motion from a pair of range images: a review,” Comput. Vis. Graphics Image Process. Image Understand. 54, 309–324 (1991).

R. Kumar, A. R. Hanson, “Robust methods for estimating pose and a sensitivity analysis,” Comput. Vis. Graphics Image Process. Image Understand. 60, 313–342 (1994).

Comput. Vis. Image Understand. (1)

D. Oberkampf, D. DeMenthon, L. Davis, “Iterative pose estimation using coplanar feature points,” Comput. Vis. Image Understand. 63, 495–511 (1996).

IEEE Trans. Pattern Anal. Mach. Intell. (4)

H. Shariat, K. E. Price, “Motion estimation with more than two frames,” IEEE Trans. Pattern Anal. Mach. Intell. 12, 417–434 (1990).

D. G. Lowe, “Fitting parameterized three-dimensional models to images,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 441–450 (1991).

T. P. Wallace, O. R. Mitchell, “Analysis of three-dimensional movement using Fourier descriptors,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2, 583–588 (1980).

D. Murray, A. Basu, “Motion tracking with an active camera,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 449–459 (1994).

IEEE Trans. Robotics Automat. (1)

R. Talluri, J. K. Aggarwal, “Mobile robot self-location using model-image feature correspondence,” IEEE Trans. Robotics Automat. 12, 63–77 (1996).

IEEE Trans. Signal Process. (1)

T. J. Hebert, R. M. Leahy, “Statistic-based MAP image restoration from Poisson data using Gibbs priors,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).

Int. J. Comput. Vis. (1)

O. Faugeras, S. Maybank, “Motion from point matches: multiplicity of solutions,” Int. J. Comput. Vis. 4, 225–246 (1990).

Int. J. Robotics Res. (1)

P. Brou, “Using the Gaussian image to find the orientation of objects,” Int. J. Robotics Res. 3, 89–125 (1984).

Opt. Eng. (3)

K. H. Fielding, J. L. Horner, “1 - f binary joint transform correlator,” Opt. Eng. 29, 1081–1087 (1990).

B. V. K. Kumar, A. J. Lee, J. M. Connelly, “Estimating object rotation and scale using correlation filters,” Opt. Eng. 28, 474–481 (1989).

J. Figue, P. Réfrégier, “Angle determination of airplanes by multicorrelation technique with optimal trade-off synthetic discriminant filters,” Opt. Eng. 33, 1821–1828 (1994).

Other (6)

J. B. Bailey, “Stability simulations of a visual tracking algorithm for automated docking of space vehicles,” M.S. thesis (University of New Hampshire, Durham, N.H., December1989).

T. J. Hebert, X. Yang, “A sequential algorithm for motion estimation from point correspondences with intermittent occlusions,” in Proceedings of the 1995 IEEE International Conference on Image Processing (ICIP) (Institute of Electrical and Electronics Engineers, New York, 1995), Vol. 2, pp. 221–224.

S. S. Cotariu, S. E. Monroe, J. Knopp, “A live input, live filter, liquid crystal correlator,” in Advances in Optical Information Processing V, D. R. Pape, ed., Proc. SPIE1704, 248–256 (1992).

S. A. Serati, G. D. Sharp, R. A. Serati, D. J. Mcknight, J. E. Stockley, “128 × 128 analog liquid crystal spatial light modulator,” in Optical Pattern Recognition VI, D. P. Casasent, T. Chao, eds., Proc. SPIE2490, 378–387 (1995).

D. H. Ballard, C. M. Brown, Computer Vision (Prentice-Hall, Englewood Cliffs, N.J., 1982).

J. M. Mendel, Lessons in Estimation Theory for Signal Processing, Communications and Control, Vol. 2/e of Signal Processing Series (Prentice-Hall, Englewood Cliffs, N.J., 1995).

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Figures (15)

Fig. 1
Fig. 1

Images of the Spartan spacecraft in a reference position (top) and six new positions, demonstrating the effect of changing the two out-of-plane rotation angles, the in-plane rotation angle, the magnification, the azimuth, and the elevation.

Fig. 2
Fig. 2

Diagram of an optical correlator tracking system. FIFO, first-in–first-out.

Fig. 3
Fig. 3

Flowchart describing the regularized nonlinear least-squares method for estimating the object pose from a video sequence by use of optical correlators.

Fig. 4
Fig. 4

Left-hand column: Frames 4, 10, 16, and 22 from a 100-frame image sequence showing the Spartan spacecraft. Righthand column: The same frames after preprocessing to approximately center, scale, and rotate the object to a reference 0° in-plane rotation.

Fig. 5
Fig. 5

Image showing an icosahedron tiling of the out-of-plane rotation-angle parameter space. Each vertex of the icosahedron provides a viewing perspective (note cameras). Here the icosahedron has a subdividing integer of n = 2 with 42 vertices.

Fig. 6
Fig. 6

Icosahedron tiling of a portion of the out-of-plane rotation parameter space. The tiles are numbered 1–16 and a few ordered pairs (α, β) denoting the locations of tile vertices in the parameter space are labeled.

Fig. 7
Fig. 7

Diagram showing points {x} in the parameter space at which anticipated values of x associated with one particular tile are precomputed and stored. Note the three filters H1, H2, and H3 at the vertices of the triangular tile, the tile center at x q , and the sampling of the parameter space in 1° increments about x q . A similar set of values { x } are precomputed and stored for each tile.

Fig. 8
Fig. 8

Plot showing the out-of-plane rotation angles for a single trajectory and the corresponding estimates from a single run of the Monte Carlo simulation with error- and noise-free video images.

Fig. 9
Fig. 9

Preprocessed video images of the Spartan spacecraft with (top left) no noise, (top right) 20 dB of noise, (bottom left) 10 dB of noise, and (bottom right) 0 dB of noise.

Fig. 10
Fig. 10

Sensitivity to image noise: rms errors (RMSE’s) in estimates of the out-of-plane rotation angles (α, β) versus the image SNR from five Monte Carlo trials. (a) All other error sources are set to zero. (b) Uniformly distributed errors of magnification of ±2%, in-plane rotation of ±1°, and camera pan–tilt of ±0.5°.

Fig. 11
Fig. 11

Sensitivity to camera pan–tilt error: rms error (RMSE’s) in degrees (five Monte Carlo trials) versus the standard deviation of the camera pan–tilt error in tracking the out-of-plane rotation (α, β).

Fig. 12
Fig. 12

Sensitivity to preprocessing magnification error: rms error (RMSE’s) (five Monte Carlo trials) versus the image magnification error in tracking the out-of-plane rotation (α, β). (a) All other error sources are set to zero. (b) An image SNR of 20 dB and uniformly distributed errors of image rotation of ±1° and camera pan–tilt of ±0.5°.

Fig. 13
Fig. 13

Sensitivity to preprocessing rotation error: rms error (RMSE’s) (five Monte Carlo trials) versus the image rotation error in tracking the out-of-plane rotation (α, β). (a) All other error sources are set to zero. (b) An image SNR of 20 dB and uniformly distributed errors of image rotation of ±1° and camera pan–tilt of ±0.5°.

Fig. 14
Fig. 14

(Top left) The Spartan spacecraft with a SNR of 20 dB. (Top right, bottom left, bottom right, respectively) The Spartan spacecraft with a SNR of 20 dB, partially occluded by a 55 × 55 pixel square centered over randomly selected points on the spacecraft.

Fig. 15
Fig. 15

Sensitivity to partial occlusion of the object: rms errors (RMSE’s) (five Monte Carlo trials) in estimates of the out-of-plane rotation (α, β) versus the size of a randomly positioned occluding square. (a) All other error sources are set to zero. (b) An image SNR of 20 dB and uniformly distributed errors of image rotation of ±1° and camera pan–tilt of ±0.5°.

Equations (8)

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

i x ,   y = | H * X ,   Y G X ,   Y | 2 ,
v t k = f I g y t k ,   x ˆ t k - 1 ,   H i x ˆ t k - 1 .
min x ˆ v t k - f ˆ x ˆ t W v t k - f ˆ x ˆ + λ x ˆ - x ˆ t k - 1 t G x ˆ - x ˆ t k - 1 .
cos - 1 5 5 2 n
f 1 = m 1 m 2 ,     f 2 = m 2 m 3 ,     f 3 = m 3 m 1 , f 4 = 1 / f 1 ,   f 5 = 1 / f 2 ,   f 6 = 1 / f 3 .
W i , i = 1 σ i 2 .
σ x / y = w x 2 / 3 + m x 2 m y 2 - w y 2 - m x 2 4 w y 2 ln m y + w y m y - w y .
α = - 13   sin 6 100 π k exp - 1.5 100 k + 13 , β = 11   cos 6 100 π k exp - 1.5 100 k + 17 , k = 1 ,   2 , ,   100 .

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