Abstract

We present a homography-based method for calibrating an omnidirectional vision system with a parabolic mirror. Assuming that the intrinsic parameters of the camera are known a priori, we focus on finding the solution for the mirror parameter and its positions. We first estimate the homographic matrix partially using six or more point correspondences. Then the rotation matrix and two components of the translation vector can be estimated. Finally, the remaining parameters are computed. In this method, a closed-form solution for all the variables is obtained using the homographic matrix. Another advantage is the enhanced robustness in implementation via the use of two over-constrained linear systems. Numerical simulations and real data experiments are also performed to validate the proposed algorithm.

© 2008 Optical Society of America

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References

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  1. B. Zhang and Y. F. Li, “Dynamic calibration of the relative pose and error analysis in a structured light system,” J. Opt. Soc. Am. A 25, 612-622 (2008).
    [CrossRef]
  2. B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
    [CrossRef]
  3. S. Baker and K. Nayar, “A theory of catadioptric image formation,” Int. J. Comput. Vis. 35, 175-196 (1999).
    [CrossRef]
  4. C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223-243 (2001).
    [CrossRef]
  5. P. Barreto, “A unifying geometric representation for central projection systems,” Comput. Vis. Image Underst. 103, 208-217 (2006).
    [CrossRef]
  6. X. Ying and Z. Hu, “Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model,” in Proc. European Conference on Computer Vision (ECCV04) (Springer-Verlag, 2004), Vol. 1, pp. 442-455.
  7. Y. H. Wu, Y. F. Li, and Z. Y. Hu, “Easy calibration for para-catadioptric-like cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5719-5724.
    [CrossRef]
  8. C. Geyer and K. Daniilidis, “Catadioptric camera calibration,” in IEEE Proc. of the Seventh International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 398-404.
    [CrossRef]
  9. P. Barreto and H. Araujo, “Issues on the geometry of central catadioptric image information,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 422-427.
  10. C. Mei and E. Malis, “Fast central catadioptric line extraction, estimation, tracking and structure from motion,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 4774-4779.
    [CrossRef]
  11. C. Geyer and K. Daniilidis, “Paracatadioptric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
    [CrossRef]
  12. P. Barreto and H. Araujo, “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 1327-1333 (2005).
    [CrossRef] [PubMed]
  13. P. Barreto and H. Araujo, “Paracatadioptric camera calibration using lines,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 1359-1365.
    [CrossRef]
  14. P. Barreto and K. Daniilidis, “Unifying image plane liftings for central catadioptric and dioptric cameras,” presented at the Workshop on Omnidirectional Vision, Camera Networks and Non-Classical Cameras, Prague, Czech Republic, May 2004.
  15. R. Orghidan, E. Mouaddib, and J. Salvi, “Omnidirectional depth computation from a single image,” in IEEE Proc. of International Conference on Robotics and Automation (IEEE, 2005), pp. 1222-1227.
    [CrossRef]
  16. G. Aliega, “Accurate catadioptric calibration for real-time pose estimation in room-size environments,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2001), Vol. 1, pp. 127-134.
  17. Y. H. Wu and Z. Y. Hu, “Geometric invariants and applications under catadioptric camera model,” in IEEE Proc. of the 10th International Conference on Computer Vision (IEEE, 2005), pp. 1547-1554.
  18. S. B. Kang, “Catadioptric self-calibration,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2000), Vol. 1, pp. 201-207.
  19. C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 279-286.
  20. T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
    [CrossRef]
  21. J. Barreto and K. Daniilidis, “Epipolar geometry of central projection systems using Veronese maps,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 1258-1265.
  22. B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. 485-490.
  23. B. Micusik and T. Pajdla, “Structure from motion with wide circular field of view cameras,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1135-1149 (2006).
    [CrossRef] [PubMed]
  24. B. Micusik and T. Pajdla, “Omnidirectional camera model and epipolar geometry estimation by RANSAC with bucketing,” Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 83-90.
    [CrossRef]
  25. J. Y. Bouguet, Camera Calibration Toolbox for Matlab, http://www.vision.caltech.edu/bouguetj/calib-doc/, public domain internet software.

2008 (1)

2007 (1)

B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
[CrossRef]

2006 (2)

P. Barreto, “A unifying geometric representation for central projection systems,” Comput. Vis. Image Underst. 103, 208-217 (2006).
[CrossRef]

B. Micusik and T. Pajdla, “Structure from motion with wide circular field of view cameras,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1135-1149 (2006).
[CrossRef] [PubMed]

2005 (1)

P. Barreto and H. Araujo, “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 1327-1333 (2005).
[CrossRef] [PubMed]

2002 (2)

T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
[CrossRef]

C. Geyer and K. Daniilidis, “Paracatadioptric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

2001 (1)

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223-243 (2001).
[CrossRef]

1999 (1)

S. Baker and K. Nayar, “A theory of catadioptric image formation,” Int. J. Comput. Vis. 35, 175-196 (1999).
[CrossRef]

Aliega, G.

G. Aliega, “Accurate catadioptric calibration for real-time pose estimation in room-size environments,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2001), Vol. 1, pp. 127-134.

Araujo, H.

P. Barreto and H. Araujo, “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 1327-1333 (2005).
[CrossRef] [PubMed]

P. Barreto and H. Araujo, “Paracatadioptric camera calibration using lines,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 1359-1365.
[CrossRef]

P. Barreto and H. Araujo, “Issues on the geometry of central catadioptric image information,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 422-427.

Baker, S.

S. Baker and K. Nayar, “A theory of catadioptric image formation,” Int. J. Comput. Vis. 35, 175-196 (1999).
[CrossRef]

Barreto, J.

J. Barreto and K. Daniilidis, “Epipolar geometry of central projection systems using Veronese maps,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 1258-1265.

Barreto, P.

P. Barreto, “A unifying geometric representation for central projection systems,” Comput. Vis. Image Underst. 103, 208-217 (2006).
[CrossRef]

P. Barreto and H. Araujo, “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 1327-1333 (2005).
[CrossRef] [PubMed]

P. Barreto and H. Araujo, “Issues on the geometry of central catadioptric image information,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 422-427.

P. Barreto and H. Araujo, “Paracatadioptric camera calibration using lines,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 1359-1365.
[CrossRef]

P. Barreto and K. Daniilidis, “Unifying image plane liftings for central catadioptric and dioptric cameras,” presented at the Workshop on Omnidirectional Vision, Camera Networks and Non-Classical Cameras, Prague, Czech Republic, May 2004.

Bouguet, J. Y.

J. Y. Bouguet, Camera Calibration Toolbox for Matlab, http://www.vision.caltech.edu/bouguetj/calib-doc/, public domain internet software.

Daniilidis, K.

C. Geyer and K. Daniilidis, “Paracatadioptric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223-243 (2001).
[CrossRef]

P. Barreto and K. Daniilidis, “Unifying image plane liftings for central catadioptric and dioptric cameras,” presented at the Workshop on Omnidirectional Vision, Camera Networks and Non-Classical Cameras, Prague, Czech Republic, May 2004.

C. Geyer and K. Daniilidis, “Catadioptric camera calibration,” in IEEE Proc. of the Seventh International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 398-404.
[CrossRef]

C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 279-286.

J. Barreto and K. Daniilidis, “Epipolar geometry of central projection systems using Veronese maps,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 1258-1265.

Geyer, C.

C. Geyer and K. Daniilidis, “Paracatadioptric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223-243 (2001).
[CrossRef]

C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 279-286.

C. Geyer and K. Daniilidis, “Catadioptric camera calibration,” in IEEE Proc. of the Seventh International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 398-404.
[CrossRef]

Hlavac, V.

T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
[CrossRef]

Hu, Z.

X. Ying and Z. Hu, “Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model,” in Proc. European Conference on Computer Vision (ECCV04) (Springer-Verlag, 2004), Vol. 1, pp. 442-455.

Hu, Z. Y.

Y. H. Wu and Z. Y. Hu, “Geometric invariants and applications under catadioptric camera model,” in IEEE Proc. of the 10th International Conference on Computer Vision (IEEE, 2005), pp. 1547-1554.

Y. H. Wu, Y. F. Li, and Z. Y. Hu, “Easy calibration for para-catadioptric-like cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5719-5724.
[CrossRef]

Kang, S. B.

S. B. Kang, “Catadioptric self-calibration,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2000), Vol. 1, pp. 201-207.

Li, Y. F.

B. Zhang and Y. F. Li, “Dynamic calibration of the relative pose and error analysis in a structured light system,” J. Opt. Soc. Am. A 25, 612-622 (2008).
[CrossRef]

B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
[CrossRef]

Y. H. Wu, Y. F. Li, and Z. Y. Hu, “Easy calibration for para-catadioptric-like cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5719-5724.
[CrossRef]

Malis, E.

C. Mei and E. Malis, “Fast central catadioptric line extraction, estimation, tracking and structure from motion,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 4774-4779.
[CrossRef]

Mei, C.

C. Mei and E. Malis, “Fast central catadioptric line extraction, estimation, tracking and structure from motion,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 4774-4779.
[CrossRef]

Micusik, B.

B. Micusik and T. Pajdla, “Structure from motion with wide circular field of view cameras,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1135-1149 (2006).
[CrossRef] [PubMed]

B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. 485-490.

B. Micusik and T. Pajdla, “Omnidirectional camera model and epipolar geometry estimation by RANSAC with bucketing,” Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 83-90.
[CrossRef]

Mouaddib, E.

R. Orghidan, E. Mouaddib, and J. Salvi, “Omnidirectional depth computation from a single image,” in IEEE Proc. of International Conference on Robotics and Automation (IEEE, 2005), pp. 1222-1227.
[CrossRef]

Nayar, K.

S. Baker and K. Nayar, “A theory of catadioptric image formation,” Int. J. Comput. Vis. 35, 175-196 (1999).
[CrossRef]

Orghidan, R.

R. Orghidan, E. Mouaddib, and J. Salvi, “Omnidirectional depth computation from a single image,” in IEEE Proc. of International Conference on Robotics and Automation (IEEE, 2005), pp. 1222-1227.
[CrossRef]

Padjla, T.

T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
[CrossRef]

Pajdla, T.

B. Micusik and T. Pajdla, “Structure from motion with wide circular field of view cameras,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1135-1149 (2006).
[CrossRef] [PubMed]

B. Micusik and T. Pajdla, “Omnidirectional camera model and epipolar geometry estimation by RANSAC with bucketing,” Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 83-90.
[CrossRef]

B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. 485-490.

Salvi, J.

R. Orghidan, E. Mouaddib, and J. Salvi, “Omnidirectional depth computation from a single image,” in IEEE Proc. of International Conference on Robotics and Automation (IEEE, 2005), pp. 1222-1227.
[CrossRef]

Svoboda, T.

T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
[CrossRef]

Wu, Y.

B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
[CrossRef]

Wu, Y. H.

Y. H. Wu and Z. Y. Hu, “Geometric invariants and applications under catadioptric camera model,” in IEEE Proc. of the 10th International Conference on Computer Vision (IEEE, 2005), pp. 1547-1554.

Y. H. Wu, Y. F. Li, and Z. Y. Hu, “Easy calibration for para-catadioptric-like cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5719-5724.
[CrossRef]

Ying, X.

X. Ying and Z. Hu, “Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model,” in Proc. European Conference on Computer Vision (ECCV04) (Springer-Verlag, 2004), Vol. 1, pp. 442-455.

Zhang, B.

B. Zhang and Y. F. Li, “Dynamic calibration of the relative pose and error analysis in a structured light system,” J. Opt. Soc. Am. A 25, 612-622 (2008).
[CrossRef]

B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
[CrossRef]

Comput. Vis. Image Underst. (1)

P. Barreto, “A unifying geometric representation for central projection systems,” Comput. Vis. Image Underst. 103, 208-217 (2006).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (3)

C. Geyer and K. Daniilidis, “Paracatadioptric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 24, 687-695 (2002).
[CrossRef]

P. Barreto and H. Araujo, “Geometric properties of central catadioptric line images and their application in calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 27, 1327-1333 (2005).
[CrossRef] [PubMed]

B. Micusik and T. Pajdla, “Structure from motion with wide circular field of view cameras,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1135-1149 (2006).
[CrossRef] [PubMed]

Int. J. Comput. Vis. (3)

S. Baker and K. Nayar, “A theory of catadioptric image formation,” Int. J. Comput. Vis. 35, 175-196 (1999).
[CrossRef]

C. Geyer and K. Daniilidis, “Catadioptric projective geometry,” Int. J. Comput. Vis. 45, 223-243 (2001).
[CrossRef]

T. Svoboda, T. Padjla, and V. Hlavac, “Epipolar geometry for panoramic cameras,” Int. J. Comput. Vis. 49, 23-37 (2002).
[CrossRef]

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

Pattern Recogn. (1)

B. Zhang, Y. F. Li, and Y. Wu, “Self-recalibration of a structured light system via plane-based homography,” Pattern Recogn. 40, 1368-1377 (2007).
[CrossRef]

Other (16)

B. Micusik and T. Pajdla, “Omnidirectional camera model and epipolar geometry estimation by RANSAC with bucketing,” Lecture Notes in Computer Science (Springer-Verlag, 2003), pp. 83-90.
[CrossRef]

J. Y. Bouguet, Camera Calibration Toolbox for Matlab, http://www.vision.caltech.edu/bouguetj/calib-doc/, public domain internet software.

J. Barreto and K. Daniilidis, “Epipolar geometry of central projection systems using Veronese maps,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2006), Vol. 1, pp. 1258-1265.

B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2003), Vol. 1, pp. 485-490.

X. Ying and Z. Hu, “Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model,” in Proc. European Conference on Computer Vision (ECCV04) (Springer-Verlag, 2004), Vol. 1, pp. 442-455.

Y. H. Wu, Y. F. Li, and Z. Y. Hu, “Easy calibration for para-catadioptric-like cameras,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 5719-5724.
[CrossRef]

C. Geyer and K. Daniilidis, “Catadioptric camera calibration,” in IEEE Proc. of the Seventh International Conference on Computer Vision (IEEE, 1999), Vol. 1, pp. 398-404.
[CrossRef]

P. Barreto and H. Araujo, “Issues on the geometry of central catadioptric image information,” in IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2001), pp. 422-427.

C. Mei and E. Malis, “Fast central catadioptric line extraction, estimation, tracking and structure from motion,” in IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2006), pp. 4774-4779.
[CrossRef]

P. Barreto and H. Araujo, “Paracatadioptric camera calibration using lines,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2003), Vol. 2, pp. 1359-1365.
[CrossRef]

P. Barreto and K. Daniilidis, “Unifying image plane liftings for central catadioptric and dioptric cameras,” presented at the Workshop on Omnidirectional Vision, Camera Networks and Non-Classical Cameras, Prague, Czech Republic, May 2004.

R. Orghidan, E. Mouaddib, and J. Salvi, “Omnidirectional depth computation from a single image,” in IEEE Proc. of International Conference on Robotics and Automation (IEEE, 2005), pp. 1222-1227.
[CrossRef]

G. Aliega, “Accurate catadioptric calibration for real-time pose estimation in room-size environments,” in IEEE Proc. of the Ninth International Conference on Computer Vision (IEEE, 2001), Vol. 1, pp. 127-134.

Y. H. Wu and Z. Y. Hu, “Geometric invariants and applications under catadioptric camera model,” in IEEE Proc. of the 10th International Conference on Computer Vision (IEEE, 2005), pp. 1547-1554.

S. B. Kang, “Catadioptric self-calibration,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2000), Vol. 1, pp. 201-207.

C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in IEEE Proc. of the Conference on Computer Vision and Pattern Recognition (IEEE, 2001), Vol. 1, pp. 279-286.

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Figures (6)

Fig. 1
Fig. 1

Parabolic camera system.

Fig. 2
Fig. 2

One of the simulated parabolic camera images.

Fig. 3
Fig. 3

Relative errors versus different levels of noise.

Fig. 4
Fig. 4

System setup in real data experiments.

Fig. 5
Fig. 5

Pattern in six different positions: they are labeled 1–6 from left to right and from top down in this experiment.

Fig. 6
Fig. 6

Reconstructed positions of the pattern.

Tables (2)

Tables Icon

Table 1 Relative Error Percentage in Cases of Arbitrary Motions

Tables Icon

Table 2 Reprojection Errors in the Image Space (Pixels)

Equations (22)

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

z = x 2 + y 2 a 2 2 a ,
p = λ ( R m P + t m ) ,
u = α [ ( 1 0 0 0 1 0 ) p 1 ] ,
p = [ u v u 2 + v 2 a 2 2 a ] ,
p = σ HP ,
H = [ h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 ]
[ u v u 2 + v 2 a 2 2 a ] = σ H [ x y 1 ] .
( h 1 x + h 2 y + h 3 ) v ( h 4 x + h 5 y + h 6 ) u = 0 .
Ah = 0 ,
A = [ v 1 x 1 v 1 y 1 v 1 u 1 x 1 u 1 y 1 u 1 v n x n v n y n v n u n x n u n y n u n ]
S H = [ r 11 r 12 t 1 r 21 r 22 t 2 r 31 r 32 t 3 ] ,
s = 1 norm ( [ h 1 h 2 h 4 h 5 ] ) .
R 11 = s h 1 , R 12 = s h 2 , R 21 = s h 4 , R 22 = s h 5 .
t 1 = s h 3 , t 2 = s h 6 .
( h 1 x + h 2 y + h 3 ) u 2 + v 2 a 2 2 a ( h 7 x + h 8 y + h 9 ) u = 0 .
k 1 a 2 + 2 u a h 9 + k 2 a + k 3 = 0 ,
k 1 = h 1 x + h 2 y + h 3 , k 2 = 2 u ( h 7 x + h 8 y ) ,
k 3 = ( u 2 + v 2 ) ( h 1 x + h 2 y + h 3 ) .
Bb = 0 ,
B = [ k 1 1 2 u 1 k 1 2 k 1 3 k n 1 2 u n k n 2 k n 3 ]
b = [ a 2 a h 9 a 1 ] T .
a = b 1 b 3 and h 9 = b 2 b 3 ,

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