Abstract

Analytical reconstruction of 3D curves from their stereo images is an important issue in computer vision. We present an optimization framework for such a problem based on a nonuniform rational B-spline (NURBS) curve model that converts reconstruction of a 3D curve into reconstruction of control points and weights of a NURBS representation of the curve, accordingly bypassing the error-prone point-to-point correspondence matching. Perspective invariance of NURBS curves and constraints deduced on stereo NURBS curves are employed to formulate the 3D curve reconstruction problem into a constrained nonlinear optimization. A parallel rectification technique is then adopted to simplify the constraints, and the Levenberg–Marquardt algorithm is applied to search for the optimal solution of the simplified problem. The results from our experiments show that the proposed framework works stably in the presence of different data samplings, randomly posed noise, and partial loss of data and is potentially suitable for real scenes.

© 2005 Optical Society of America

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  1. O. Faugeras, Three-dimensional Computer Vision: a Geometric Viewpoint (MIT Press, 1993).
  2. D. Scharstein, R. Szeliski, “Taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
    [CrossRef]
  3. M. Z. Brown, D. Burschka, G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
    [CrossRef]
  4. M. Pollefeys, S. Sinha, “Iso-disparity surfaces for general stereo configurations,” presented at the Eighth European Conference on Computer Vision, Prague, Czech Republic, May 11–14, 2004.
  5. M. H. Lin, C. Tomasi, “Surfaces with occlusions from layered stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1073–1078 (2004).
    [CrossRef]
  6. O. Faugeras, R. Keriven, “Variational principles, surface evolution, PDEs, level-set methods, and the stereo problem,” IEEE Trans. Image Process. 7, 336–344 (1998).
    [CrossRef]
  7. N. Ayache, F. Lustman, “Trinocular stereo vision for robotics,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 73–85 (1991).
    [CrossRef]
  8. D. Q. Huynh, R. A. Owens, “Line labeling and region-segmentation in stereo image pairs,” Image Vis. Comput. 12, 213–225 (1994).
    [CrossRef]
  9. G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Stereo matching using Hebbian learning,” IEEE Trans. Syst. Man Cybern. 29, 553–559 (1999).
    [CrossRef]
  10. N. Ayache, B. Faverjon, “Efficient registration of stereo images by matching graph descriptions of edge segments,” Int. J. Comput. Vis. 1, 107–131 (1987).
    [CrossRef]
  11. S. D. Ma, “Conic-based stereo, motion estimation, and pose determination,” Int. J. Comput. Vis. 10, 7–25 (1993).
    [CrossRef]
  12. L. Quan, “Conic reconstruction and correspondence from two views,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 151–160 (1996).
    [CrossRef]
  13. L. Li, S. D. Ma, “3D pose estimation from a N-degree planar curve in two perspective views,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.
  14. M. H. An, C. N. Lee, “Stereo vision based on algebraic curves,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.
  15. L. Robert, O. D. Faugeras, “Curve-based stereo: figural continuity and curvature,” presented at the International Conference on Computer Vision and Pattern Recognition, Maui, Hawaii, June 3–6, 1991.
  16. K. Kedem, Y. Yarmovski, “Curve based stereo matching using the minimum Hausdorff distance,” presented at the 12th Symposium on Computational Geometry, Philadelphia, Pennsylvania, May 24–26, 1996.
  17. A. T. Brant, M. Brady, “Stereo matching of curves by least deformation,” presented at the International Workshop on Intelligent Robots and Systems ’89, Tsukuba, Japan, September 4–6, 1989.
  18. N. M. Nasrabadi, “A stereo vision technique using curve segments and relaxation matching,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 566–572 (1992).
    [CrossRef]
  19. N. M. Nasrabadi, Y. Liu, “Stereo vision correspondence using a multi-channel graph matching technique,” Image Vis. Comput. 7, 237–245 (1989).
    [CrossRef]
  20. G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Relaxation labeling in stereo image matching,” Pattern Recogn. 33, 53–68 (2000).
    [CrossRef]
  21. J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
    [CrossRef]
  22. Y. Shan, Z. Zhang, “New measurements and corner-guidance for curve matching with probabilistic relaxation,” Int. J. Comput. Vis. 46, 157–171 (2002).
    [CrossRef]
  23. C. Schmid, A. Zisserman, “The geometry and matching of lines and curves over multiple views,” Int. J. Comput. Vis. 40, 199–233 (2000).
    [CrossRef]
  24. J. Sato, R. Cipolla, “Quasi-invariant parameterisations and matching of curves in images,” Int. J. Comput. Vis. 28, 117–136 (1998).
    [CrossRef]
  25. R. Berthilsson, K. Astrom, A. Heyden, “Reconstruction of general curves using factorization and bundle adjustment,” Int. J. Comput. Vis. 41, 171–182 (2002).
    [CrossRef]
  26. Y. J. Xiao, M. Y. Ding, J. X. Peng, “B-spline based stereo for 3D reconstruction of line-like objects using affine camera model,” Int. J. Pattern Recognit. Artif. Intell. 15, 347–358 (2001).
    [CrossRef]
  27. D. F. Rogers, N. G. Fog, “Constrained B-spline curve and surface fitting,” Comput. Aided Des. 21, 641–648 (1989).
    [CrossRef]
  28. S. Hu, Y. Li, T. Ju, X. Zhu, “Modifying the shape of NURBS surfaces with geometric constraints,” Comput.-Aided Des. 33, 903–912 (2001).
    [CrossRef]
  29. H. Qin, D. Terzopoulos, “D-NURBS: a physics-based framework for geometric design,” IEEE Trans. Vis. Comput. Graph. 2, 85–96 (1996).
  30. F. S. Cohen, J. Y. Wang, “Modeling image curves using invariant 3D curve models—a path to 3D recognition and shape estimation from image contours,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 1–12 (1994).
    [CrossRef]
  31. L. Piegl, “On NURBS: a survey,” IEEE Comput. Graphics Appl. 13(1), 55–71 (1991).
    [CrossRef]
  32. J. Aloimonos, “Perspective approximations,” Image Vis. Comput. 8, 179–192 (1990).
    [CrossRef]
  33. N. Ayache, C. Hansen, “Rectification of images for binocular and trinocular stereovision,” presented at the 9th International Conference on Pattern Recognition, Beijing, China, November 14–17, 1988.
  34. W. Ma, J.-P. Kruth, “NURBS curve and surface fitting for reverse engineering,” Int. J. Adv. Manuf. Technol. 14, 918–927 (1998).
    [CrossRef]
  35. D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
    [CrossRef]
  36. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), pp. 681–688.
  37. P. J. Besl, N. D. McKay, “A method for registration of 3-d shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
    [CrossRef]
  38. R. M. Haralick, L. G. Shapiro, Computer and Robot Vision (Addison-Wesley, 1992), Vol. I, pp. 233–236.
  39. H. P. Yang, W. P. Wang, J. G. Sun, “Control point adjustment for B-spline curve approximation,” Comput. Aided Des. 36, 639–652 (2004).
    [CrossRef]
  40. T. J. Cham, R. Cipolla, “Automated B-spline curve representation incorporating MDL and error-minimizing control point insertion strategies,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 49–53 (1999).
    [CrossRef]

2004 (2)

M. H. Lin, C. Tomasi, “Surfaces with occlusions from layered stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1073–1078 (2004).
[CrossRef]

H. P. Yang, W. P. Wang, J. G. Sun, “Control point adjustment for B-spline curve approximation,” Comput. Aided Des. 36, 639–652 (2004).
[CrossRef]

2003 (1)

M. Z. Brown, D. Burschka, G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

2002 (3)

D. Scharstein, R. Szeliski, “Taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Y. Shan, Z. Zhang, “New measurements and corner-guidance for curve matching with probabilistic relaxation,” Int. J. Comput. Vis. 46, 157–171 (2002).
[CrossRef]

R. Berthilsson, K. Astrom, A. Heyden, “Reconstruction of general curves using factorization and bundle adjustment,” Int. J. Comput. Vis. 41, 171–182 (2002).
[CrossRef]

2001 (2)

Y. J. Xiao, M. Y. Ding, J. X. Peng, “B-spline based stereo for 3D reconstruction of line-like objects using affine camera model,” Int. J. Pattern Recognit. Artif. Intell. 15, 347–358 (2001).
[CrossRef]

S. Hu, Y. Li, T. Ju, X. Zhu, “Modifying the shape of NURBS surfaces with geometric constraints,” Comput.-Aided Des. 33, 903–912 (2001).
[CrossRef]

2000 (2)

C. Schmid, A. Zisserman, “The geometry and matching of lines and curves over multiple views,” Int. J. Comput. Vis. 40, 199–233 (2000).
[CrossRef]

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Relaxation labeling in stereo image matching,” Pattern Recogn. 33, 53–68 (2000).
[CrossRef]

1999 (2)

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Stereo matching using Hebbian learning,” IEEE Trans. Syst. Man Cybern. 29, 553–559 (1999).
[CrossRef]

T. J. Cham, R. Cipolla, “Automated B-spline curve representation incorporating MDL and error-minimizing control point insertion strategies,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 49–53 (1999).
[CrossRef]

1998 (3)

W. Ma, J.-P. Kruth, “NURBS curve and surface fitting for reverse engineering,” Int. J. Adv. Manuf. Technol. 14, 918–927 (1998).
[CrossRef]

O. Faugeras, R. Keriven, “Variational principles, surface evolution, PDEs, level-set methods, and the stereo problem,” IEEE Trans. Image Process. 7, 336–344 (1998).
[CrossRef]

J. Sato, R. Cipolla, “Quasi-invariant parameterisations and matching of curves in images,” Int. J. Comput. Vis. 28, 117–136 (1998).
[CrossRef]

1996 (2)

L. Quan, “Conic reconstruction and correspondence from two views,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 151–160 (1996).
[CrossRef]

H. Qin, D. Terzopoulos, “D-NURBS: a physics-based framework for geometric design,” IEEE Trans. Vis. Comput. Graph. 2, 85–96 (1996).

1994 (2)

F. S. Cohen, J. Y. Wang, “Modeling image curves using invariant 3D curve models—a path to 3D recognition and shape estimation from image contours,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 1–12 (1994).
[CrossRef]

D. Q. Huynh, R. A. Owens, “Line labeling and region-segmentation in stereo image pairs,” Image Vis. Comput. 12, 213–225 (1994).
[CrossRef]

1993 (1)

S. D. Ma, “Conic-based stereo, motion estimation, and pose determination,” Int. J. Comput. Vis. 10, 7–25 (1993).
[CrossRef]

1992 (2)

N. M. Nasrabadi, “A stereo vision technique using curve segments and relaxation matching,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 566–572 (1992).
[CrossRef]

P. J. Besl, N. D. McKay, “A method for registration of 3-d shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

1991 (3)

L. Piegl, “On NURBS: a survey,” IEEE Comput. Graphics Appl. 13(1), 55–71 (1991).
[CrossRef]

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

N. Ayache, F. Lustman, “Trinocular stereo vision for robotics,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 73–85 (1991).
[CrossRef]

1990 (1)

J. Aloimonos, “Perspective approximations,” Image Vis. Comput. 8, 179–192 (1990).
[CrossRef]

1989 (2)

N. M. Nasrabadi, Y. Liu, “Stereo vision correspondence using a multi-channel graph matching technique,” Image Vis. Comput. 7, 237–245 (1989).
[CrossRef]

D. F. Rogers, N. G. Fog, “Constrained B-spline curve and surface fitting,” Comput. Aided Des. 21, 641–648 (1989).
[CrossRef]

1987 (1)

N. Ayache, B. Faverjon, “Efficient registration of stereo images by matching graph descriptions of edge segments,” Int. J. Comput. Vis. 1, 107–131 (1987).
[CrossRef]

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[CrossRef]

Aloimonos, J.

J. Aloimonos, “Perspective approximations,” Image Vis. Comput. 8, 179–192 (1990).
[CrossRef]

An, M. H.

M. H. An, C. N. Lee, “Stereo vision based on algebraic curves,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.

Astrom, K.

R. Berthilsson, K. Astrom, A. Heyden, “Reconstruction of general curves using factorization and bundle adjustment,” Int. J. Comput. Vis. 41, 171–182 (2002).
[CrossRef]

Ayache, N.

N. Ayache, F. Lustman, “Trinocular stereo vision for robotics,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 73–85 (1991).
[CrossRef]

N. Ayache, B. Faverjon, “Efficient registration of stereo images by matching graph descriptions of edge segments,” Int. J. Comput. Vis. 1, 107–131 (1987).
[CrossRef]

N. Ayache, C. Hansen, “Rectification of images for binocular and trinocular stereovision,” presented at the 9th International Conference on Pattern Recognition, Beijing, China, November 14–17, 1988.

Berthilsson, R.

R. Berthilsson, K. Astrom, A. Heyden, “Reconstruction of general curves using factorization and bundle adjustment,” Int. J. Comput. Vis. 41, 171–182 (2002).
[CrossRef]

Besl, P. J.

P. J. Besl, N. D. McKay, “A method for registration of 3-d shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Brady, M.

A. T. Brant, M. Brady, “Stereo matching of curves by least deformation,” presented at the International Workshop on Intelligent Robots and Systems ’89, Tsukuba, Japan, September 4–6, 1989.

Brant, A. T.

A. T. Brant, M. Brady, “Stereo matching of curves by least deformation,” presented at the International Workshop on Intelligent Robots and Systems ’89, Tsukuba, Japan, September 4–6, 1989.

Brown, M. Z.

M. Z. Brown, D. Burschka, G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Burschka, D.

M. Z. Brown, D. Burschka, G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Cham, T. J.

T. J. Cham, R. Cipolla, “Automated B-spline curve representation incorporating MDL and error-minimizing control point insertion strategies,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 49–53 (1999).
[CrossRef]

Cipolla, R.

T. J. Cham, R. Cipolla, “Automated B-spline curve representation incorporating MDL and error-minimizing control point insertion strategies,” IEEE Trans. Pattern Anal. Mach. Intell. 21, 49–53 (1999).
[CrossRef]

J. Sato, R. Cipolla, “Quasi-invariant parameterisations and matching of curves in images,” Int. J. Comput. Vis. 28, 117–136 (1998).
[CrossRef]

Cohen, F. S.

F. S. Cohen, J. Y. Wang, “Modeling image curves using invariant 3D curve models—a path to 3D recognition and shape estimation from image contours,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 1–12 (1994).
[CrossRef]

Cruz, J. M.

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Relaxation labeling in stereo image matching,” Pattern Recogn. 33, 53–68 (2000).
[CrossRef]

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Stereo matching using Hebbian learning,” IEEE Trans. Syst. Man Cybern. 29, 553–559 (1999).
[CrossRef]

Ding, M. Y.

Y. J. Xiao, M. Y. Ding, J. X. Peng, “B-spline based stereo for 3D reconstruction of line-like objects using affine camera model,” Int. J. Pattern Recognit. Artif. Intell. 15, 347–358 (2001).
[CrossRef]

Faugeras, O.

O. Faugeras, R. Keriven, “Variational principles, surface evolution, PDEs, level-set methods, and the stereo problem,” IEEE Trans. Image Process. 7, 336–344 (1998).
[CrossRef]

O. Faugeras, Three-dimensional Computer Vision: a Geometric Viewpoint (MIT Press, 1993).

Faugeras, O. D.

L. Robert, O. D. Faugeras, “Curve-based stereo: figural continuity and curvature,” presented at the International Conference on Computer Vision and Pattern Recognition, Maui, Hawaii, June 3–6, 1991.

Faverjon, B.

N. Ayache, B. Faverjon, “Efficient registration of stereo images by matching graph descriptions of edge segments,” Int. J. Comput. Vis. 1, 107–131 (1987).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), pp. 681–688.

Fog, N. G.

D. F. Rogers, N. G. Fog, “Constrained B-spline curve and surface fitting,” Comput. Aided Des. 21, 641–648 (1989).
[CrossRef]

Hager, G. D.

M. Z. Brown, D. Burschka, G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Hansen, C.

N. Ayache, C. Hansen, “Rectification of images for binocular and trinocular stereovision,” presented at the 9th International Conference on Pattern Recognition, Beijing, China, November 14–17, 1988.

Haralick, R. M.

R. M. Haralick, L. G. Shapiro, Computer and Robot Vision (Addison-Wesley, 1992), Vol. I, pp. 233–236.

Heyden, A.

R. Berthilsson, K. Astrom, A. Heyden, “Reconstruction of general curves using factorization and bundle adjustment,” Int. J. Comput. Vis. 41, 171–182 (2002).
[CrossRef]

Hu, S.

S. Hu, Y. Li, T. Ju, X. Zhu, “Modifying the shape of NURBS surfaces with geometric constraints,” Comput.-Aided Des. 33, 903–912 (2001).
[CrossRef]

Huynh, D. Q.

D. Q. Huynh, R. A. Owens, “Line labeling and region-segmentation in stereo image pairs,” Image Vis. Comput. 12, 213–225 (1994).
[CrossRef]

Ju, T.

S. Hu, Y. Li, T. Ju, X. Zhu, “Modifying the shape of NURBS surfaces with geometric constraints,” Comput.-Aided Des. 33, 903–912 (2001).
[CrossRef]

Kedem, K.

K. Kedem, Y. Yarmovski, “Curve based stereo matching using the minimum Hausdorff distance,” presented at the 12th Symposium on Computational Geometry, Philadelphia, Pennsylvania, May 24–26, 1996.

Keriven, R.

O. Faugeras, R. Keriven, “Variational principles, surface evolution, PDEs, level-set methods, and the stereo problem,” IEEE Trans. Image Process. 7, 336–344 (1998).
[CrossRef]

Kruth, J.-P.

W. Ma, J.-P. Kruth, “NURBS curve and surface fitting for reverse engineering,” Int. J. Adv. Manuf. Technol. 14, 918–927 (1998).
[CrossRef]

Lee, C. N.

M. H. An, C. N. Lee, “Stereo vision based on algebraic curves,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.

Li, L.

L. Li, S. D. Ma, “3D pose estimation from a N-degree planar curve in two perspective views,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.

Li, Y.

S. Hu, Y. Li, T. Ju, X. Zhu, “Modifying the shape of NURBS surfaces with geometric constraints,” Comput.-Aided Des. 33, 903–912 (2001).
[CrossRef]

Lin, M. H.

M. H. Lin, C. Tomasi, “Surfaces with occlusions from layered stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 26, 1073–1078 (2004).
[CrossRef]

Liu, Y.

N. M. Nasrabadi, Y. Liu, “Stereo vision correspondence using a multi-channel graph matching technique,” Image Vis. Comput. 7, 237–245 (1989).
[CrossRef]

Lopez-Orozco, J. A.

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Relaxation labeling in stereo image matching,” Pattern Recogn. 33, 53–68 (2000).
[CrossRef]

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Stereo matching using Hebbian learning,” IEEE Trans. Syst. Man Cybern. 29, 553–559 (1999).
[CrossRef]

Lustman, F.

N. Ayache, F. Lustman, “Trinocular stereo vision for robotics,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 73–85 (1991).
[CrossRef]

Ma, S. D.

S. D. Ma, “Conic-based stereo, motion estimation, and pose determination,” Int. J. Comput. Vis. 10, 7–25 (1993).
[CrossRef]

L. Li, S. D. Ma, “3D pose estimation from a N-degree planar curve in two perspective views,” presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25–30, 1996.

Ma, W.

W. Ma, J.-P. Kruth, “NURBS curve and surface fitting for reverse engineering,” Int. J. Adv. Manuf. Technol. 14, 918–927 (1998).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math. 11, 431–441 (1963).
[CrossRef]

McKay, N. D.

P. J. Besl, N. D. McKay, “A method for registration of 3-d shapes,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 239–256 (1992).
[CrossRef]

Nasrabadi, N. M.

N. M. Nasrabadi, “A stereo vision technique using curve segments and relaxation matching,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 566–572 (1992).
[CrossRef]

N. M. Nasrabadi, Y. Liu, “Stereo vision correspondence using a multi-channel graph matching technique,” Image Vis. Comput. 7, 237–245 (1989).
[CrossRef]

Owens, R. A.

D. Q. Huynh, R. A. Owens, “Line labeling and region-segmentation in stereo image pairs,” Image Vis. Comput. 12, 213–225 (1994).
[CrossRef]

Pajares, G.

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Relaxation labeling in stereo image matching,” Pattern Recogn. 33, 53–68 (2000).
[CrossRef]

G. Pajares, J. M. Cruz, J. A. Lopez-Orozco, “Stereo matching using Hebbian learning,” IEEE Trans. Syst. Man Cybern. 29, 553–559 (1999).
[CrossRef]

Peng, J. X.

Y. J. Xiao, M. Y. Ding, J. X. Peng, “B-spline based stereo for 3D reconstruction of line-like objects using affine camera model,” Int. J. Pattern Recognit. Artif. Intell. 15, 347–358 (2001).
[CrossRef]

Piegl, L.

L. Piegl, “On NURBS: a survey,” IEEE Comput. Graphics Appl. 13(1), 55–71 (1991).
[CrossRef]

Pollard, S.

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

Pollefeys, M.

M. Pollefeys, S. Sinha, “Iso-disparity surfaces for general stereo configurations,” presented at the Eighth European Conference on Computer Vision, Prague, Czech Republic, May 11–14, 2004.

Porrill, J.

J. Porrill, S. Pollard, “Curve matching and stereo calibration,” Image Vis. Comput. 9, 45–50 (1991).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), pp. 681–688.

Qin, H.

H. Qin, D. Terzopoulos, “D-NURBS: a physics-based framework for geometric design,” IEEE Trans. Vis. Comput. Graph. 2, 85–96 (1996).

Quan, L.

L. Quan, “Conic reconstruction and correspondence from two views,” IEEE Trans. Pattern Anal. Mach. Intell. 18, 151–160 (1996).
[CrossRef]

Robert, L.

L. Robert, O. D. Faugeras, “Curve-based stereo: figural continuity and curvature,” presented at the International Conference on Computer Vision and Pattern Recognition, Maui, Hawaii, June 3–6, 1991.

Rogers, D. F.

D. F. Rogers, N. G. Fog, “Constrained B-spline curve and surface fitting,” Comput. Aided Des. 21, 641–648 (1989).
[CrossRef]

Sato, J.

J. Sato, R. Cipolla, “Quasi-invariant parameterisations and matching of curves in images,” Int. J. Comput. Vis. 28, 117–136 (1998).
[CrossRef]

Scharstein, D.

D. Scharstein, R. Szeliski, “Taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Schmid, C.

C. Schmid, A. Zisserman, “The geometry and matching of lines and curves over multiple views,” Int. J. Comput. Vis. 40, 199–233 (2000).
[CrossRef]

Shan, Y.

Y. Shan, Z. Zhang, “New measurements and corner-guidance for curve matching with probabilistic relaxation,” Int. J. Comput. Vis. 46, 157–171 (2002).
[CrossRef]

Shapiro, L. G.

R. M. Haralick, L. G. Shapiro, Computer and Robot Vision (Addison-Wesley, 1992), Vol. I, pp. 233–236.

Sinha, S.

M. Pollefeys, S. Sinha, “Iso-disparity surfaces for general stereo configurations,” presented at the Eighth European Conference on Computer Vision, Prague, Czech Republic, May 11–14, 2004.

Sun, J. G.

H. P. Yang, W. P. Wang, J. G. Sun, “Control point adjustment for B-spline curve approximation,” Comput. Aided Des. 36, 639–652 (2004).
[CrossRef]

Szeliski, R.

D. Scharstein, R. Szeliski, “Taxonomy and evaluation of dense two-frame stereo correspondence algorithms,” Int. J. Comput. Vis. 47, 7–42 (2002).
[CrossRef]

Terzopoulos, D.

H. Qin, D. Terzopoulos, “D-NURBS: a physics-based framework for geometric design,” IEEE Trans. Vis. Comput. Graph. 2, 85–96 (1996).

Teukolsky, S. A.

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Figures (17)

Fig. 1
Fig. 1

Space curve and its binocular perspective projections.

Fig. 2
Fig. 2

Jacobian matrix of formalism (15).

Fig. 3
Fig. 3

Simulated space curve.

Fig. 4
Fig. 4

Simulated stereo projections of the curve in Fig. 3.

Fig. 5
Fig. 5

3D reconstruction from projections in Fig. 4. (a) Point-based, (b) NURBS-based.

Fig. 6
Fig. 6

Corrupted stereo projections ( σ = [ 0.6 0.6 ] T ) of the curve in Fig. 3 and the back projections of the reconstructed curve.

Fig. 7
Fig. 7

Reconstructed curve from corrupted stereo projections.

Fig. 8
Fig. 8

Broken stereo projections of the curve in Fig. 3.

Fig. 9
Fig. 9

Reconstructed curve from data in Fig. 8.

Fig. 10
Fig. 10

Real stereo images of a fan model and back projections of the reconstruction result.

Fig. 11
Fig. 11

Reconstructed curves of the fan model.

Fig. 12
Fig. 12

Reconstructed line segments of fan model in Fig. 10.

Fig. 13
Fig. 13

Stereo images of bent wire objects and the back projections of the reconstructed curves. (a) Left image, (b) right image.

Fig. 14
Fig. 14

Reconstructed curves of bent wire objects.

Fig. 15
Fig. 15

Residual errors with iterations in reconstructing curves of fan model.

Fig. 16
Fig. 16

Residual errors with iterations in reconstructing curves of bent wire objects.

Fig. 17
Fig. 17

Trace of control points in the optimization of curve 2 in Fig. 10a.

Tables (5)

Tables Icon

Table 1 Errors of Reconstruction a with Sampling Differences, in Pixels

Tables Icon

Table 2 Errors of Reconstruction a with Corrupted Data, in Pixels

Tables Icon

Table 3 Errors of Reconstruction a with Fragmented Data, in Pixels

Tables Icon

Table 4 Reconstruction Errors a of the Fan Model, in Pixels

Tables Icon

Table 5 Reconstruction Errors a of the Bent Wire Objects, in Pixels

Equations (51)

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

C ( t ) = i = 0 m W i V i B i , k ( t ) i = 0 m W i B i , k ( t ) .
B i , 0 ( t ) { = 1 if u i t u i + 1 = 0 otherwise } ,
B i , k ( t ) = t u i u i + k u i B i , k 1 ( t ) + u i + k + 1 t u i + k + 1 u i + 1 B i + 1 , k 1 ( t ) .
C ( t ) = i = 0 m V i R i , k ( t ) ,
R i , k ( t ) = W i B i , k ( t ) j = 0 m W j B j , k ( t ) ,
C ( t ) = i V i R i , k ( t ) ,
V i = A V i + t A ,
T ( X ) = x S [ x 1 ] = [ T 1 T 2 T 3 ] [ X 1 ] ,
c ( t ) = i = 0 m w i v i B i , k ( t ) i = 0 m w i B i , k ( t ) ,
w i = W i T 3 [ V i 1 ]
[ v i ( R ) 1 ] T F ( R L ) [ V i ( L ) 1 ] = 0 ,
w i ( L ) w i ( R ) = T 3 ( L ) [ V i 1 ] T 3 ( R ) [ V i 1 ] .
W i = w i ( L ) T 3 ( L ) [ V i 1 ] ,
W i = w i ( R ) T 3 ( R ) [ V i 1 ] ,
min ( j 1 = 1 n 1 p j 1 ( L ) T ( L ) ( C ( { V i } , { W i } , t j 1 ) ) 2 + j 2 = 1 n 2 p j 2 ( R ) T ( R ) ( C ( { V i } , { W i } , s j 2 ) ) 2 )
t j 1 : j 1 = 1 , 2 , , n 1 ;
s j 2 : j 2 = 1 , 2 , , n 2 ; { V i } ; { W i } :
i = 0 , 1 , , m ,
min ( j 1 = 1 n 1 p j 1 ( L ) c ( { v i ( L ) } , { w i ( L ) } , t j 1 ) 2 + j 2 = 1 n 2 ) p j 2 ( R )
( 1 2 c ( { v i ( R ) } , { w i ( R ) } , s j 2 ) 2 )
t j 1 : j 1 = 1 , 2 , , n 1 ; s j 2 : j 2 = 1 , 2 , , n 2 ;
{ v i ( L ) } ; { v i ( R ) } ; { w i ( L ) } ; { w i ( R ) } : i = 0 , 1 , , m ;
[ v i ( R ) 1 ] T F ( R L ) [ v i ( L ) 1 ] = 0 , i = 0 , 1 , , m ;
w i ( L ) w i ( R ) = T 3 ( L ) [ V i 1 ] T 3 ( R ) [ V i 1 ] , i = 0 , 1 , , m .
T 3 ( L ) [ V i 1 ] = T 3 ( R ) [ V i 1 ] .
w i ( L ) = w i ( R )
y i ( L ) ( v ) = y i ( R ) ( v )
min ( j = 1 2 ( n 1 + n 2 ) f i 2 ) ,
x i ( L ) ( v ) , x i ( R ) ( v ) , y i ( v ) , i = 0 , 1 , m ,
t i 1 : i 1 = 1 , 2 , , n 1 , s i 2 : i 2 = 1 , 2 , , n 2 ,
f j = { x j ( L ) ( p ) i = 0 m x i ( L ) ( v ) R i , 3 ( t j ) , j = 1 , 2 , n 1 y j n 1 ( L ) ( p ) i = 0 m y i ( v ) R i , 3 ( t j n 1 ) , j = n 1 + 1 , 2 , 2 n 1 x j 2 n 1 ( R ) ( p ) i = 0 m x i ( R ) ( v ) R i , 3 ( s j 2 n 1 ) , j = 2 n 1 + 1 , 2 , 2 n 2 y j 2 n 1 n 2 ( R ) ( p ) i = 0 m y i ( v ) R i , 3 ( s j 2 n 1 n 2 ) , j = 2 n 1 + n 2 + 1 , 2 , 2 ( n 1 + n 2 ) ; }
ψ ( t ) = i = 0 m ψ i R i , k ( t )
ψ ( t ) ψ i = R i , k ( t ) .
[ J ( d ) T J ( d ) + λ I ] δ d = g ( d ) ,
t j 1 = { t 1 = t 0 r = 2 j 1 p r ( L ) p r 1 ( L ) r = 2 n 1 p r ( L ) p r 1 ( L ) , j 1 = 2 , 3 , , n 1 , }
s j 2 = { s 1 = s 0 r = 2 j 2 p r ( R ) p r ( R ) r = 2 n 2 p r ( R ) p r ( R ) , j 2 = 2 , 3 , , n 2 , }
min { y i , i = 0 , 1 , , m } ( j 1 = 1 n 1 ( y j 1 ( L ) ( p ) i = 0 m y i ( v ) R i , 3 ( t j 1 ) ) 2 + j 2 = 1 n 2 ( y j 2 ( R ) ( p ) i = 0 m y i ( v ) R i , 3 ( s j 2 ) ) 2 ) ,
min { x i ( L ) , i = 0 , 1 , , m } j 1 = 1 n 1 ( x j 1 ( L ) ( p ) i = 0 m x i ( L ) ( v ) R i , 3 ( t j 1 ) ) 2 ,
min { x i ( R ) , i = 0 , 1 , , m } j 2 = 1 n 2 ( x j 2 ( R ) ( p ) i = 0 m x i ( R ) ( v ) R i , 3 ( s j 2 ) ) 2 .
Φ ( t ) { X = 2 cos t Y = 2 sin t t [ 0 , 5 4 π ] Z = 2 ( t + 1 ) } .
T ( L ) = [ 100 0 0 1 0 100 0 0 0 0 1 1 ] , T ( R ) = [ 100 0 0 1 0 100 0 0 0 0 1 1 ] .
T ( L ) = [ 16.2 0.3 4.5 1455.9 1 16.6 2.5 773.7 0.3 0.2 1 538.1 ] ,
T ( R ) = [ 16.1 0.3 4.4 2042.6 0.4 16.5 2.6 980.8 0.3 0.2 1 510.3 ] .
T ( L ) = [ 2.5465 1.6975 14.6749 90.1389 2.8606 21.9431 0.5021 6414.6 0.0176 0.0065 0.0022 1 ] ,
T ( R ) = [ 8.3536 1.3601 8.5493 4096.7 2.1115 17.6109 0.9637 4891.9 0.0127 0.0052 0.0065 1 ] .
{ x ( t ) = i = 0 m T 1 [ V i 1 ] R i , k ( t ) i = 0 m T 3 [ V i 1 ] R i , k ( t ) y ( t ) = i = 0 m T 2 [ V i 1 ] R i , k ( t ) i = 0 m T 3 [ V i 1 ] R i , k ( t ) } ,
x ( t ) = i = 0 m T 1 [ V i 1 ] W i B i , k ( t ) i = 0 m T 3 [ V i 1 ] W i B i , k ( t ) .
x ( t ) = i = 0 m T 1 [ V i 1 ] T 3 [ V i 1 ] T 3 [ V i 1 ] W i B i , k ( t ) i = 0 m T 3 [ V i 1 ] W i B i , k ( t ) .
y ( t ) = i = 0 m T 2 [ V i 1 ] T 3 [ V i 1 ] T 3 [ V i 1 ] W i B i , k ( t ) i = 0 m T 3 [ V i 1 ] W i B i , k ( t ) .
c ( t ) = i = 0 m w i v i B i , k ( t ) i = 0 m w i B i , k ( t )
w i = W i T 3 [ V i 1 ] .

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