## Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

JOSA A, Vol. 22, Issue 9, pp. 1746-1762 (2005)

http://dx.doi.org/10.1364/JOSAA.22.001746

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### 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

**OCIS Codes**

(000.3870) General : Mathematics

(100.2960) Image processing : Image analysis

(100.3010) Image processing : Image reconstruction techniques

(100.3190) Image processing : Inverse problems

(150.5670) Machine vision : Range finding

(150.6910) Machine vision : Three-dimensional sensing

**Citation**

Yi Jun Xiao and Y. F. Li, "Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model," J. Opt. Soc. Am. A **22**, 1746-1762 (2005)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-22-9-1746

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### References

- O. Faugeras, Three-dimensional Computer Vision: a Geometric Viewpoint (MIT Press, 1993).
- D. Scharstein and R. Szeliski, "Taxonomy and evaluation of dense two-frame stereo correspondence algorithms," Int. J. Comput. Vis. 47, 7-42 (2002).
- M. Z. Brown, D. Burschka, and G. D. Hager, "Advances in computational stereo," IEEE Trans. Pattern Anal. Mach. Intell. 25, 993-1008 (2003).
- M. Pollefeys and 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.
- M. H. Lin and C. Tomasi, "Surfaces with occlusions from layered stereo," IEEE Trans. Pattern Anal. Mach. Intell. 26, 1073-1078 (2004).
- O. Faugeras and R. Keriven, "Variational principles, surface evolution, PDEs, level-set methods, and the stereo problem," IEEE Trans. Image Process. 7, 336-344 (1998).
- N. Ayache and F. Lustman, "Trinocular stereo vision for robotics," IEEE Trans. Pattern Anal. Mach. Intell. 13, 73-85 (1991).
- D. Q. Huynh and R. A. Owens, "Line labeling and region-segmentation in stereo image pairs," Image Vis. Comput. 12, 213-225 (1994).
- G. Pajares, J. M. Cruz, and J. A. Lopez-Orozco, "Stereo matching using Hebbian learning," IEEE Trans. Syst. Man Cybern. 29, 553-559 (1999).
- N. Ayache and B. Faverjon, "Efficient registration of stereo images by matching graph descriptions of edge segments," Int. J. Comput. Vis. 1, 107-131 (1987).
- S. D. Ma, "Conic-based stereo, motion estimation, and pose determination," Int. J. Comput. Vis. 10, 7-25 (1993).
- L. Quan, "Conic reconstruction and correspondence from two views," IEEE Trans. Pattern Anal. Mach. Intell. 18, 151-160 (1996).
- L. Li and 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.
- M. H. An and C. N. Lee, "Stereo vision based on algebraic curves," presented at the 13th International Conference on Pattern Recognition, Vienna, Austria, August 25-30, 1996.
- L. Robert and 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.
- K. Kedem and 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.
- A. T. Brant and 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.
- N. M. Nasrabadi, "A stereo vision technique using curve segments and relaxation matching," IEEE Trans. Pattern Anal. Mach. Intell. 14, 566-572 (1992).
- N. M. Nasrabadi and Y. Liu, "Stereo vision correspondence using a multi-channel graph matching technique," Image Vis. Comput. 7, 237-245 (1989).
- G. Pajares, J. M. Cruz, and J. A. Lopez-Orozco, "Relaxation labeling in stereo image matching," Pattern Recogn. 33, 53-68 (2000).
- J. Porrill and S. Pollard, "Curve matching and stereo calibration," Image Vis. Comput. 9, 45-50 (1991).
- Y. Shan and Z. Zhang, "New measurements and corner-guidance for curve matching with probabilistic relaxation," Int. J. Comput. Vis. 46, 157-171 (2002).
- C. Schmid and A. Zisserman, "The geometry and matching of lines and curves over multiple views," Int. J. Comput. Vis. 40, 199-233 (2000).
- J. Sato and R. Cipolla, "Quasi-invariant parameterisations and matching of curves in images," Int. J. Comput. Vis. 28, 117-136 (1998).
- R. Berthilsson, K. Astrom, and A. Heyden, "Reconstruction of general curves using factorization and bundle adjustment," Int. J. Comput. Vis. 41, 171-182 (2002).
- Y. J. Xiao, M. Y. Ding, and 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).
- D. F. Rogers and N. G. Fog, "Constrained B-spline curve and surface fitting," Comput. Aided Des. 21, 641-648 (1989).
- S. Hu, Y. Li, T. Ju, and X. Zhu, "Modifying the shape of NURBS surfaces with geometric constraints," Comput.-Aided Des. 33, 903-912 (2001).
- H. Qin and D. Terzopoulos, "D-NURBS: a physics-based framework for geometric design," IEEE Trans. Vis. Comput. Graph. 2, 85-96 (1996).
- F. S. Cohen and 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).
- L. Piegl, "On NURBS: a survey," IEEE Comput. Graphics Appl. 13(1), 55-71 (1991).
- J. Aloimonos, "Perspective approximations," Image Vis. Comput. 8, 179-192 (1990).
- N. Ayache and 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.
- W. Ma and J.-P. Kruth, "NURBS curve and surface fitting for reverse engineering," Int. J. Adv. Manuf. Technol. 14, 918-927 (1998).
- D. W. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," J. Soc. Ind. Appl. Math. 11, 431-441 (1963).
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992), pp. 681-688.
- P. J. Besl and N. D. McKay, "A method for registration of 3-d shapes," IEEE Trans. Pattern Anal. Mach. Intell. 14, 239-256 (1992).
- R. M. Haralick and L. G. Shapiro, Computer and Robot Vision (Addison-Wesley, 1992), Vol. I, pp. 233-236.
- H. P. Yang, W. P. Wang, and J. G. Sun, "Control point adjustment for B-spline curve approximation," Comput. Aided Des. 36, 639-652 (2004).
- T. J. Cham and 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).

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