Reconstruction of quadric surfaces from disparity measurements

Techniques for the reconstruction of quadric surfaces (cones and cylinders) from disparity measurements obtained by a stereo vision system are presented. For the initial estimation of the geometrical parameters of cones, a regression based approach is adopted from Newman, Flynn, and Jain, CVGIP: Image Understanding, Vol. 58, pp. 235-249]. A new method based on surface normal estimation is developed for the cylindrical case. For the reenement of the parameter estimates, the Levenberg-Marquardt nonlinear least squares algorithm is used as in Flynn and Jain, Proc. 1988 IEEE Comput. Soc. Conf. Computer Vision and Pattern Recognition, pp. 261-267] but two mistaken formulas (one for a cone and the same for a cylinder) are corrected. The prior approaches for quadrics in stereo vision have been more or less approximative. Our contribution is to t the exact quadric shape to disparity data using the existing methods for range data. The disparity space ooers a convenient frame to analyze the diierences between the measurements and the reconstructed model. In stereo vision, much attention has been paid to the determination of binocular disparity from two images taken simultaneously from diierent viewpoints. The nal step in the process is to build a model for a 3-D object surface from the disparity measurements. In this paper, the reconstruction of quadric surfaces is considered. The prior approaches for quadrics in stereo vision have been more or less approximative. The paper lls this gap by tting the exact quadric shape to disparity data. We rst introduce the concept of disparity and describe the measuring system used for disparity measurements. Related work in stereo vision and in 3-D computer vision generally are then reviewed. Our approach for quadrics applies existing techniques for range data to the special case of disparity data. The method for quadric surfaces is presented in Section 2. This section is divided into subsections according to the diierent stages of the procedure starting from rough data segmentation and formulation of the problem, followed by parameter initial estimation and reenement, and ending up to model building and analysis. Section 3 contains the conclusions and Section 4 acknowledgments. The paper should be considered as a part of a larger work where the disparity map is used as a basic tool during the modeling process. In the course of our presentation, we consequently perform some stages of the procedure in the disparity frame although they could be performed in a Cartesian …