A Degenerate Conic-Based Method for a Direct Fitting and 3-D Pose of Cylinders with a Single Perspective View Christophe Doignon and Michel de Mathelin Abstract—In this paper, we address the problem of the pose recovery of a straight homogeneous circular cylinder (SHCC) from its apparent contour in a single 2-D image. In many real-world situations, one may encounter cylindrical objects especially in man-made structures Object models based on surfaces of revolution, in particular cylinders, are suitable for many fields, like the automatic assembly, the human motion capture and also in medical image-based robotic guidance. To model the geometry of a SHCC, we first introduce a singular matrix which represents this degenerate quadric and we show it can be expressed with the Pl¨ ucker coordinates of its symmetry axis. We demonstrate that the perspective projection of a SHCC is related to the pose parameters and for this model-based pose estimation problem, we present a degenerate conic-based fitting method which has some connections with the estimation of the Fundamental matrix. We provide a closed-form solution for the pose determination (4 dof). Compared to earlier works in this field, the proposed approach exhibits some geometric properties of SHCCs, it can deal with partial occlusions of apparent contours and it provides an efficient direct pose solution for the symmetry axis. Simulated data and real images are used to validate the fitting and pose computation. Finally, to highlight the effectiveness of the proposed method to deal with apparent contours in a poor structured environment, we apply this work to the localization of instruments used in laparoscopic surgery. I. I NTRODUCTION A. Motivations The recovery of the 3-D information from 2-D images is a fundamental problem in computer vision and for the past four decades, the model-based pose estimation has been widely addressed. Monocular scene analysis based on perspective projection can be successfully used to solve the determi- nation of 3-D object attitude if an a priori knowledge is available. The choice of the involved primitives is a key point since they must be robust to noise and efficiently detectable. In the man-made mechanical object recognition field, ge- ometrical features like straight lines, circles or cylinders are often encountered and for cylindrical objects, e.g., food cans, missiles, containers, pipes and circular pillars, quadrics of revolution can be thought as important components for object modeling, tracking or grasping. Such visual cue occurs in many areas like assembly, human motion capture [8], mobile robot guidance and in medical image analysis, e.g. for estimating geometrical transformations between fragments of a broken cylindrical structure [17]. Extremal contours and discontinuities are salient features for localisation purposes. To this end, the line fitting is usually C. Doignon and M. de Mathelin are with the LSIIT (UMR ULP-CNRS 7005), University of Strasbourg, Boulevard Brant, 67412 Illkirch, France. e-mail: {name}@lsiit.u-strasbg.fr applied on both side of the imaged cylinder axis for each set of classified contours. But in a complex environment, if many pixels of one of the two sets are occluded, the classification of apparent contours may fail, the resulting pose determination is then inaccurate or intractable. The current work aims at providing cylinder modeling and fitting for the overall apparent contours with the objectives of avoiding a tricky classification and to get a more robust results in presence of outliers. Finally, we expect this work bring contributions to the recovery and decoupling of the dof constrained to by visual tasks in robotics. B. Related work In the early 90s, shape from contour approaches have been developed in an attempt to determine constraints on a 3-D scene structure based on assumptions about the shape. The understanding of the relations between image contours geometry, the shape of the observed object and the viewing parameters is still a challenging problem and it is essential that special shapes are not represented by freeform surfaces without regard to their special properties, but treated in a way more appropriate to their simple nature. Explicit relations from occluding contours to model a curved three- dimensional object have been presented for generalized cylinders (GC), straight homogeneous generalized cylinders (SHGC) or surfaces of revolution (SOR) ( [2], [4], [9], [10], [16]). More recent works are based on the image contour of a cylinder cross-section. Puech et al. [11] used the image of two cross-sections to locate a straight uniform generalized cylinder in 3-D space and Shiu and Huang [14] solve the problem for a finite and known cylinder height, that is for 5 dof. Huang et al. [7] solve the pose determination of a cylinder through a reprojection transformation which may be viewed as a rectification. The computed transformation brings the camera optical axis to perpendicularly intersect the cylinder axis, which is then parallel to one of the two image axes. The new image (called ”canonical” image) is a symmetrical pattern which symplifies the pose computation. It is an interesting method which provides an analytical solution, including the recovery of the height of the cylinder. However, it requires a prior image transformation and errors for estimating the reprojection transformation may lead to a significant bias in the contours location of the resulting canonical image and consequently to the pose parameters. In a similar way, Wong et al. [18] take advantage of the invariance of surfaces of revolution (SOR) to harmonic homology and have proposed to recover the depth and the focal length (by assuming that the principal point is 2007 IEEE International Conference on Robotics and Automation Roma, Italy, 10-14 April 2007 FrC11.2 1-4244-0602-1/07/$20.00 ©2007 IEEE. 4220