Mirror shape recovery from image curves and intrinsic parameters: Rotationally symmetric and conic mirrors Nuno Gonc ¸alves and Helder Ara´ ujo Institute of Systems and Robotics University of Coimbra Pinhal de Marrocos - POLO II - 3030 Coimbra PORTUGAL Abstract This paper analyzes the problem of the estimation of the local mirror shape in a catadioptric imaging system. We propose a method to recover the 3D coordinates of mirror surface points as long as there are images of those points, i.e., as long as there is an image of a 3D geometric element that is reflected by those points. For that purpose the in- formation required is the image, the intrinsic parameters of the camera, and the 3D coordinates of 3 points in the scene. The estimation of the local shape can be used to calibrate the system even though that problem is not addressed in this paper. We address the problem of the shape recovery for conic shaped mirrors and rotationally symmetric mirrors. Experimental results for synthetic images are presented. 1. Introduction Panoramic and omnidirectional images are being increas- ingly used in many applications. New and interesting ap- plications are being developed. Omnidirectional images are obtained by combining cameras with mirrors. Many of these systems use configurations that assure that the projec- tion is central. In what concerns the type of mirror employed, rotation- ally symmetric conic mirrors are usually used. However, those mirrors provide central projection only for special po- sitions of the camera [2, 5, 9]. Suppose that the type of the mirror surface is not known. Can the mirror surface be recovered? Using which information? Previous work on this topic includes some works on reflectance models, po- larization, color, photometric characteristics and structured light [1, 6–8, 14, 15, 17]. Other approaches to the problem include using stereo or multiple views [7, 16] and also a moving observer or moving surfaces [10, 12]. In this paper we are interested in non-central projection systems with a perspective camera and a rotationally sym- metric mirror. The special case of conic mirrors is also ad- (nunogon,helder)@isr.uc.pt dressed. In this paper we propose a method to recover the mirror surface locally using the following a priori informa- tion: the image, the intrinsic parameters of the camera and 3 3D points in the scene. In the next section the problem of the mirror surface recovery is addressed and in section 3 we address the initial value problem. This problem results from the fact that the surface reconstruction is obtained from the integration of an ordinary differential equation. In section 4 the validity of the model is demonstrated with experimental results and then we draw the conclusions. 2. Mirror shape recovery In this section the problem of estimating the 3D mirror points corresponding to the image of a moving point in the scene is addressed. The point can describe any curve in the real scene. The corresponding curve on the image (af- ter the reflection) is tracked. The only a priori information required are the image of that very point, the intrinsic pa- rameters of the camera and the 3D coordinates of two or three points in the scene (depending on the model used). Let be the point on the mirror surface that we wish to recover. A curve in 3D space will be projected in the im- age plane and let the curve be parameterized by the variable which should not be confused with time (see figure 1). It is possible to express as the sum of its two perpendic- ular components (see figure 2). (1) where is the unitary reflected ray and is the inner product. Vector represents the distance vector from the origin of coordinates to the reflected ray (notice that . Let us now differentiate equation 1 with respect to the 1