Estimation of Relative Camera Positions for Uncalibrated Cameras Richard I. Hartley G.E. CRD, Schenectady, NY, 12301. Abstract. This paper considers, the determination of internal camera pa- rameters from two views of a point set in three dimensions. A non-iterative algorithm is given for determining the focal lengths of the two cameras, as well as their relative placement, assuming all other internal camera param- eters to be known. It is shown that this is all the information that may be deduced from a set of image correspondences. 1 Introduction A non-iterative algorithm to solve the problem of relative camera placement was given by Longuet-Higgins ([4]). However, Longuet-Higgins’s solution made assumptions about the camera that may not be justified in practice. In particular, it is assumed implicitly in his paper that the focal length of each camera is known, as is the principal point (the point where the focal axis of the camera intersects the image plane). Whereas it is often a safe assumption that the principal point of an image is at the center pixel, the focal length of the camera is not easily deduced, and will generally be unknown for images of unknown origin. In this paper a non-iterative algorithm is given for finding the focal lengths of the two cameras along with their relative placement, as long as other internal parameters of the cameras are known. It follows from the derivation of the algorithm, as well as from counting degrees of freedom that this is all the information that may be deduced about camera parameters from a set of image correspondences. In this paper, the term magnification will be used instead of focal length, since it includes the equivalent effect of image enlargement. 2 The 8-Point Algorithm First, I will derive the 8-point algorithm of Longuet-Higgins in order to fix notation and to gain some insight into its properties. Alternative derivations were given in [4] and [5]. Since we are dealing with homogeneous coordinates, we are interested only in values determined up to scale. Consequently we introduce the notation A ≈ B (where A and B are vectors or matrices) to indicate equality up to multiplication by a scale factor. Image space coordinates will usually be given in homogeneous coordinates as (u,v,w) ⊤ . 2.1 Algorithm Derivation We consider the case of two cameras, one which is situated at the origin of object space coordinates, and one which is displaced from it. The two cameras may be represented by the transformation that they perform translating points from object space into image space coordinates. The two transformations are assumed to be (u,v,w) ⊤ =(x,y,z ) ⊤ (1)