Finding the epipole is a useful starting point for uncalibrated reconstruction of motion and structure [ 2 , 8 , 9 , 3 ]. In this paper we propose a differential technique based on optical flow for estimating the location of the instantaneous epipole in a sequence of images. The advantage of differential techniques is that they work in the presence of small disparities, a situation in which methods based on correspondences tend to fail, as they need, in general, larger disparities. Representative examples of correspondence-based methods rely on estimates of the fundamental or the essential matrix [ 2 , 4 , 6 , 7 ], or higher-order tensors [ 1 , 11 ]. Our method assumes full perspective, imposing no restrictions at all on the viewing geometry. In the ideal, noise-free case the method is exact; with noisy sequences, since it is based on the minimization of a least-squares residual function, it proves accurate and robust. Although in practice the method is run with many points, in principle it only requires estimates of the optical flow at six points.
The key idea is to write the optical flow equations in terms of a quantity, termed generalized time-to-impact , which contains all the information available about the observed 3-D structure and motion, and five parameters which only depend on the 3-D motion. This is achieved by waiving the customary separation of the translational and rotational components of the optical flow. The epipole is the only point in which the generalized time-to-impact can be eliminated from the optical flow equations, leaving an expression that depends only on the intrinsic parameters and the observed 3-D motion. As discussed in Section 5 , this property has been used in the past [ 5 , 6 ] but in the calibrated case (intrinsic parameters known), and within different mathematical frameworks.
This work establishes a first bridge between optical flow methods, which determine the direction of translation assuming invariably the intrinsic parameters known, and uncalibrated approaches, in which knowledge of the epipole replaces that of calibration parameters.
The paper is organized as follows. First, we revisit the motion field equations and introduce the generalized time-to-impact (Section 2 ). We then use the resulting equations to compute the instantaneous epipole from optical flow estimates at six or more image points (Section 3 ), and report initial tests proving the consistency and robustness of the proposed method (Section 4 ). We conclude with a critical comparison of our method with the most relevant related work, and sketch the developments being pursued and planned (Section 5 ).
Adrian F Clark