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Abstract. The main result of this paper is a procedure for self-calibration of a moving camera from instantaneous optical ow. Under certain assumptions, this procedure allows the ego-motion and some intrinsic parameters of the camera to be determined solely from the instantaneous positions and velocities of a set of image features. The proposed method relies upon the use of a di erential epipolar equation that relates optical ow to the ego-motion and internal geometry of the camera. The paper presents a detailed derivation of this equation. This aspect of the work may be seen as a recasting into an analytical framework of the pivotal research o f V i eville and Faugeras. 1 The information about the camera's ego-motion and internal geometry enters the di erential epipolar equation via two matrices. It emerges that the optical ow determines the composite ratio of some of the entries of the two matrices. It is shown that a camera with unknown focal length undergoing arbitrary motion can be self-calibrated via closed-form expressions in the composite ratio. The corresponding formulae specify ve ego-motion parameters, as well as the focal length and its derivative. An accompanying procedure is presented for reconstructing the viewed scene, up to scale, from the derived self-calibration data and the optical ow data. Experimental results are given to demonstrate the correctness of the approach. IntroductionOf considerable interest in recent y ears has been to generate computer vision algorithms able to operate with uncalibrated cameras. One challenge has been to reconstruct a scene, up to scale, from a stereo pair of images obtained by cameras whose internal geometry is not fully known, and whose relative orientation is unknown. Remarkably, such a reconstruction is sometimes attainable solely by consideration of corresponding points (that depict a common scene point) identied within the two images. A key process involved here is that of self-calibration, whereby the unknown relative orientation and intrinsic parameters of the cameras are automatically determined. 2,3 In this paper, we d e v elop a method for self-calibration of a single moving camera from instantaneous optical ow. Here self-calibration amounts to automatically determining the unknown instantaneous ego-motion and intrinsic parameters of the camera, and is analogous to self-calibration of a stereo vision set-up using corresponding points.The proposed method of self-calibration rests on a di erential epipolar equation that relates optical ow to the ego-motion and intrinsic parameters of the camera. A substantial portion of the paper is devoted to a detailed derivation of this equation. The di erential epipolar equation has as its counterpart in stereo vision

Abstract-Hartley's eight-point algorithm has maintained an important place in computer vision, notably as a means of providing an initial value of the fundamental matrix for use in iterative estimation methods. In this paper, a novel explanation is given for the improvement in performance of the eight-point algorithm that results from using normalized data. It is first established that the normalized algorithm acts to minimize a specific cost function. It is then shown that this cost function is statistically better founded than the cost function associated with the nonnormalized algorithm. This augments the original argument that improved performance is due to the better conditioning of a pivotal matrix. Experimental results are given that support the adopted approach. This work continues a wider effort to place a variety of estimation techniques within a coherent framework.

Many parameter estimation methods used in computer vi

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