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In this paper we consider robust techniques for estimating structure from motion in the uncalibrated case. We show how information describing the uncertainty of the data may be incorporated into the formulation of the problem, and we explore the situations in which this appears to be advantageous. The structure recovery technique is based on a method for self-calibrating a single moving camera from instantaneous optical flow developed in previous work of some of the authors. The method of self-calibration rests upon an equation that we term the differential epipolar equation for uncalibrated optical flow. This equation incorporates two matrices (analogous to the fundamental matrix in stereo vision) which encode information about the ego-motion and internal geometry of the camera. Any sufficiently large, non- degenerate optical flow field enables the ratio of the entries of the two matrices to be estimated. Under certain assumptions, the moving camera can be self-calibrated by means of closed-form expressions in the entries of these matrices. Reconstruction of the scene, up to a scalar factor, may then proceed using a straightforward method. The critical step in this whole approach is therefore the accurate estimation of the aforementioned ratio. To this end, the problem is couched in a least-squares minimization framework whereby candidate cost functions are derived via ordinary least squares, total least squares, and weighted least squares techniques. Various computational schemes are adopted for minimizing the cost functions. Carefully devised synthetic experiments reveal that when the optical flow field is contaminated with inhomogeneous and anisotropic Gaussian noise, the best performer is the weighted least squares approach with renormalization.
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