In this paper, we study three-dimensional motion estimation using optical flow. We construct a weighted quotient-form objective function that provides an unbiased estimator. Using this objective function with a certain projection operator as a weight drastically reduces the computational cost for estimation compared with using the maximum likelihood estimator. To reduce the variance of the estimator, we examine the weight, and we show by theoretical evaluations and simulations that, with an appropriate projection function, and when the noise variance is not too small, this objective function provides an estimator whose variance is smaller than that of the maximum likelihood estimator. The use of this projection is based on the knowledge that the depth function has a positive value (i. e., the object is in front of the camera) and that it is generally smooth.

This paper presents a theoretically best algorithm within the framework of our image noise model for reconstructing 3-D from two views when all the feature points are on a planar surface. Pointing out that statistical bias is introduced if the least-squares scheme is used in the presence of image noise, we propose a scheme called renormalization, which automatically removes statistical bias. We also present an optimal correction scheme for canceling the effect of image noise in individual feature points. Finally, we show numerical simulation and confirm the effectiveness of our method.

This paper describes a noise resistant algorithm for estimating 3-D rigid motion from optical flow. We first discuss the problem of constructing the objective function to be minimized. If a Gaussian distribution is assumed for the niose, it is well-known that the least-squares minimization becomes the maximum likelihood estimation. However, the use of this objective function makes the minimization procedure more expensive because the program has to go through all the points in the image at each iteration. We therefore introduce an objective function that provides unbiased estimators. Using this function reduces computational costs. Furthermore, since good approximations can be analytically obtained for the function, using them as an initial guess we can apply an iterative minimization method to the function, which is expected to be stable. The effectiveness of this method is demonstrated by computer simulation.