This paper describes a noise resistant algorithm for recovering the three-dimensional motion of a rigid object from optical flow. First, it is shown that in the absence of noise three-demensional motion can be obtained exactly by a linear algorithm except in the special case in which the surface of the object is on a general quadratic surface passing through the viewpoint, and the normal vector of the surface at the viewpoint is perpendicular to the translation velocity vector. In the presence of noise, an evaluation function is introduced based on the least squares method. It is shown, however, that the solution which minimizes the evaluation function is not always optimal due to statistical bias. To deal with this problem, a method to eliminate the statistical bias in the evaluation function is proposed for zero mean white noise. Once the statistical bias is eliminated, the solution of the linear algorithm coincides with the correct solution by means of expectation. In this linear algorithm, only the eigenvector corresponding to the zero eigenvalue of a 33 matrix is necessary to find the translational velocity. Once the translational velocity is obtained, the rotational velocity can be computed directly. This method is also shown to be noise resistant by computer simulation.

A new approach is presented for the detection and computation of a two-dimensional motion field in image sequences. This computational model has a multi-channel motion detector and an optimal motion selector. In the motion detector, each channel has an inherent spatial resolution. The detector computes a two-dimensional motion field by the gradient-based method in parallel. The motion selector compares those candidates of the motion field by a correlation value of the intensity patterns hierarchically arranged from low to high resolution. It then determines the most probable motion for each image point. Experimental results are shown for synthetic images. This model can detect more reliable motion fields than the conventional one-chanel model.

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.

The present paper clarifies that the variance of the maximum likelihood estimator (MLE) of a parameter does not reach the Cramer-Rao lower bound (CRLB) when fitting a straight-line to observed two-dimensional data. In addition, the variance of the MLE can be shown to be equal to the CRLB only if observed noise reduces to a one-dimensional Gaussian variable. For most practical applications, it can be assumed that noise is added only to the range direction. In this case, the MLE is clearly an asymptotically effective estimator. However, even if we assume such a noise model, ML line-fitting to the data from many points of view has a high computational cost. The present paper proposes an alternative fitting method in order to provide a cost-effective unbiased estimator. The reliability of this new method is analyzed statistically and by computer simulation.

Since the detection of optical flow (two-dimensional motion field on an image) from image sequences is essentially an ill-posed problem, most of the conventional methods use a smoothness constraint for optical flow heuristically and detect reasonable optical flow. However, little discussion exists regarding the degree of smoothness. Furthermore, to recover the relative three-dimensional motion and depth between a camera and a rigid object, in general at first, the optical flow is detected without a rigid motion constraint, and next, the motion and depth are estimated using the detected optical flow. Rigorously speaking, the optical flow should be detected with such a constraint, and consequently three-dimensional motion and depth should be determined. To solve these problems, in this paper, we apply a parametric model to an optical flow, and construct an estimation algorithm based on this model.

In conventional methods for detecting vanishing points and vanishing lines, the observed feature points are clustered into collections that represent different lines. The multiple lines are then detected and the vanishing points are detected as points of intersection of the lines. The vanishing line is then detected based on the points of intersection. However, for the purpose of optimization, these processes should be integrated and be achieved simultaneously. In the present paper, we assume that the observed noise model for the feature points is a two-dimensional Gaussian mixture and define the likelihood function, including obvious vanishing points and a vanishing line parameters. As a result, the above described simultaneous detection can be formulated as a maximum likelihood estimation problem. In addition, an iterative computation method for achieving this estimation is proposed based on the EM (Expectation Maximization) algorithm. The proposed method involves new techniques by which stable convergence is achieved and computational cost is reduced. The effectiveness of the proposed method that includes these techniques can be confirmed by computer simulations and real images.

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.

We prove that the maximum likelihood estimator for estimating 3-D motion from noisy optical flow is not optimal", i.e., there is an unbiased estimator whose covariance matrix is smaller than that of the maximum likelihood estimator when a Gaussian noise distribution is assumed for a sufficiently large number of observed points. Since Gaussian assumption for the noise is given, the maximum likelihood estimator minimizes the mean square error of the observed optical flow. Though the maximum likehood estimator's covariance matrix usually reaches the Cramér-Rao lower bound in many statistical problems when the number of observed points is infinitely large, we show that the maximum likelihood estimator's covariance matrix does not reach the Cramér-Rao lower bound for the estimation of 3-D motion from noisy optical flow under such conditions. We formulate a superior estimator, whose covariance matrix is smaller than that of the maximum likelihood estimator, when the variance of the Gaussian noise is not very small.