Man yc lustering problems in Computer Vision group data points that are the result of statistical estimation and these data points can have a great amount of uncertainty . Motion segmentation by clustering of optical flo wi ss uch an example because very often optical flo wc annot be estimated without significant uncertainty .W ep resent a EM based clustering algorithm for incomplete data and we apply it to the problem of motion segmentation. The input to the algorithm are the velocity likelihoods and the number of clusters. The algorithm is mathematically very elegant because it does not impose any constraints on the velocity likelihood thus multi-modal likelihood is modeled without difficulty .C oupled with a sophisticated correlated image noise model, the algorithm can handle substantial deviations from the intensity constanc ya ssumption. Experiments with real image sequences sho we xcellent results. 1. Intr oduction The process of grouping pixels having similar motion characteristics is called motion segmentation .Ap opular approach for describing motion similarity within a se gment/layer [12] is by their optical flow. T he computation of optical flo wa tap ixel is an under-constrained problem and the classical solutions [2] almost exclusively use constraints from neighboring pixels by assuming one of the several smoothness constraints which usually do not hold on object boundaries. Motion segmentation based on optical flo wi st hus a chicken and egg problem: In order to compute flo wa ccurately ,w en eed to kno wm otion boundaries but locating the motion boundaries amounts to doing segmentation which requires flo wa si nput. Our approach subscribes to the paradigm [4] that does motion segmentation without computing the full optical flo wf irst.

The discontinuities and the large image displacements pose some of the hardest problems in flow estimation. This paper uses a set of filters that change shape to avoid blending of the constraints across discontinuity boundaries. This is done by using an incompatibility measure of the constraints of neighbouring pixels. The algorithm is embedded in a coarse to fine multigrid scheme to address the problem of large displacements. We report results on real images which show that the algorithm works very well.

Estimating 3-D motion parameters from three-dimensional point correspondences is considered. A minimum errors-in-variables formulation of the motion estimation problem is developed to obtain stable and accurate solutions and is connected with the ordinary least squares formulation of the problem. When covariance matrices of 3-D points are known, a closed-form matrix-weighted solution is derived for estimating the motion parameters in the presence of independent and identically distributed (i.i.d.) Gaussian noise. In a general case, a closed-form approximate solution is presented. The experimental results demonstrate that the authors' solutions are accurate and reliable in the presence of noise with different deviations.>

The problem of estimating 3D motion in an optimal manner using correspondences of features in two views is analyzed. The importance of having an optimal estimator is twofold: first, for the estimation itself and, second, for the bound it offers on how much sensitivity one can expect from a two-frame, point-based motion algorithm. The optimal estimator turns out to be nonlinear, and for that reason, techniques that provide very good initial guesses for the iterative computation of the optimal estimator are developed.>

One of the problems associated with any approach to the structure from motion problem using point correspondence, i.e. recovering the structure of a moving object from its successive images, is the use of least squares on dependent variables. We formulate the problem as a quadratic minimization problem with a non-linear constraint. Then we derive the condition for i,he solution to be optimal under the assumption of Gaussian noise in the input, in the Maximum Likelihood Principle sense. This constraint minimization reduces to the solution of a nonlinear system which in the presence of modest noise is easy to approximate. We present two efficient ways to approximate it and we discuss some inherent limitations of the structure from motion problem when two frames are used that should be taken into account in robotics applications that involve dynamic imagery. In addition, our formulation introduces a framework in which previous works on the subject become special cases.