| Accurate Statistical Models were recognized as essential for Computer Vision long ago. The main difficulties related to the application of such models are devising the model itself, computing the model parameters, applying the model efficiently, conditioning the data, evaluating the performance and applicability of the model and adapting powerful statistical techniques to the needs of the problem at hand. Using the problem of motion segmentation and tracking as testbed we developed a suite of statistical techniques that deal with all the above problems. Every attempt is made to keep the model simple by exploiting knowledge about the problem domain. The data is conditioned by preselecting good features using several techniques, together or separately, to select coherently moving features, select features that result in good segmentations in RANdom SAmpling Consensus (RANSAC) fashion, as well as ones based on Expectation Maximization (EM) clustering. We developed a noise model that accounts for misalignments, change in illumination, aliasing, etc that results in the computation of Mahalanobis distance. A well known hindrance to the efficient computation of Mahalanobis distance is that one needs O( k6) time for k x k patches but we developed an exact technique to do it in constant time. Since no model is useful without a method to compute the model parameters we developed Maximum Likelihood technique to compute the parameters offline. We used the EM method in two novel ways. In the first, we derived affine motion parameters directly from the clustering parameters obtained from EM using the optical flow as input and in the second we treat the flow as a hidden variable rather than input. We tested several algorithms built on this statistical framework on a variety of image sequences (indoor and outdoor, real and synthetic, constant and varying lighting, stationary and moving camera, some of them with known ground truth) with very good results even under difficult situations like varying illumination when the images were taken, moving background etc. As an application of our motion segmentation algorithm, we used it in optical flow computation, the resulting optical flow is more accurate than that obtained by a state of the art optical flow algorithm under extremely large inter-frame motion. |
