Contour Motion Estimation Using Relaxation Matching with a Smoothness Constraint on the Velocity Field

We estimate optical flow from a sequence of 2-D images by computing the velocity field along moving contours in the scene. This new approach is different from others in that it combines displacements computed by feature matching with a smoothness constraint on the second derivative of velocity. First, we use our previously reported relaxation matching technique to find correspondences between contour features in adjacent image frames. Displacements for discrete points along the contours are interpolated from the magnitudes and directions of neighboring matched points. The displacements so-computed are used as initial estimates for the velocity (magnitude and direction) along contours. The final estimated velocities are required to yield components which are close in a least-squares sense to these initial velocity magnitudes, when projected along the same directions. We also constrain the second derivative of velocity to be a minimum when integratedalong the contour, leading to a unique solution for the motion of a straight line undergoing an affine transformation. The second-derivative constraint gives better results than the first-derivative constraint in this case. Our method also gives better results for most second-order flows. In cases where it does not, a combination of first- and second-derivative constraints can be used. Computation of velocities at discrete points along the contour is achieved by solving linear equations via the conjugate gradient algorithm. The image flow technique is applied to examples of rigid and nonrigid motion.