Eecient Multiscale Regularization with Applications to the Computation of Optical Flow 1

A new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical ow elds in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial diierential equation which arises from the often used \smoothness constraint" type regularization. We utilize the interpretation of the smoothness constraint as a \fractal prior" to motivate regularization based on a recently introduced class of multiscale stochastic models. The solution of the new problem formulation is computed with an eecient multiscale algorithm. Experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is non-iterative and in fact has a per pixel computational complexity which is independent of image size. The new approach also has a number of other important advantages. Speciically, multiresolution ow eld estimates are available, allowing great exibility in dealing with the tradeoo between resolution and accuracy. Multiscale error covariance information is also available, which is of considerable use in assessing the accuracy of the estimates. In particular, these error statistics can be used as the basis for a rational procedure for determining the spatially-varying optimal reconstruction resolution. Furthermore, if there are compelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of our approach should extend to a wide variety of ill-posed inverse problems in which variational techniques seeking a \smooth" solution are generally used.