Efficient multiscale regularization with applications to the computation of optical flow

Abstract

Absfruet-A new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence. Standard formulations of this problem require the computationally intensive solution of an elliptic partial differential equation that arises from the often used "smoothness constraint" 'yl". regularization. The interpretation of the smoothness constraint is utilized 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 efficient multiscale algorithm. Experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size. The new approach also has a number of other important advantages. Specifically, multiresolution flow field estimates are available, allowing great flexibility in dealing with the tradeoff 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.

A Wavelet-Based Method for Multiscale Tomographic Reconstruction

“…Combining the object approximation and detail definitions (13) and (14) with the scale recursive relationship (10) we see that the object itself satisfies the following scale recursive relationship, whereby the object approximation at the next finer scale is obtained from the approximation at the current (coarser) scale through the addition of the incremental detail at this scale, just as for the 1-D case treated in Section 2.3:…”

“…Multiresolution object estimates and the detail between them can then be easily obtained by using the resulting MAP coefficient estimates ~k at multiple scales in the multiscale object definitions (13) and (14), as was done previously in Section 3.…”

“…Such scale-space based prior models are desirable for a number of reasons, e.g. they have been shown to lead to extremely efficient scale-recursive algorithms [9,13] and they parsimoniously capture self-similar behavior, thus providing realistic models for a wide range of natural phenomenon. In particular, such self-similar models can be obtained by choosing the detail components ((m) (i.e.…”

“…We now have a natural multiscale object representation framework through (13), (14), and (15) that is similar in spirit to the FBP case (3). To complete the process and create multiscale object estimates from data we must find the multiscale object coefficients ~k (which contain all the information we need).…”

“…Using conventional techniques we would first have to reconstruct the entire field and then use post-processing to extract such features. A final compelling motivation for the use of multiresolution methods in estimation and reconstruction problems, is that they generally lead to extremely efficient algorithms, as in [13].…”