A new aplproach to regularization methods for image processing is iltro(duced and developed using as a. vehicle the problem of coml)uting dleilse optical flow fields in an image sequence. Standard formulations of this problemi re(luire the computationally intensive solution of an ellil)tic pa.rtial differential equation which arises froin the often used "smootlhness constraint' type regularization. We utilize the illterl)reta.tion of' .he smoot.hness constraint as a. "fractal prior" to motivate regularization based on a. recently introduced class of multiscale stochastic models. The solution of the new l)robleln formulat.ion is computed with an efficient multiscale algorithm. Experiiments on several ima.ge sequences (demonstrate the sutl)sta.nftial compult.a.tional savings tlia.t. canll I)e achieved due to the fact tha.t the algorit.hm is non-iterative and in fact has a. per pixel computational complexity which is independent. of imiage size. Tlie new a.pp)l)roach also has a. numbler of other important. advantages. Specifically, nitltiresolution flow field estimates are availabl)le, allowing grea.t. flexibility in dealing wvithl the t.radeoff b.)et.ween resolution a.nd(l accuracy. NIultiscale error covariance information is also availabLle. which is of considerable use inll assessing the accuracy of the estimates. Ini particular, these error statistics calln be used as the b)asis for a rational procedure for dletermlining the spatially-varying optimal reconstruction resolution. Furthermore, if there are conipelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algoritlhms associated with the smoot. ness constraint l)roblemi formulation. Finally, the usefulness of otir appI)roach should extend t-o a. wide variety of ill-posed inverse problems inll which variational techniques seeking a. "smooth" solution are generally usedl. EDICS category 1.11. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Abstract Recently, a. framework for multiscale stochastic modeling was introduced based on coarse-to-fine scale-recursive dynamics defined on trees. This model class has some attractive characteristics which lead to extremely efficient, statistically optimal signal and image processing algorithms. In this paper, we show that this model class is also quite rich. In particular, we describe how 1-D Markov processes and 2-D Ma.rkov random fields (MRF's) can be represented within this framework. Markov processes in one-dimension and Markov randoml fields in two-dimensions are widely used classes of models for analysis, design and statistical inference. The recursive structure of 1-D Markov processes makes them simple to analyze, and generally leads to computationally efficient algorithms for statistical inference. On the other hand, 2-D MIR,F's are well known to be very difficult to analyze due to their non-causal structure, and thus their use typically leads to colmputationally intensive algoritlllhms for smoothing and parameter idlentification. Our multiscale representations are based on scale-recursive models, thus providing a framework for the development of new and efficient algorithms for Markov processes and MRF's. In 1-D, the representation generalizes the mid-point deflection construction of Brownian motion. In 2-D, we use a further generalization to a. "mid-line" deflection construction. Our exact representations of 2-D MRF's are of potentially high dimension, and this motivates a class of approximate models ba.sed on one-dim7ensional wavelet transforms. We demonstrate the use of these models in the context of texture representation and in particular, we show how they can be used as approximations for or alternatives to well-known MRF texture models: We illustrate how the quality of the representations varies a.s a. function of the underlying MRF and the complexity of the wavelet-based approximate representation.