A class of multiscale stochastic models based on scale-recursive dynamics on trees has previously been introduced. Theoretical and experimental results have shown that these models provide an extremely rich framework for representing both processes which are intrinsically multiscale, e.g., 1/f processes, as well as 1D Markov processes and 2D Markov random fields. Moreover, efficient optimal estimation algorithms have been developed for these models by exploiting their scale-recursive structure. The authors exploit this structure in order to develop a computationally efficient and parallelizable algorithm for likelihood calculation. They illustrate one possible application to texture discrimination and demonstrate that likelihood-based methods using the algorithm achieve performance comparable to that of Gaussian Markov random field based techniques, which in general are prohibitively complex computationally.

A class of multiscale models describing stochastic processes indexed by the nodes of a tree has recently been introduced by Chou et al. (1994). Experimental and theoretical results indicate that this class of models is quite rich, and moreover these models lead to extremely efficient algorithms for optimal estimation based on noisy observations. This motivates further algorithmic development, and in particular, in this paper we present a likelihood calculation algorithm for this class of multiscale models. That is, we consider the problem of computing the log of the conditional probability of a set of data assuming that they correspond to a particular multiscale model. We exploit the structure of the multiscale models to develop an efficient, scale recursive algorithm that allows for multiresolution data and parameters which vary in both space and scale. We illustrate one possible application of the algorithm to a texture classification problem in which one must choose from a given set of models that model which best represents or most likely corresponds to a given set of random field measurements. Texture modeling with Gaussian Markov random field (GMRF) models is well documented. One difficulty in using GMRF models, however, is that the calculation of likelihoods may be prohibitively complex computationally if there is an irregular sampling pattern. It is shown here that GMRF models can be represented within our multiscale model class which allows us to approximately compute likelihoods for GMRF models based on measurements over arbitrarily sampled regions. As we demonstrate in the context of texture discrimination problems, the multiscale approach not only leads to computationally efficient implementations, but also to virtually the same performance as the optimal GMRF-based likelihood ratio test. We discuss further applications in the area of synthetic aperture radar imagery processing.>

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. 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, the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation. Finally, the usefulness of the approach should extend to a wide variety of ill-posed inverse problems in which variational techniques seeking a "smooth" solution are generally used.