MODELING AND ESTIMATION OF MULTIRESOLUTION STOCHASTIC-PROCESSES

An overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing. As described, a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image. In particular, it will be seen how the wavelet transform directly suggests such a modeling paradigm. This perspective then leads directly to the investigation of several classes of dynamic models and related notions of "multiscale stationarity" in which scale plays the role of a time-like variable. Focus is primarily on the investigation of models on homogenous trees. In particular, the elements of a dynamic system theory on trees are described and two notions of stationarity are introduced. One of these leads naturally to the development of a theory of multiscale autoregressive modeling including a generalization of the celebrated Schur and Levinson algorithms for order-recursive model building. The second, weaker notion of stationarity leads directly to a class of state space models on homogenous trees. Several of the elements of the system theory for such models are described and also the natural, extremely efficient algorithmic structures for optimal estimation are described that these models suggest: one class of algorithms has a multigrid relaxation structure; a second uses the scale-to-scale whitening property of wavelet transforms for our models; and a third leads to a new class of Riccati equations involving the usual predict and update steps and a new "fusion" step as information is propagated from fine to coarse scales. This framework allows for consideration, in a very natural way, the fusion of data from sensors with differing resolutions. Also, thanks to the fact that wavelet transforms do an excellent job of "compressing" large classes of covariance kernels, it will be seen that these modeling paradigms appear to have promise in a far broader context than one might expect.