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. 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 is shown 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. The investigation of models on homogeneous trees is emphasized. The framework examined here allows for consideration, in a very natural way, of 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 is seen that these modeling paradigms appear to have promise in a far broader context than one might expect.>

A novel framework for multiscale statistical signal processing is introduced. Its purpose is to provide a statistical toolbox to analyze properties of signals involving time and scale simultaneously. Stationary processes over the dyadic tree are borrowed from harmonic analysts for this purpose, and a new partial order is proposed to model causality in scale. Autoregressive processes are investigated, and it is shown that Schur-Levinson parameterizations play a crucial role. As expected from the model, the restriction at a given scale (level) of a sample of such processes looks like a fractal, i.e. a random signal appearing similar whether seen from close or far away.>

New "instrumental" tests for detecting and diagnosing changes in the poles of a signal having unknown time-varying zeros are proposed. Numerical results for nonstationary scalar signals are given. The extension of these tests to the vector case may be used for vibration monitoring.

The problem of sequential detection of abrupt changes in the spectral behavior of a digital signal is addressed. This problem arises, for example, in the sequential segmentation of nonstationary digital signals such as speech, (EEG) electroencepholograms, (ECG) electrocardiogram, and geophysical signals. The limitations of a classical test will be emphasized, and some new algorithms will be presented and compared via a simulation study and from a theoretical point of view.

One possible approach for the analysis of nonstationary signals (segmentation, identification, filtering...) consists in the joint use of a suitable estimation algorithm together with a detector of sudden changes in a convenient statistical model, and sequential detectors of jump in mean are of particular interest in this framework. Three types of such detectors are investigated here, and some new algorithms are presented together with a comparative study based upon application to real data (geographysical signals). Finally, the updating scheme for the filter estimates after detection of a jump, is discussed.