Multiresolution stochastic models, data fusion, and wavelet transforms

Abstract In this paper we describe and analyze a class of multiscale stochastic processes which are modeled using dynamic representations evolving in scale based on the wavelet transform. The statistical structure of these models is Markovian in scale, and in addition the eigenstructure of these models is given by the wavelet transform. The implication of this is that by using the wavelet transform we can convert the apparently complicated problem of fusing noisy measurements of our process at several different resolutions into a set of decoupled, standard recursive estimation problems in which scale plays the role of the time-like variable. In addition we show how the wavelet transform, which is defined for signals that extend from −∞ to +∞, can be adapted to yield a modified transform matched to the eigenstructure of our multiscale stochastic models over finite intervals. Finally, we illustrate the promise of this methodology by applying it to estimation problems, involving single and multi-scale data, for a first-order Gauss-Markov process. As we show, while this process is not precisely in the class we define, it can be well-approximated by our models, leading to new, highly parallel and scale-recursive estimation algorithms for multi-scale data fusion. In addition our framework extends immediately to 2D signals where the computational benefits are even more significant.