Maximum-likelihood estimation of multiscale stochastic model parameters

We consider the class of multiscale stochastic models developed by Chou, Willsky and Benveniste (see IEEE Trans. on Automatic Control, vol.39, no.3, 1994) and by Luettgen, Karl, Willsky and Tenney (see IEEE Trans. Signal Processing, vol.41, no.12, 1993) for signal and image modeling. These are Markov random field models on trees that describe signals in a scale-recursive way. In particular, they are state-space models with dynamics with respect to scale and have available fast algorithms for smoothing data. We present a maximum likelihood (ML) procedure for estimating the state-space parameters of these models from data. The procedure uses the expectation-maximization (EM) algorithm to iteratively solve for the ML estimates. Each iteration consists of (1) an expectation step that takes advantage of the fast smoother available for these multiscale models and (2) a maximization step that is also fast. We present an example of using this procedure to identify parameters based on imagery data and, subsequently, to perform multiscale target detection.