FILTERING IMAGE RECORDS USING WAVELETS AND THE ZAKAI EQUATION

Consider the problem of detecting and localizing a faint object moving in an ''essentially stationary'' background, using a sequence of two- dimensional low-SNR images of the scene. A natural approach consists o f ''digitizing'' each snapshot into a discrete set of observations, su fficiently (perhaps not exactly) matched to the object in question, th en tracking the object using an appropriate stochastic filter. The tra cking would be expected to make up for the low signal-to-noise ratio, thus allowing one to ''coherently'' process successive images in order to beat down the noise and localize the object. Thus, ''tracking'' he re does not refer to the ususal notion of detecting then tracking: rat her, we track in order to detect. The problem then becomes one of choo sing the appropriate image representation as well as the optimal (and necessarily nonlinear) filter. We propose exact and approximate soluti ons using wavelets and the Zakai equation. The smoothness of the wavel ets used is required in the derivation of the evolution equation for t he conditional density giving the filter, and their orthogonality make s it possible to carry out actual computations of the Ito- and change- of-gauge-terms in the algorithm effectively.