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 2D low S/N ratio images of the scene. A natural approach consists of “digitizing” each snapshot into a discrete set of observations, sufficiently (perhaps not exactly) matched to the object in question, then tracking the object using an appropriate stochastic filter. The tracking would be expected to make up for the low S/N ratio, thus allowing one to “coherently” process successive images in order to beat down the noise and localize the object. The problem then becomes one of choosing the appropriate image representation as well as the optimal (and necessarily nonlinear) filter. We propose exact and approximate solutions using wavelets and the Zakai equation. The smoothness of the wavelets used is required in the derivation of the evolution equation for the conditional density giving the filter, and their orthogonality makes it possible to carry out actual computations of the Ito- and change-of-gauge-terms in the algorithm effectively