Discrete optimal filtering

We consider a problem of discrete optimal filtering: using the symbols of an observed binary sequence $\{\eta_{t}\}$, to construct a binary sequence $\{w_{t}^*\}$ which is in a sense the best estimate of a non-observable deterministic (non-random) binary sequence $\{\vartheta_{t}\}$ related to the sequence $\{\eta_{t}\}$ by the equalities
$$ \eta _{t}= \xi_{t}\oplus \vartheta _{t}, \qquad t=1,2,\ldots,N, $$
where $\{\xi_{t}\}$ is a random stationary binary sequence and $\oplus$ means the addition modulo 2.
We demonstrate an applications of the discrete optimal filtering in the cases where the sequence $\{\vartheta_{t}\}$ is an encoded black-and-white facsimile or television image transmitted through some channel with noise.