Optimal nonlinear pattern restoration from noisy binary figures

A mathematical framework for the solution of statistical inference problems on a class of random sets is proposed. It is based on a new definition of expected pattern. The least-mean-difference estimator (restoration filter) is proved, under certain conditions, to be equivalent to the minimization of the measure of size (area) of the set-difference between the original pattern and the expected pattern of the estimated (restored) pattern. Consequently, it is proved that, under certain conditions, if the estimator (restoration filter) is unbiased, then it is the least mean difference estimator (restoration filter).<>