Except in the case of very low-bit signals (images), unconstrained design of mean-square-error optimal digital window-based filters from sample signals is hampered by the inability to obtain sufficient data to make acceptably precise estimates of the large number of conditional expectations required for the filter. Even under typical nonlinear constraints such as increasingness or iterative decomposition, estimation remains intractable. This paper mitigates the estimation dilemma by windowing in the range, as well as in the domain. At each point, the signal is viewed through an aperture, which is the product between a domain window and a gray-range window that is chosen according to the signal values in the domain window. Signal values above and below the range window are projected into the top and bottom of the aperture, respectively. This projection compresses the probability mass of the observed signal into a smaller set of variables in such a way as not to alter the mass of observations within the aperture (which carry the most mass) and minimally alter the mass of those outside the aperture. Experiments show that, as opposed to commonly employed increasing nonlinear filters, aperture filters can outperform linear filters for deblurring, especially in the restoration of edges. This paper addresses several issues concerning aperture filters: positioning of the aperture, the effect of range constraint on probability mass, the size of the aperture relative to estimation precision and the amount of training data, estimation of conditional probabilities, and representation by decision trees. A sampling is provided of the many experiments carried out to study the effects of aperture filters on corruption by additive noise and blurring.
