The present paper introduces a digital gray-scale morphological filtering technique that is based on postfiltering a given filter so the invariant class of the new filter is larger than the invariant class of the original filter. More specifically, the invariant class of the new filter contains the original invariant class together with all signals whose variation is below a chosen threshold. The postfiltering includes a single erosion and a single dilation by a one-parameter structuring element, the choice of parameter determining the resulting invariant class. While the methodology is quite general, the two applications considered pertain to moving averages with nonnegative weights and moving means, both of which are morphological filters. Both suppress noise in a signal and each possesses specific advantages and disadvantages. In brief, means tend to give better noise suppression while blurring edges, whereas medians preserve edges, while at the same time flattening small background variation in the underlying signal. The new one parameter family of filters derived from a moving average, called pseudoconvolutions, preserve steps beneath a predetermined threshold. The filters derived from medians, called pseudomedians, preserve uncorrupted low background variation. Because they preserve small variation while at the same time behaving like their mother filters, both filters are especially effective when the noise occurs in bursts, rather than uniformly across the signal.