For given binary image and degradation processes, an optimal mean-absolute-error translation- invariant filter can be designed via the representation of such filters as a union of morphological hit-or-miss transforms. The present paper investigates a different optimization methodology by representing translation-invariant filters as differencing filters. Rather than employing structuring templates to build the entire output image, as is done with direct hit-or- miss representation, differencing filters only employ templates that locate value flips (black-to- white or white-to-black). Differencing filters play a central role in several digital document processing tasks. It is shown how differencing filters are statistically designed and applied for image restoration and resolution conversion.