When applying a filter to an image, it often makes practical sense to maintain the local brightness level from input to output image. This is achieved by normalizing the filer coefficients so that they sum to one. This concept is generally taken for granted, but is particularly important where non-linear filters such as the bilateral or and non-local means are concerned, where the effect on local brightness and contrast can be complex. Here we present a method for achieving the same level of control over the local filter behavior without the need for this normalization. Namely, we show how to closely approximate any normalized filter without in fact needing this normalization step. This yields a new class of filters. We derive a closed-form expression for the approximating filter and analyze its behavior, showing it to be easily controlled for quality and nearness to the exact filter, with a single parameter. Our experiments demonstrate that he un-normalized affinity weights can be effectively used in applications such as image smoothing, sharpening and detail enhancement.