A new class of rank-order-based filters, called lower-upper-middle (LUM) filters, is introduced. The output of these filters is determined by comparing a lower- and an upper-order statistic to the middle sample in the filter window. These filters can be designed for smoothing and sharpening, or outlier rejection. The level of smoothing done by the filter can range from no smoothing to that of the medianfilter. This flexibility allows the LUM filter to be designed to best balance the tradeoffs between noisesmoothing and signal detail preservation. LUM filters for enhancing edge gradients can be designed to be insensitive to low levels of additive noise and to remove impulsive noise. Furthermore, LUM filters do not cause overshoot or undershoot. Some statistical and deterministic properties of the LUM filters are developed, and a number of experimental results are presented to illustrate the performance. These experiments include applications to 1D signals and to images.