We introduce a new class of linear-nonlinear combinational filters, called FIR stack hybrid (FSH) filters, which greatly enlarges the family of FIR median hybrid (FMH) filters. The optimal FSH filtering theory under the mean absolute error (MAE) criterion is studied. A general two-step method to synthesize optimal FSH filters is developed. In the first step, the probabilities needed in the optimal filter design are estimated based on images. In the second step, the linear program (LP) required to solve the best filter is avoided by using a good suboptimal algorithm that only needs data comparisons. A sufficient condition under which the suboptimal routine can result in optimal solutions is given, and this condition is shown to hold in most practical cases. To demonstrate the efficiency of the proposed approach, a group of FSH filters are synthesized for the standard image "Bridge." The task is to restore the image from impulsive or Gaussian noise. By checking the sufficient condition, each filter is found to be optimal in the minimum MAE sense. Testing results of some synthesized filters show that the optimal FSH filters can do a much better job than the corresponding FMH filters in both noise environments.