Tuning the smoothness of the recursive median filter

The median filter is a special case of nonlinear filters used for smoothing signals. Since the output of the median filter is always one of the input samples, it is conceivable that certain signals could pass through the median filter unaltered. These signals define the signature of a filter and are referred to as root signals. Median filters are known to possess the convergence property, meaning that by repeating median filtering a root signal will be found, starting from any input signal. By associating the nonlinear operation of median filtering with a two terms cost function, an optimization process that minimizes that function is obtained. Cost functions of the same type are associated with different recursive median filtering schemes by replacing the actual values inside the filter's window with the original signal. The convergence behavior of these filters and their smoothness are studied. By changing the positions of the replacements during filtering, a tuning effect of the smoothness is obtained. Simulation results show that the proposed filtering schemes provide improved performance over the standard recursive median filter, succeeding in preserving. small details and fine textures.