Median filtering by threshold decomposition

Median filters are a special class of ranked order filters used for smoothing signals. Repeated application of the filter on a quantized signal of finite length ultimately results in a sequence, termed a root signal, which is invariant to further passes of the median filter. In this paper, it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals, filtering each binary signal to a root with a binary median filter, and then reversing the decomposition. This equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals. Since the effects of median filters on binary signals are well understood, this technique is a powerful new tool.