Threshold decomposition of soft morphological filters

Soft morphological filters are used for smoothing signals with the advantage of being less sensitive to additive noises and to small variations in the shape of the objects to be transformed as compared to standard morphological filters. In this paper, we present the properties of soft morphological operations and the new definitions of binary soft morphological operations. It is shown that soft morphological filtering an arbitrary signal is equivalent to decomposing the signal into binary signals, filtering each binary signal with a binary soft morphological filter, and then reversing the decomposition. This equivalence allows problems in the analysis and the implementation of soft morphological operations in real time by using only logic gates for binary signals instead of sorting numbers.