Analysis of the properties of soft morphological filtering using threshold decomposition

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 the numbers. The architectures of logic-gate implementation of soft morphological operations are also presented. Furthermore, unlike standard morphological filters, the soft morphological closing and opening are in general not idempotent. We develop the conditions and properties for a new class of idempotent soft morphological filters