ROOT PROPERTIES OF MORPHOLOGICAL FILTERS

The characterization of the root signal sets of the most common digital morphological filters is presented in this paper. This has proven very crucial in understanding the utility and usefulness of these nonlinear operators. Important deterministic properties of root signals of morphological opening, closing, open-closing and clos-opening are derived leading to a complete characterization of output signals. Recall that the output signals of these morphological filters are root signals since the filters are idempotent. A system of difference equations is then derived to compute the number of root signals for these morphological filters by a structuring element of width k and for signals of length n with m quantization levels. The derivation is based on the state description for these root signals. Simplified recursive equations have been obtained for binary root signals. An application example where the root signals are used in block truncation coding (BTC) for image compression is discussed