Conversion between center-weighted medians and positive Boolean functions

The use of positive Boolean functions for the representation of weighted median filters simplifies the statistical and deterministic analysis of their properties and, in some situations, the hardware realization, too. To convert the weight coefficients of the weighted median to the corresponding positive Boolean function (PBF), there is usually used a well-known conversion algorithm. It includes generation of appropriate logical products, their consistent logical summing and minimizing of the resulting disjunctive form in the sense of Boolean algebra rules. However, in the case of center-weighted median (CWM) filters, concerning center weight only, the conversion can be simplified and minimal PBF are produced directly, without the above more complex steps. For that reason, we propose a new conversion algorithm between CWM and minimal PBF with complete proof of the method. Considering the center weight of CWM filters, the conversion is very fast and effective, and fewer operations must be performed in comparison with the traditional conversion algorithm.