Threshold Boolean Filters

Abstract

A class of nonlinear digital filters, called the threshold Boolean filter (TBF), is introduced. The TBF is defined by a Boolean function on the binary domain and is a natural extension of stack filters. Multilevel representations of a TBF corresponding to a Boolean function are derived; a TBF can be represented either as a sum of “local minimum-local maximum” terms or as an adaptive linear combination of ordered input data. It is shown that TBF’s may be neither translation invariant nor scale invariant and that any TBF can be expressed as a linear combination of stack filters. A subclass of TBF’s, called linearly separable (LS) TBF’s, defined by the threshold logic is introduced as a direct extension of weighted-order statistic (WOS) filters. Implementation and design of a TBF and an LS TBF is investigated. The procedure for designing TBF’s (LS TBF’s) is shown to be considerably simpler than designing stack (WOS) filters, and the former can outperform the latter at marginal increase in computational cost. Finally, experimental results are presented to illustrate the performance characteristics of TBF’s and LS TBF’s.