Stack filters, stack smoothers, and mirrored threshold decomposition

Stack smoothers have received considerable attention in signal processing in the past decade. Stack smoothers define a large class of nonlinear smoothers based on positive Boolean functions (PBF) applied in the binary domain of threshold decomposition. Although stack smoothers can offer some advantages over traditional linear FIR filters, they are in essence smoothers lacking the flexibility to adequately address a number of signal processing problems that require bandpass or highpass filtering characteristics. In this paper, mirrored threshold decomposition is introduced, which, together with the associated binary PBF, define the significantly richer class of stack filters. Using threshold logic representation, a number of properties of stack filters are derived. Notably, stack filters defined in the binary domain of mirrored threshold decomposition require the use of double weighting of each sample in the integer domain. The class of recursive stack filters and the corresponding recursive weighted median (RWM) filters in the integer domain admitting negative weights are introduced. The new stack filter formulation leads to a more powerful class of estimators capable of effectively addressing a number of fundamental problems in signal processing that could not adequately be addressed by prior stack smoother structures