Analysis and optimization of the threshold boolean filter and its extension

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. Multi-level 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 TBFs 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 TBFs, defined by a threshold logic and called the linearly separable (LS) TBF, 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 TBFs (LS TBFs) is shown to be considerably simpler than designing stack (WOS) filters, and the former can outperform the latter at marginal increase in computational cost. In this thesis, the TBF is further extended to a larger class of filters, called the extended TBF (ETBF), which encompasses linear FIR, linear combination of order statistic (LOS), and linear combination of weighted order statistic (LWOS) filters as well as the TBF. The optimal design of the ETBF under the mean square error (MSE) criterion is studied. It is shown that the optimization of the TBF can be formulated as a quadratic zero-one programming, and that the optimization of the ETBF as a classical quadratic problem. Thus, the linear and nonlinear filters under consideration can be analyzed and designed in the unified framework of the ETBF. Computer simulations and experiments with real images show that the ETBF has advantages of both linear and nonlinear filters.