| Nonlinear filters have been widely used in many multimedia and communication applications since they can effectively deal with noise-corrupted situations that involve impulsive, multiplicative or signal dependent noise. In particular, the class of nonlinear filters that have gained much popularity is the class based on Positive Boolean Functions (PBFs) defined in the binary domain of threshold decomposition, the so-called stack smoothers. Although stack smoothers can offer some advantages over traditional linear finite impulse response (FIR) filters, they are in essence smoothers lacking the flexibility to adequately address a number of signal processing and communication applications that require bandpass and highpass filtering characteristics.; In this dissertation, mirrored threshold decomposition is introduced which, together with the associated binary PBF, define the significantly richer class of stack filters. This new class of stack filters have been empowered not only with lowpass filtering characteristics but with bandpass and highpass filtering characteristics as well. Therefore, these filters can be effectively used in signal processing and communication applications where robust frequency selection is critical. Using the threshold logic representation, a number of properties of stack filters are derived, analyzed and studied. Furthermore, an adaptive optimization algorithm for the design of the proposed filters, and statistical tools to derive the output distribution function for the new class of stack filters are also developed.; Much like the stack smoother framework is used in the definition of weighted order statistic (WOS) and weighted median (WM) smoothers admitting positive weights, the new stack filter framework leads naturally to the definition of WOS and WM filters that admit positive and negative weights. In particular, we also focus in this dissertation on the development of recursive WM filter structure admitting negative weights. As the sample median is analogous to the sample mean, the proposed class of recursive WM filters is analogous to the class of infinite impulse response (IIR) linear filters. A novel “recursive decoupling” adaptive optimization algorithm for the design of recursive WM filters is also developed. Several properties of recursive WM filters are presented and a number of simulations are included to illustrate the advantages of recursive WM filters over their non-recursive counterparts and linear IIR filters. |
