A new class of filters, called linear combination of weighted order statistic (LWOS) filters, is introduced as a mutilevel representation of the threshold extended Boolean (TEB) filters. This filter is a combination of L-filters and weighted order statistic (WOS) filters. Based on the observation that this filter possesses the threshold decomposition property, a representation of LWOS filters, named the canonical representation, is developed. It is shown that most nonrecursive filters having the threshold decomposition property can be thought of as special cases of the canonical LWOS filter. This result indicates that this class of LWOS filters encompasses a variety of filters which include median-type nonlinear filters and linear FIR filters. A procedure for designing an optimal canonical LWOS filter under the mean square error (MSE) criterion has been developed. The optimization of LWOS filters yields an FIR Wiener filter when the input is zero-mean Gaussian and a median-type nonlinear filter for non-Gaussian inputs. Experimental results in image restoration are presented to compare the relative performances of the LWOS and existing filters.
We derive an alternative multilevel representation of the TEB filters, showing that a TEB filter can be expressed a weighted sum of subfiltered outputs. This representation naturally leads to a subclass of TEB filter, called K-th order subset averaged (SA) filter that employs only those subfilters whose window size are less than or equal to K. They are called SA minimum (SAMIN) filters, SA maximum (SAMAX) filters, SA exclusive-OR (SAXOR) filters, and SA median (SAMED) filters according to the subfilters used. It is shown that the K-th order SAMAX, SAMIN, and SAXOR filters are all equivalent class, and the self-duality or its extension is closely related to the scale-preserving property. By designing K-th order SA filters under the mean square error criterion for various values of K and applying them to restore noisy signals, the tradeoff between the performance and complexity of this class of filters is examined.