Linearly separable stack-like architecture for the design of weighted order statistic filters with application in image processing

In this thesis, the problem of designing a weighted order statistic (WOS) filter which is approximately optimal in the mean absolute error (MAE) sense, for estimating a signal from the noise-corrupted observation of the same, is considered. A stack filter configuration described by a linearly separable positive Boolean function (LSPBF) and referred to as linearly separable stack (LSS) filter, has been traditionally used for this design. The design of WOS filters in the domain of LSS filter architecture is a constrained design, in the sense that the weights defining the LSPBF can assume only positive or zero values.This thesis introduces a new approach for the design of WOS filters which are approximately optimal for estimation in the MAE sense, by defining a more general type of filter configuration than that of LSS filters. This new type of architecture is characterized by a linearly separable Boolean function (LSBF), and is designated as linearly separable stack-like (LSSL) filter. In the case of LSSL filters, the weights may assume any value, positive or negative. It is shown that LSSL filters satisfy a new property referred to as generalized stacking property. It is established that due to this property, the fundamentals of the optimality theory that has been developed for the class of stack filters, remain valid in the framework of LSSL filters as well. It is demonstrated that in the multilevel signal domain, an LSSL filter architecture performs the operation of WOS filtering.An adaptive algorithm for the design of WOS filters in the domain of LSSL filter architecture is derived. Since the filter weights in this architecture can assume any real values, the algorithm is less constrained than that in the case of LSS filter architecture. Consequently, the proposed design is expected to give better results in the sense of mean absolute error in signal estimation problems. An implementation of the proposed design algorithm is constructed by using a binary-level LMS algorithm.The proposed design and implementation is applied to the problem of restoring images corrupted with impulsive noise. Simulation results show that the WOS filters designed with the new method provide better results compared with those obtained by using the LSS filter architecture.