Weighted median filters and related nonlinear filters with applications in image processing

In the last few decades, a number of nonlinear filters were presented, especially in the electrical engineering literature. One of the most popular nonlinear filters is weighted median filtering. The main objective of this thesis is to analyse the statistical and deterministic properties of the weighted median filter and some related nonlinear filters with applications in image processing. The output distribution of the weighted median filter with independent but not identically distributed inputs is derived. The joint distributions of the outputs of the median and weighted median filters with independent but not identically distributed inputs on overlapping windows are obtained. As an application, these results are used to compute the power function for detecting a step change in images by using a range of existing window shapes for the median filter and weighting schemes for the weighted median filter. As an extension of max/median filtering, generalized max/median filtering is proposed. The relationship between them is given. The output distributions of generalized max/median filtering and some other modified median filtering are derived. Using these results, statistical evaluations are obtained to show how well these filters can suppress noise and preserve image details. Simulation results for these modified median filters are presented. Stack filtering is briefly reviewed. Some new expressions of the output distribution and joint distribution for stack filtering are derived. Applications of stack filtering are illustrated by examples. Statistical comparisons between the weighted median and modified median filters are presented. Exponential weighted median filters, a general parametric class of the weighted median filter, are evaluated for their ability to approximate the behaviour of the other filters considered. Statistical analyses are given to compare some well known weighted median filters. The final chapter reviews some methods for edge detection. A generalized real root signal and a generalized root signal of two dimensional median filtering are defined. A new edge detection method based on them is proposed and compared with conventional edge detectors