In this letter, we provide a method for deriving the output distribution function of any recursive stack filter. In particular, we give the output distribution of the recursive median filter. The method used relies on finite automata and Markov chain theory. The distribution of any recursive stack filter is expressed as a vector multiplication of steady-state probabilities by the truth table vector of the Boolean function defining the filter. Index Terms---Finite automata, Markov chain, output distribution, recursive median filter, stack filter. I. INTRODUCTION A CLASS of nonlinear filters useful for many signal and image processing applications is the class of median based filters. The success of these filters is due to their ability to preserve edges as well as efficiently attenuate noise in impulsive noise environments (see [4] and [5] for overviews). Many median-type filters, such as weighted median filters and order statistic filters, are special cases of stack filters [6]. If...

Abstract: This paper examines approaches to the realisation of nonlinear filters as used in signal and image processing. The design of adaptive nonlinear processors is examined and their application as adaptive equalisers to alleviate bandlimiting, distortion and interference in a typical communications channel is investigated. This paper re-examines the equalisation process as one which seeks to correctly classify the channel output into one of a finite and known alphabet of symbols encompassing the data at the channel input. The optimal solution for this classification problem is shown to be inherently nonlinear. Several nonlinear structures are examined, which allow much more complex classification boundaries, and provide greatly enhanced performance for the nonlinear filter over the more conventional linear filter. Finally the use of a nonlinear predictor is investigated for time series analysis.

This dissertation is dedicated to Sandy. Without you blocking all obstacles in front of me, as well as pushing me along gently from behind, I never could have achieved this dream. To “Four Whatevers”! Also, this dissertation is dedicated to Emily, Tyler, and Stephen. I promise that I will now start saying “Yes, I have time to do that with you ” much more often than before. Don’t ever forget that you can accomplish anything if you set your heart and mind to it.iii Acknowledgements Without the guidance, mentoring, and advice of my Advisor, Professor Karla Hoffman, this dissertation and this project would never have been attainable. Thank you Karla. Special thanks go to the President of DECISIVE ANALYTICS Corp., John Donnellon, who took the risk in funding this project, which has now become a software product called Pythagoras. Thanks to my fellow Directors, who supported me as this project moved along, albeit slower than originally anticipated. Also, thanks go to Matt, Dean, and Tom for the long hours they put in to turn this crazy idea into a reality, and to

Within the last two decades a useful, nontrivial theory of nonlinear signal processing has been built around the median filter. We outline the development of this theory from its beginnings in the study of the noise removal properties and structural behavior of the median filter to the recently developed theory of optimal stack filtering. A recent application of stack filters is provided to demonstrate the effectiveness of this new theory. 1.

Abstract- Weighted Median (WM) filters have attracted a growing number of interest in the past few years. They inherent the robustness and edge preserving capability of the classical median filter and resemble linear FIR filters in certain properties. Furthermore, WM filters belong to the broad class of nonlinear filters called stack filters. This enables the use of the tools developed for the latter class in characterizing and analyzing the behavior and properties of WM filters, e.g. noise attenuation capability. The fact that WM filters are threshold functions allows the use of neural network training methods to obtain adaptive WM filters. In this tutorial paper we trace the development of the theory of WM filtering from its beginnings in the median filter to the recently developed theory of optimal weighted median filtering. The following one and multidimensional applications are presented in this paper: idempotent weighted median filters for speech processing, adaptive weighted median and optimal weighted median filters for image and image sequence restoration, weighted medians as robust predictors in DPCM coding and Quincunx coding, and weighted median filters in scan rate conversion in normal TV and HDTV systems. I.

Abstract — Recent works in multispectral image processing advocate the employment of vector approaches for this class of signals. Vector processing operators that involve the minimization of a suitable error criterion have been proposed and shown appropriate for this task. In this framework, two main classes of vector processing filters have been reported in the literature. In [1], Astola et al. introduce the well-known class of vector median filters (VMF), which are derived as maximum likelihood (ML) estimates from exponential distributions. In [2] and [3], the authors study the processing of color image data using directional information, considering the class of vector directional filters (VDF). In this paper, we introduce a new filter structure, the directional-distance filters (DDF), which combine both VDF and VMF in a novel way. We show that DDF are robust signal estimators under various noise distributions, they have the property of chromaticity preservation and, finally, compare favorably to other multichannel image processing filters. I.