Two different formal definitions of gray-scale reconstruction are presented. The use of gray-scale reconstruction in various image processing applications discussed to illustrate the usefulness of this transformation for image filtering and segmentation tasks. The standard parallel and sequential approaches to reconstruction are reviewed. It is shown that their common drawback is their inefficiency on conventional computers. To improve this situation, an algorithm that is based on the notion of regional maxima and makes use of breadth-first image scannings implemented using a queue of pixels is introduced. Its combination with the sequential technique results in a hybrid gray-scale reconstruction algorithm which is an order of magnitude faster than any previously known algorithm. >

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Morphological grayscale reconstruction in image analysis: applications and efficient algorithms

Two nonlinear algorithms for processing vector-valued signals are introduced. The algorithms, called vector median operations, are derived from two multidimensional probability density functions using the maximum-likelihood-estimate approach. The underlying probability densities are exponential, and the resulting operations have properties very similar to those of the median filter. In the vector median approach, the samples of the vector-valued input signal are processed as vectors. The operation inherently utilizes the correlation between the signal components, giving the filters some desirable properties. General properties as well as the root signals of the vector median filters are studied. The vector median operation is combined with linear filtering, resulting in filters with improved noise attenuation and filters with very good edge response. An efficient algorithm for implementing long vector median filters is presented. The noise attenuation of the filters is discussed, and an application to velocity filtering is shown. >

Vector median filters

Weighted Median (WM) filters have 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. Applications discussed include: 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.

Weighted median filters: a tutorial

The median and other rank-order operators possess two properties called the threshold decomposition and the stacking properties. The first is a limited superposition property which leads to a new architecture for these filters; the second is an ordering property which allows an efficient VLSI implementation of the threshold decomposition architecture. Motivated by the success of rank-order filters in a wide variety of applications and by the ease with which they can now be implemented, we consider in this paper a new class of filters called stack filters. They share the threshold decomposition and stacking properties of rank-order filters but are otherwise unconstrained. They are shown to form a very large class of easily implemented nonlinear filters which includes the rank-order operators as well as all compositions of morphological operators. The convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters. The results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals. The output distribution for these filters is also found. All the stack filters of window width 3 are determined along with their convergence properties. Among these filters are found two which we have named asymmetric median filters. They share all the properties of median filters except that they remove impulses of one sign only; that is, one removes only positive going edges, the other removes only negative going edges, while the median filter removes impulses of both signs. This investigation of the properties of stack filters thus produces several new, useful, and easily implemented filters.

Stack filters

Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrative and holistic manner. It also paves the way toward the development of systematic approaches for effective therapeutic intervention in disease. The central theme in this paper is the Boolean formalism as a building block for modeling complex, large-scale, and dynamical networks of genetic interactions. We discuss the goals of modeling genetic networks as well as the data requirements. The Boolean formalism is justified from several points of view. We then introduce Boolean networks and discuss their relationships to nonlinear digital filters. The role of Boolean networks in understanding cell differentiation and cellular functional states is discussed. The inference of Boolean networks from real gene expression data is considered from the viewpoints of computational learning theory and nonlinear signal processing, touching on computational complexity of learning and robustness. Then, a discussion of the need to handle uncertainty in a probabilistic framework is presented, leading to an introduction of probabilistic Boolean networks and their relationships to Markov chains. Methods for quantifying the influence of genes on other genes are presented. The general question of the potential effect of individual genes on the global dynamical network behavior is considered using stochastic perturbation analysis. This discussion then leads into the problem of target identification for therapeutic intervention via the development of several computational tools based on first-passage times in Markov chains. Examples from biology are presented throughout the paper.

From Boolean to probabilistic Boolean networks as models of genetic regulatory networks

A median filter is a nonlinear digital filter which consists of a window of length 2N + 1 that moves over a signal of finite length. For each input sample, the corresponding output point is the median of all samples in the window centered on that input sample. Any finite length, M -level, signal that ends with constant regions of length N + 1 will converge to an invariant signal in a finite number of passes of this median filter. Such an invariant signal is called a root. The concept of a root signal has proved to be crucial in understanding the properties of the median filter, root signals are to median filters what passband signals are to linear signals. In this paper, two results concerning the rate at which a signal is filtered to a root are developed. For a window of width 3, we derive a recursive formula to count the number of binary signals of length L that converge to a root in exactly m passes of a median filter. Also, we show that, given a window of width 2N + 1 , any signal of length L will converge to a root in at most 3\lceil\frac{(L-2)}{2(N + 2)}\rceil passes of the filter.

P. Wendt

Papers

Stack filters

Some convergence properties of median filters