Mathematical and computational modeling of genetic regulatory networks promises to uncover the fundamental principles governing biological systems in an integrarive 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. 1

Nonlinear filtering techniques based on the theory of robust estimation are introduced. Some deterministic and asymptotic properties are derived. The proposed denoising methods are optimal over the Huber-contaminated normal neighborhood and are highly resistant to outliers. Experimental results showing a much improved performance of the proposed filters in the presence of Gaussian and heavy-tailed noise are analyzed and illustrated.

Abstruct- A new expression for the output moments of weighted mcdian filtered data is derived in this paper. The noise attenuation capability of a weighted median filter can now be assessed using the L-vector and AI-vector parameters in the new expression. The second major contribution of the paper is the development of a new optimality theory for weighted median filters. This theory is based on the new expression for the output moments, and combines the noise attenuation and some structural constraints on the filter’s behavior. In certain special cases, the optimal weighted median filter can be obtained by merely solving a set of linear inequalities. This leads in some cases to closed form solutions for optimal weighted median filters. Some applications of the theory developed in this paper, in 1-D signal processing and image processing are discussed. Throughout the analysis, some striking similarities are pointed out between linear FIR filters and weighted median filters. I.

Computer architectures are quickly changing toward heterogeneous many-core systems. Such a trend opens up interesting opportunities but also raises immense challenges since the efficient use of heterogeneous many-core systems is not a trivial problem. In this paper, we explore how to program data processing operators on top of field-programmable gate arrays (FPGAs). FPGAs are very versatile in terms of how they can be used and can also be added as additional processing units in standard CPU sockets. In the paper, we study how data processing can be accelerated using an FPGA. Our results indicate that efficient usage of FPGAs involves non-trivial aspects such as having the right computation model (an asynchronous sorting network in this case); a careful implementation that balances all the design constraints in an FPGA; and the proper integration strategy to link the FPGA to the rest of the system. Once these issues are properly addressed, our experiments show that FPGAs exhibit performance figures competitive with those of modern general-purpose CPUs while offering significant advantages in terms of power consumption and parallel stream evaluation. 1.

Abstract. Within the last two decades a small group of researchers has built a useful, nontrivial theory of nonlinear signal processing around the median-related filters known as rank-order filters, order-statistic filters, weighted median filters, and stack filters. This required significant effort to overcome the bias, both in education and research, toward linear theory, which has been dominant since the days of Fourier, Laplace, and "Convolute." We trace the development of this theory of nonlinear filtering from its beginnings in the study of noise-removal properties and structural behavior of the median filter to the recently developed theory of optimal stack filtering. The theory of stack filtering provides a point of view which unifies many different filter classes, including morphological filters, so it is discussed in detail. Of particular importance is the way this theory has brought together, in a single analytical framework, both the estimation-based and the structural-based approaches to the design of these filters. Some recent applications of median and stack filters are provided to demonstrate the effectiveness of this approach to nonlinear filtering. They include: the design of an optimal stack filter for image restoration; the use of vector median filters to attenuate impulsive noise in color images and to eliminate cross luminance and cross color in TV images; and the use of median-based filters for image sequence coding, reconstruction, and scan rate conversion in normal TV and HDTV systems. 1.

Simple nonlinear filters are often used to enforce "hard" syntactic constraints while remaining close to the observation data; e.g., in the binary case it is common practice to employ iterations of a suitable median, or a one-pass recursive median, openclose, or closopen filter to impose a minimum symbol run-length constraint while remaining "faithful" to the observation. Unfortunately, these filters are - in general - suboptimal. Motivated by this observation, we pose the following optimization: Given a finite-alphabet sequence of finite extent, y = fy(n)g N \Gamma1 n=0 , find a sequence, b x = fbx(n)g N \Gamma1 n=0 , which minimizes d(x; y) = P N \Gamma1 n=0 dn (y(n); x(n)) subject to: x is piecewise constant of plateau run-length M . We show how a suitable reformulation of the problem naturally leads to a simple and efficient Viterbi-type optimal algorithmic solution. We call the resulting nonlinear input-output operator the Viterbi Optimal Runlength-Constrained Approximation...

Abstract In this paper, we do not aim at new applications or algorithms, but at a formalism as simple and as practical as possible to deal with the several useful concepts in image and shape analysis, namely: level sets of a function, reconstruction of a function from its level sets, monotone set operators, contrast invariant monotone image operators, threshold superposition principle, sup-inf operators and flat morphology, image operators commuting with thresholds. We prove that five slightly different terminologies or formalisms can be merged into a single simple presentation. Namely: the operators of “flat Mathematical Morphology”, the “contrast invariant image operators”, the “monotone set operators”, the “sup-inf ” operators and finally the “contrast invariant image operators defined on continuous images” are fully equivalent. In this equivalence statement, set functions are defined from set operators by the threshold superposition principle and set operators are defined from contrast invariant operators by the so-called Evans-Spruck extension. All that we prove may be known in different contexts but has not been formalized, to our knowledge, in a simple unified format. The closest theory to what we present, in Mathematical Morphology, is in the abstract framework of complete lattices. We do not request any completeness requirement in what follows and the statements apply to operators defined on any set of functions or any set of sets. As illustration, we show how the unified formalism permits to define easily several image operators by giving their simpler set operator definition and conversely how we also easily deal with set operators defined from P.D.E.’s, as it occurs with geodesic snakes. The formal presentation of contrast invariant mathematical morphology given here will be developed in the book [5] in project.

The continuous threshold decomposition is a segmentation operator used to split a signal into a set of multilevel components. This decomposition method can be used to represent continuous multivariate piecewise linear (PWL) functions, and therefore can be employed to describe PWL systems defined over a rectangular lattice. The resulting filters are canonical and have a multichannel structure that can be exploited for the development of rapidly convergent algorithms. The optimum design of the class of PWL filters introduced in the paper can be postulated as a least squares problem whose variables separate into a linear and a nonlinear part. Based on this feature, parameter estimation algorithms are developed. First, a block data processing algorithm that combines linear least-squares with grid localization through recursive partitioning is introduced. Second, a time-adaptive method based on the combination of an RLS algorithm for coefficient updating and a signed gradient descent module...

In this paper we introduce and analyze a new class of non--linear filters which have their roots in permutation theory. We show that a large body of non--linear filters proposed to date constitute a proper subset of Permutation Filters (P Filters). In particular, rank--order filters, weighted rank--order filters, and stack filters embody limited permutation transformations of a set. Indeed, by using the full potential of a permutation group transformation we can design very efficient estimation algorithms. Permutation groups inherently utilize both rank--order and temporal--order information; thus, the estimation of non--stationary processes in Gaussian/non--Gaussian environments with frequency selection can be effectively addressed. An adaptive design algorithm which minimizes the mean absolute error criterion is described as well as a more flexible adaptive algorithm which attains the optimal permutation filter under a deterministic least normed error criterion. Simulation results ar...

This paper describes the definition and testing of a new type of median filter for images. The topological median filter implements some existing ideas and some new ideas on fuzzy connectedness to improve, over a conventional median filter, the extraction of edges in noise. The concept of #-connectivity is defined and used to create an algorithm for computing the degree of connectedness of a pixel to all the other pixels in an arbitrary neighborhood. The resulting connectivity map of the neighborhood e#ectively disconnects peaks in the neighborhood that are separated from the center pixel by a valley in the brightness topology. The median of the connectivity map is an estimate of the median of the peak or plateau to which the center pixel belongs. Unlike the conventional median filter, the topological median is relatively unaffected by disconnected features in the neighborhood of the center pixel. Four topological median filters are defined. Qualitative and statistical analyses of the four filters are presented. It is demonstrated that edge detection can be more accurate on topologically median filtered images than on conventionally median filtered images. Index Terms --- median filters, noise reduction, fuzzy digital topology, topological median filters. I.