Fundamental problems in electrophysiology can be studied by computationally modeling and simulating the associated microscopic and macroscopic bioelectric fields. To study such fields computationally, researchers have developed a number of numerical and computational techniques. Advances in computer architectures have allowed researchers to model increasingly complex biophysical system. Modeling such systems requires a researcher to apply a wide variety of computational and numerical methods to describe the underlying physics and physiology of the associated three-dimensional geometries. Issues naturally arise as to the accuracy and efficiency of such methods. In this paper we review computational and numerical methods for solving bioelectric field problems. The motivating applications represent a class of bioelectric field problems that arise in electrocardiography and

This paper is concerned with regression under a "sum" of partial order constraints. Examples include locally monotonic, piecewise monotonic, runlength constrained, and unimodal and oligomodal regression. These are of interest not only in nonlinear filtering but also in density estimation and chromatographic analysis. It is shown that under a least absolute error criterion, these problems can be transformed into appropriate finite problems, which can then be efficiently solved via dynamic programming techniques. Although the result does not carry over to least squares regression, hybrid programming algorithms can be developed to solve least squares counterparts of certain problems in the class. Index Terms--- Dynamic programming, locally monotonic, monotone regression, nonlinear filtering, oligomodal, piecewise monotonic, regression under order constraints, runlength constrained, unimodal. I.

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.

Locally monotonic regression is the optimal counterpart of iterated median filtering. In [1], Restrepo and Bovik developed an elegant mathematical framework in which they studied locally monotonic regressions in R N . The drawback is that the complexity of their algorithms is exponential in N . In this paper, we consider digital locally monotonic regressions, in which the output symbols are drawn from a finite alphabet, and, by making a connection to Viterbi decoding, provide a fast O(jAj 2 ffN) algorithm that computes any such regression, where jAj is the size of the digital output alphabet, ff stands for lomo-degree, and N is sample size. This is linear in N , and it renders the technique applicable in practice. I. Introduction Local monotonicity is a property that appears in the study of the set of root signals of the median filter [2], [3], [4], [5], [6], [7], [8]; it constraints the roughness of a signal by limiting the rate at which the signal undergoes changes of trend (inc...

This paper presents a new algorithm for locating the boundaries of textured regions (both step changes and outliers) using a robust estimator. Previous robust image filters perform poorly on binary images, blur edges, round corners, and run slowly. I avoid artifacts on binary images by modelling them as continuous and interpolating values. Information is combined directly between non-adjacent locations to prevent blurring. Corners are sharpened by relabelling mis-classified pixels. The algorithm is made as fast as a Marr-Hildreth edge finder by restructuring the estimator as a series of 2D image operations, using new multi-ring order statistic operators, and running most of the estimator on a randomly sampled image. 1 Introduction Standard edge finders do not work well on images containing texture or similar fine-scale variation. They detect numerous micro-edges within the texture, omitting the boundaries between regions or burying them in clutter (figure 1). Gaussian smoothing remove...

MATHEMATICAL PROPERTIES OF THE PSEUDOMEDIAN FILTER by MARK ALLEN SCHULZE, B.A., B.S.E.E.

Anisotropic diffusion affords an efficient, adaptive signal smoothing technique that can be used for signal enhancement, signal segmentation, and signal scale-space creation. This paper introduces a novel partial differential equation (PDE)-based diffusion method for generating locally monotonic signals. Unlike previous diffusion techniques that diverge or converge to trivial signals, locally monotonic (LOMO) diffusion converges rapidly to welldefined LOMO signals of the desired degree. The property of local monotonicity allows both slow and rapid signal transitions (ramp and step edges) while excluding outliers due to noise. In contrast with other diffusion methods, LOMO diffusion does not require an additional regularization step to process a noisy signal and uses no ad hoc thresholds or parameters. In the paper, we develop the LOMO diffusion technique and provide several salient properties, including stability and a characterization of the root signals. The convergence of the algorithm is well behaved (nonoscillatory) and is independent of signal length, in contrast with the median filter. A special case of LOMO diffusion is identical to the optimal solution achieved via regression. Experimental results validate the claim that LOMO diffusion can produce denoised LOMO signals with low error using less computation than the median-order statistic approach.

We introduce a continuous time method to analyze the response of median, pseudomedian, average (mean), and midrange filters to certain periodic signals. The filter definitions are generalized to continuous time, and these definitions are applied to periodic signals such as triangle, square, and sinusoidal waves of varying frequencies. These operations yield "amplitude response" measures which are analytic functions of the frequency of the input signal. In addition, a "correlation" measure is defined to indicate the level of distortion introduced by each filter. Examples of this analysis for the median, pseudomedian, average, and midrange filters show similarities and differences among them.

The median operator is a nonlinear (morphological) image transformation which has become very popular because it can suppress noise while preserving the edges. It treats the foreground and background of an image in an identical way, that is, it is a self-dual operator. Unfortunately, the median operator lacks the idempotence property: it is not a morphological filter. This paper gives a complete characterization of morphological operators on discrete binary images which are increasing, translation invariant, and self-dual. Furthermore, it presents a general method for the modification of an increasing operator such that it becomes activity-extensive. Such modifications lead to idempotent operators under iteration. The general procedure is illustrated by giving several modifications of the 3 \Theta 3 median operator. 1. INTRODUCTION A well-known operator in digital image processing is the median operator. In this paper we will restrict ourselves to 2D discrete binary images modelled b...

This paper investigates the relationship between anisotropic diffusion and local monotonicity. A diffusion technique that has locally monotonic root signals is presented. The enhancement algorithm rapidly converges to a locally monotonic signal of the desired degree. It is shown that the diffusion coefficient used here is the only formation that guarantees idempotence for locally monotonic signals. The signals resulting from locally monotonic diffusion are closer to the original signals than the corresponding median root signals. Furthermore, the diffusion algorithm does not have a difficulty with alternating signals, as does the median filter. In contrast to other anisotropic diffusion techniques, the diffusion method given here does not preserve outliers and does not require a gradient magnitude threshold in the diffusion coefficient.