R. Shonkwiler

An image algorithm for computing the Hausdorff distance efficiently in linear time

In this paper, the above equation is used as the definition of Hausdorff distance. (The classical Hausdorff metric as described in ‘Kelley [l] is, in our notation, max(df( A, B), db( A, B)). It is easy to show that the two are topologically equivalent.) The Hausdorff distance is useful in computer graphics as a measure of the closeness of two images. As a concrete example, suppose one has a “goal” %dimensional black-and-white image known pixel by pixel in some @d of resolution m x n. Then, given an m x n “ test” image with the same resolution, the Hausdorff distance between the two provides a numerical measure of their nearness. In this case, the Hausdorff distance is in the range O..p + ((1, 1), (m, n)). In this setting, a convenient metric p is the L1 or “Manhattan” metric: the distance between pixels (xl, yl) and (x2, y2) is defined by

Annealing a genetic algorithm over constraints

A class of variable fitness genetic algorithms is studied as a technique for use on constrained optimization problems. Fitness is taken as the product of the objective with an attenuation factor which is 1 for feasible solutions but some variable fraction of 1 for infeasible ones. It is shown that this technique leads to algorithms which converge in probability to globally optimal feasible solutions. An application of the technique is made to a problem of engineering interest with excellent results: the ground water treatment problem for unconfined aquifers.

Dimensions associated with recurrent self-similar sets

The Hausdorff and box dimensions for measures associated with recurrent selfsimilar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

Computing the Hausdorff set distance in linear time for any L p point distance

Abstract A linear time minimal memory algorithm is given for constructing the “metric field” for cellularized point sets in arbitrary dimension. Resultant such metric fields are used to obtain Hausdorff set distances between sets of points. These techniques have applications to image processing.

Separating the vertices of N-cubes by hyperplanes and its application to artificial neural networks

A new sufficient condition that a region be classifiable by a two-layer feedforward network using threshold activation functions is found. Briefly, it is either a convex polytope, or that minus the removal of convex polytope from its interior, or. . .recursively. The author refers to these sets as convex recursive deletion regions. The proof of implementability exploits the equivalence of this problem with that of characterizing two set partitions of the vertices of a hypercube which are separable by a hyperplane, for which a new result is obtained.

A constructive algorithm to solve "convex recursive deletion" (CoRD) classification problems via two-layer perceptron networks

A sufficient condition that a region be classifiable by a two-layer feedforward neural net (a two-layer perceptron) using threshold activation functions is that either it be a convex polytope or that intersected with the complement of a convex polytope in its interior, or that intersected with the complement of a convex polytope in its interior or ... recursively. These have been called convex recursive deletion (CoRD) regions.We give a simple algorithm for finding the weights and thresholds in both layers for a feedforward net that implements such a region. The results of this work help in understanding the relationship between the decision region of a perceptron and its corresponding geometry in input space. Our construction extends in a simple way to the case that the decision region is the disjoint union of CoRD regions (requiring three layers). Therefore this work also helps in understanding how many neurons are needed in the second layer of a general three-layer network. In the event that the decision region of a network is known and is the union of CoRD regions, our results enable the calculation of the weights and thresholds of the implementing network directly and rapidly without the need for thousands of backpropagation iterations.

Common source epidemics II: Toxoplasmosis in Atlanta

A discrete time stochastic model formulated for the study of common source epidemics (Shonkwiler and Thompson, 1982) is implemented to study an outbreak of toxoplasmosis in Atlanta, Georgia, in 1977. A computer simulation program is described and conclusions are drawn on the basis of the simulations. Also, the detailed empirical data are organized to illuminate the roles of visitation and location patterns, and the variation in the empirical epidemic curve with different reporting schemes.

A gonorrhea model treating sensitive and resistant strains in a multigroup population

In recent years gonorrhea infection with antibiotic-resistant strains, especially PPNG, has become a significant public health problem. Drawing on the gonorrhea model of Lajmanovich and Yorke, a multigroup model that embraces both resistant and sensitive strains of the organism is introduced. It is shown that, like the Lajmanovich and Yorke (single-strain) model, in the general case the sensitive-resistant model has a unique globally asymptotic equilibrium. As a function of the interplay between contact rates, cure rates, and reversion rates, the equilibrium can lead to endemic infection with sensitive infection only, resistant infection only, or both, or to elimination of sensitive and resistant infection.

Lag time in microbe growth as an age-dependent branching process with two phase types.

Abstract We study a two-type, age-dependent branching process in which the branching probabilities of one of the types may vary with time. Specifically this modification of the Bellman-Harris process starts with a Type I particle which may either die or change to a Type II particle depending upon a time varying probability. A Type II particle may either die or reproduce with fixed probabilities but may not return to a particle of Type I. In this way the process models the lag phenomenon observed in microbe growth subsequent to transfer to a new culture medium while the organism is adapting to its new environment. We show that if the mean reproduction rate of Type II particles exceeds 1, then the population size grows exponentially. Further the extinction probability for this process is related to that of the Bellman-Harris process. Finally the governing equations are solved for several choices of the growth parameters and the solutions are graphically displayed showing that a wide variety of behavior can be modeled by this process.

Consequences of the growth and division model for mitochondria assuming no cellular control.

Under the assumption that the mitochondria of a cell are independently existing organisms their population size is modeled by a probabilistic branching process and, as with any colony of organisms, can conceivably die out. From the calculation of the probabilityP that a cell normally havingN=4 and 8 mitochondria could arise devoid of mitochondria either by extinction or by maldistribution and subsequent cell division, it is inferred that for known cell mitochondrial numbersP is vanishingly small. Therefore chance alone is adequate to explain the inclusion of mitochondria in both daughter cells during cell division.