Godfried T. Toussaint

The Relative Neighbourhood Graph of a Finite Planar Set

Abstract The relative neighbourhood graph (RNG) of a set of n points on the plane is defined. The ability of the RNG to extract a perceptually meaningful structure from the set of points is briefly discussed and compared to that of two other graph structures: the minimal spanning tree (MST) and the Delaunay (Voronoi) triangulation (DT). It is shown that the RNG is a superset of the MST and a subset of the DT. Two algorithms for obtaining the RNG of n points on the plane are presented. One algorithm runs in 0(n2) time and the other runs in 0(n3) time but works also for the d-dimensional case. Finally, several open problems concerning the RNG in several areas such as geometric complexity, computational perception, and geometric probability, are outlined.

Relative neighborhood graphs and their relatives

Results of neighborhood graphs are surveyed. Properties, bounds on the size, algorithms, and variants of the neighborhood graphs are discussed. Numerous applications including computational morphology, spatial analysis, pattern classification, and databases for computer vision are described. >

Bibliography on estimation of misclassification

Articles, books, and technical reports on the theoretical and experimental estimation of probability of misclassification are listed for the case of correctly labeled or preclassified training data. By way of introduction, the problem of estimating the probability of misclassification is discussed in order to characterize the contributions of the literature.

Computing the width of a set

For a set of points P in three-dimensional space, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in O(n log n+I) time and O(n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is the number of vertices; in the worst case, I=O(n/sup 2/). For a convex polyhedra the time complexity becomes O(n+I). If P is a set of points in the plane, the complexity can be reduced to O(nlog n). For simple polygons, linear time suffices. >

The use of context in pattern recognition

Abstract The importance of contextual information, at various different levels, for the satisfactory solution of pattern recognition problems is illustrated by examples. A tutorial survey of techniques for using contextual information in pattern recognition is presented. Emphasis is placed on the problems of image classification and text recognition, where the text is in the form of machine and handprinted characters, cursive script, and speech. The related problems of scene analysis, natural language understanding, and error-correcting compilers are only lightly touched upon.

Movable Separability of Sets

Abstract Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout, considerable attention has been devoted recently to the problem of moving sets of objects, such as line segments and polygons in the plane or polyhedra in three dimensions, without allowing collisions between the objects. One class of such problems considers the separability of sets of objects under different kinds of motions and various definitions of separation. This paper surveys this new area of research in a tutorial fashion, presents new results, and provides a list of open problems and suggestions for further research.

On a convex hull algorithm for polygons and its application to triangulation problems

Abstract A frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases. Due to its simplicity the algorithm is, nevertheless, attractive. In this paper it is shown that the algorithm does in fact work for a family of simple polygons known as weakly externally visible polygons. Some application areas where such polygons occur are briefly discussed. In addition, it is shown that with a trivial modification the algorithm can be used to internally and externally triangulate certain classes of polygons in 0( n ) time.

Experiments in Text Recognition with the Modified Viterbi Algorithm

In this paper a modification of the Viterbi algorithm is formally described, and a measure of its complexity is derived. The modified algorithm uses aheuristic to limit the search through a directed graph or trellis. The effectiveness of the algorithm is investigated via exhaustive experimentation on an input of machine-printed text. The algorithm assumes language to be a Markov chain and uses transition probabilities between characters. The results empirically answer the long-standing question of what is the benefit, if any, of using transition probabilities that depend on the length of a word and their position in it.

An efficient algorithm for decomposing a polygon into star-shaped polygons

Abstract In this paper we show how a theorem in plane geometry can be converted into a O(n log n) algorithm for decomposing a polygon into star-shaped subsets. The computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods.

Note on optimal selection of independent binary-valued features for pattern recognition (Corresp.)

Given a set of conditionally independent binary-valued features, a counter example is given to a possible claim that the best subset of features must contain the best single feature.