Frank Rhodes

Discrete Euclidean metrics

Abstract It is shown that rounding up the Euclidean metric produces an integer-valued metric with properties which enable its values on the integer plane to be calculated without taking square roots. By rounding up to an appropriate scale the absolute and proportional differences from the Euclidean metric can be contained within predetermined bounds.

Some characterizations of the chessboard metric and the city block metric

Abstract It is shown that a convex metric on a digital lattice which satisfies a condition used by Rosenfeld in an investigation of continuous functions on digital images must be the city block metric. A corresponding characterization of the chessboard metric is given. The two metrics on the digital plane are characterized among metric spaces, and the tolerance relations associated with the two metrics are characterized. In the case of the city block metric, these correct and improve some earlier results of Melter.

On the metrics of Chaudhuri, Murthy and Chaudhuri

Abstract The paper considers the approximation of Euclidean distance in n-dimensional space by linear combinations of the L1 and L∞ metrics. Maximal proportional errors for the one parameter family introduced by Chaudhuri, Murthy and Chaudhuri are calculated. Estimates of the optimal parameters for one parameter families are obtained by solving a quartic equation numerically. The maximal proportional errors for these parameters are much smaller than those for the parameters chosen by Chaudhuri et al. It is shown that for two parameter families the corresponding quartic equation can be solved algebraically. Thus the behaviour of the optimal solutions can be seen more clearly, though the approximations to the Euclidean metric are not substantially improved.

The principal part of a block map

Abstract A block map is a map f {0, 1} n {0, 1} for some n ⪖ 1. Block maps can be represented by polynomials with coefficients in Z 2 . The notion of the principal part of a block map is introduced. It is used to obtain some conditions under which block maps which are linear in the first variable but not linear in all variables are irreducible with respect to composition.

Discrete subadditive functions as Gomory functions

Our aim, in this paper, is to study a class of functions which occurs in pure integer programming, and to investigate conditions under which discrete subadditive functions belong to that class. The inspiration for the paper was the problem of classifying discrete metrics used in pattern recognition, while the methods of proof of the main theorem are those of pure integer programming.

Discrete metrics as Gomory functions

It has been shown recently that discrete, non-decreasing subadditive functions are value functions of pure integer programs and so belong to the class of Gomory functions. Some consequences of this result for discrete metrics are reported in this paper. If a discrete metric in the digital plane is invariant under translations and reflections in the axes, then it is determined by a subadditive function on the first quadrant. If it is also non-decreasing in each coordinate then its values in each finite block are determined by a Gomory function. If the values of the function throughout the first quadrant are determined by the values in a finite block, either by shift-periodicity or by a Hilbert basis, then the subadditive function is determined in the whole of the first quadrant by a unique Gomory function.

Metric subgraphs of the Chamfer metrics and the Melter-Tomescu path generated metrics

Abstract Chamfer metrics are determined by local distances which are chosen to ensure that each geodesic lies within one of the cones determined by the mask and contains only edges in the directions of the bounding rays of the cone. It is shown that the chamfer distances calculated within a set are the same as those calculated in the whole space if and only if the set is convex in each of the local distance directions. The result does not hold when the local distances allow more general geodesics. The results for chamfer metrics are related to corresponding results for the metrics generated by the two-, three- and four-direction graphs studied by Melter and Tomescu.

Digital metrics generated by families of paths

Abstract Relationships between properties of a family of paths on a graph and properties of the distance function defined by the family are studied. Types of properties which lead to the distance function being a metric are considered. The study is a response to a query by Melter and Rosenfeld about possible generalizations of their s-connections. An application to metrics on sets of interlocking sublattices is given.

The enumeration of certain sets of block maps

Abstract An n-block is a sequence b1b2…bn, where bi ϵ Z2 for 1 ⩽ i ⩽ n, and an n-block map is a function from the set of n-blocks to the ring Z2. The n-block maps form the ring GF(2) {x1, x2,…, xn | xi2 = 1}. The set of all block maps with the operations of addition and polynomial substitution form a near-ring. The general problem of searching non-linear block maps for factors with respect to polynomial substitution seems to be extremely complex. However, effective search procedures have been described for factors within certain sets of block maps which are non-linear but are linear in the first variable. In this paper we show that the smallest of these sets contains just over 85% of the block maps which are linear in the first variable. The largest of the sets, for which there is an additional step in the search procedure, contains over 99% of the block maps which are linear in the first variable.

Geodetic metrizations of graphs

Abstract A graph can be metrized by assigning a length to each of its edges. Such a graph is said to be geodetic if for each pair of vertices there is a unique geodesic joining them. It is said to be normally geodetic if each of these unique geodesics is one of the geodesics in the usual metrization of the graph in which each edge is given unit length. It is shown that every graph admits a normally geodetic metrization. Geodetic metrizations of the four- and eight-connection graphs of the digital plane which can be processed easily on a computer are investigated. Examples are given of normally geodetic integral metrizations of arbitrarily large rectangular blocks of these planes. However, it is proved that there are no normally geodetic metrizations of the whole of these planes which are periodic in each variable.