Fred C. Holroyd

Spatially referenced methods of processing raster and vector data

Abstract The authors consider a general method of constructing addressing and arithmetic systems for two-dimensional image data using the hierarchy of ‘molecular’ tilings based on an original isohedral ‘atomic’ tiling. (Each molecular title at level k is formed from a constant number of tiles at level k−1; this is termed the ‘aperture’ property of the hierarchy.) In addition they present 11 objective criteria (which are of significance in cartographic image processing), by which these hierarchies and tilings may be described and compared. Of the 11 topologically distinct types of isohedral tiling, three ([36], [44] and [63]) are composed of regular polygons, and two of these ([36] and[44]) satisfy the condition that all tiles have the same ‘orientation’. In general, although each level in a hierarchy is topologically equivalent, the tiles may differ in shape at different levels and only [63], [44], [4.82] and [4.6.12] are capable of giving rise to hierarchies in which the tiles at all levels are the same shape. The possible apertures of hierarchies obeying this condition are n2 (for any n > 1)in the cases of [63] and [44]; n2 or 2n2 in the case of [4.82]; and n2 or 3n2 in the case of [4.6.12]. In contrast the only tiling exhibiting the uniform ‘adjacency’ criterion is[36]. However, hierarchies based on this atomic tiling generate molecular tiles with different shapes at every level. If these disadvantages are accepted, hierarchies based on first-level molecular tiles referred to as the 4-shape, 4′-shape, 7-shape and 9-shape are generated. Of these the 4-shape and the 9-shape appear to satisfy many of the cartographically desirable properties in addition to having an atomic tiling which exhibits uniform adjacency. In recent years the generalized balanced ternary addressing system has been developed to exploit the image processing power of the 7-shape. The authors have generalized and extended this system as ‘tesseral addressing and arithmetic’, showing how it can be used to render a 4-shape into a spatially correct linear quadtree.

A digital geometry for hexagonal pixels

Abstract Comparison of hexagonal and square pixels and arrays for image processing shows that the former have many advantages. However, squares can be addressed with integers and orthogonal axes, while for hexagons the axes must be at an oblique angle of 60°. This paper describes a general method of producing geometrical algorithms for such axes.

Multichromatic numbers, star chromatic numbers and Kneser graphs

We investigate the relation between the multichromatic number (discussed by Stahl and by Hilton, Rado and Scott) and the star chromatic number (introduced by Vince) of a graph. Denoting these by χ* and η*, the work of the above authors shows that χ*(G) = η*(G) if G is bipartite, an odd cycle or a complete graph. We show that χ*(G) ≤ η*(G) for any finite simple graph G. We consider the Kneser graphs , for which χ* = m/n and η*(G)/χ*(G) is unbounded above. We investigate particular classes of these graphs and show that η* = 3 and η* = 4; (n ≥ 1), and η* = m - 2; (m ≥ 4).

Colouring of cubic graphs by Steiner triple systems

Let J be a Steiner triple system and G a cubic graph. We say that G is J-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of J. We show that if J is a projective system PG(n, 2), n≥2, then G is J-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an J-colouring for every Steiner triple system of order greater than 3. We establish a condition on a cubic graph with a bridge which ensures that it fails to have an J-colouring if J is an affine system, and we conjecture that this is the only obstruction to colouring any cubic graph with any non-projective system of order greater than 3.

Efficient linear quadtree construction algorithm

Abstract An algorithm for constructing a linear quadtree having a full leaf set from a raster image is investigated. This is based on methods similar to those developed by Lauzon et al. 1 for handling two-dimensional run-encoded quadtrees. It examines pixels in Morton scan order, allowing the largest possible nodes to be generated in order of increasing Morton address. It has been designed for use with disc-based images and quadtrees, and is able to cope with the very large images characteristic of geographic information processing. The algorithm is compared to that of Shaffer and Samet 2 , and is found to be faster and to require less disc space for the output quadtree.

Tesseral amalgamators and hierarchical tessellations

Abstract Hierarchical tesselations are used in many fields, for example image processing and geographic information systems (GIS). This paper describes a general method of generating hierarchical tilings 1 on a two-dimensional, optionally square lattice using the Tesseral Amalgamator Theory (TAT). Each hierarchy has an hierarchical address which may be used in the same way as that of the quadtree 2–4 , and an arithmetic for geometric transforms 5 . Thousands of new hierarchies have been listed. Examples demonstrating how the new hierarchies and TAT can be used to increase the efficiency of hierarchical methods are given, including an account of three recent successful benchmarks using TAT. References are given to the generalization of TAT to three and four dimensions.

Raster GIS: models of raster encoding

Abstract A fundamental problem for Geographical Information systems (GIS) is the need to interrelate spatial and nonspatial data into a system that can handle both spatially and object-oriented types of query. It is natural to structure the data primarily with respect either to the spatial or to the nonspatial information. The former selection leads to encodings of the raster type. There are potentially infinitely many addressing schemes for individual locations; any of these may be used as the basis for a simple raster encoding, a run-length encoding or (intermediate between these) an encoding analogous to the linear quadtree. The paper presents a unified view of such schemes, considers questions of storage and image-processing efficiency, and concludes with a brief look at the problem of integrating raster, vector, and object-oriented data types in GIS.

The overfull conjecture and the conformability conjecture

Abstract In this paper we show that under some fairly general conditions the Overfull Conjecture about the chromatic index of a graph G implies the Conformability Conjecture about the total chromatic number of G. We also show that if G has even order and high maximum degree, then G is conformable unless the deficiency is very small.

Lattice rings: coordinates for self-similar hierarchies and their relevance to geographic information systems

Abstract Spatial data or indexes can often be usefully organized in hierarchical form within Geographical Information Systems. Many thousands of different hierarchies have now been found in 2-dimensional space. A brief description of, and references to, the method of doing this are given. A benchmark is described which demonstrates how the new hierarchies can be used to improve efficiency in practice. All these hierarchies lie on lattices and have a tesseral arithmetic which relies on a mathematical construct explored in this paper and is given the name lattice ring. These lattice rings are fully classified, and the classification related to the geometrical properties of the tesseral arithmetic of any given hierarchy, and to other important geometrical properties of the hierarchy.

Cycles in the complement of a tree or other graph

Formulas are obtained for the number of m-cycles, γm(G, n), and the number of all cycles, γ(G, n), in the complement of a graph G with respect to the complete graph Kn, in terms of the ‘linear forest array’ of G. Some elementary properties of these arrays are obtained. Computer results are reported which show that, as T ranges over all trees of order p, the star graph maximizes γ(T, p) for p = 5 to 8 and the path maximizes γ(T, p) for p = 9 to 12. This corroborates a conjecture of K.B. Reid. An asymptotic result is proved, comparing γn m(F, n) with γm(G, n) and γ(F, n) with γ(G, n)as n → ∞ for fixed F, G. Finally, if F is a forest it is shown that the computational complexity of calculating γm(F, n) and γ(F, n) is polynomially bounded in the number of edges of F.