The paper is a overview of the major qualitative spatial representation and reasoning techniques. We survey the main aspects of the representation of qualitative knowledge including ontological aspects, topology, distance, orientation and shape. We also consider qualitative spatial reasoning including reasoning about spatial change. Finally there is a discussion of theoretical results and a glimpse of future work. The paper is a revised and condensed version of [33,34].

Qualitative Spatial Representation and Reasoning: An Overview

The convex hulls of sets of n points in two and three dimensions can be determined with O(n log n) operations. The presented algorithms use the “divide and conquer” technique and recursively apply a merge procedure for two nonintersecting convex hulls. Since any convex hull algorithm requires at least O(n log n) operations, the time complexity of the proposed algorithms is optimal within a multiplicative constant.

Convex hulls of finite sets of points in two and three dimensions

This paper surveys the work of the qualitative spatial reasoning group at the University of Leeds. The group has developed a number of logical calculi for representing and reasoning with qualitative spatial relations over regions. We motivate the use of regions as the primary spatial entity and show how a rich language can be built up from surprisingly few primitives. This language can distinguish between convex and a variety of concave shapes and there is also an extension which handles regions with uncertain boundaries. We also present a variety of reasoning techniques, both for static and dynamic situations. A number of possible application areas are briefly mentioned.

Qualitative Spatial Representation and Reasoning with the Region Connection Calculus

We develop optimal algorithms for forming the intersection of geometric objects in the plane and apply them to such diverse problems as linear programming, hidden-line elimination, and wire layout. Given N line segments in the plane, finding all intersecting pairs requires O(N2) time. We give an O(N log N) algorithm to determine whether any two intersect and use it to detect whether two simple plane polygons intersect. We employ an O(N log N) algorithm for finding the common intersection of N half-planes to show that the Simplex method is not optimal. The emphasis throughout is on obtaining upper and lower bounds and relating these results to other problems in computational geometry.

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Geometric intersection problems

Fast polygonal approximation of digitized curves

A theory for describing and measuring the concavities of cellular complexes (digitized silhouettes) is developed. This theory involves the use of the minimum-perimeter polygon and its convex hull.

Measuring Concavity on a Rectangular Mosaic

We describe a parallel mechanism, based on an array of circularly nutating photodetectors, that computes an approximation of the density of slopes of the boundary of any piecewise regular silhouette. This density, when properly normalized, is invariant with respect to the size, translation, and orientation of the given silhouette.

J. Sklansky

Papers

Measuring Concavity on a Rectangular Mosaic

A Parallel Mechanism for Describing Silhouettes