Modeling and Reasoning with Star Calculus

Spatial knowledge representation (KR) schemes are important in many areas of computation, e.g., geographical information systems, natural language technology, image processing, battlefield management, etc. In this paper we have discussed a spatial KR technique for the purpose of expressing angular relations between points in Euclidean space. The scheme is a hybrid of what are known as quantitative and qualitative schemes. In this section we will briefly introduce a related area for the purpose of motivation. Digital image processing sometimes deploys a method called “chain code” in order to represent a polygon as a sequence of angular relations between the respective pairs of points of the polygon [Gonzalez and Wintz, 1997]. The angles are measured with reference to an absolute direction in the space (e.g., North), in a clockwise or typically anti-clockwise sense. The measurement is discrete rather than continuous, i.e., they come from a finite discrete set of values between 0 and 360 degrees, e.g., {x=0°, x=30°, x=60°, . . ., x=330°}, for an angular direction x. Actually the ordering index is utilized rather than the angle’s value, e.g., {1 for x=0°, 2 for x=30°, . . ., 12 for x=330°} for the above set. Thus, a polygon will be represented as a sequence of these numbers for adjacent pairs of points on its perimeter (Figure 1). Obviously the discretization looses some information and thus, may distort the polygon when it is recovered. However, chain codeschemes may use different level of granularity, e.g., representation with 15° instead of 30°, depending on the requirement in an application.