Surface modelling with guaranteed consistency — An object-based approach

Abstract

There have been many interpolation methods developed over the years, each with their own problems. One of the biggest limitations in many applications is the non-correspondence of the surface with the objects used to support it — usually a set of arbitrarily distributed data points. This is due to the metric methods used to define both the zone of influence of a data point and the set of data points used to estimate surface properties at intermediate locations. Most methods are coordinate system oriented, not object oriented.

We describe here an object-oriented approach, in the sense that the data objects themselves form the spatial structure used for interpolation. This has the primary benefit that the surface is described in terms of this structure, permitting easy correspondence between the surface and data values. In addition, the spatial objects used are not restricted to points, but may include more complex objects — currently composed of line segments. The structure used is the Voronoi diagram, which provides easily understood spatial relationships between map objects, as well as the local coordinate systems needed for point/surface correspondence.

The methods described here use the Voronoi diagram to generate surfaces which pass precisely through all data objects (points and line segments) and have no inadvertent surface or slope discontinuities. In addition, discontinuities in the slope or the surface itself may be assigned to the line segment objects; and measures of surface consistency may be derived from the values of nearby data objects. This approach permits a consistent method of mapping a fragmented surface with both point and line segment information, and the surface may be guaranteed to be only locally perturbed by the modification of any one data object. While the concepts discussed are here applied to two spatial dimensions plus “elevation”, the Voronoi method may readily be applied to higher dimensions. The Voronoi approach, if used for traditional GIS topological structuring as well as for interpolation, provides a new way to handle an old challenge: to combine object and field data within the same GIS structure.