Optimal Approximation in Automated Cartography

Abstract

For several years, attempts are being made in many countries of the world to apply electronic data processing techniques in the production and updating of maps. Therein, cartographic generalization still is a vital problem.

Generalization in cartography is the process of transforming the contents of a map into a form appropriate for a map of smaller scale; it consists of several partial processes (as selection, geometric combination, qualitative summarization, simplification and displacement of map objects) which are interdependent in a non-hierarchic way.

It is shown that cartographic generalization is a problem of optimal approximation and that it may be described by a model of mathematical optimization. The basis of such a model is the calculation or semantic information content of the individual map objects in a way corresponding to the thesises of information theory and to the way people (most probably) read maps. Prediction techniques play an essential part in this respect.

The generalization model is subsequently used as a scheme for the classification and judgement of the most important contemporary partial solutions of automated map generalization.

Concluding, three other problems in automated cartography are described which are solved by using methods of approximation theory. These are the reduction of the quantity of digital cartographic data to a minimal amount allowing the ‘exact’ reconstruction of a line, the Helrnert Transformation (which is a conformal transformation — optimal in the sense of the least-squares method) for the minimization of distortions in geometric data digitized from a distorted map sheet, and finally the calculation of splines for the reconstruction of lines from a limited number of given points for output on flatbed plotters.