A Unified Methodology for the Generalisation of the Geometry of Features

Abstract

The development of generalisation (simplification) methods for the geometry of features in digital cartography in most cases involves the improvement of existing algorithms without their validation with respect to the similarity of feature geometry before and after the process. It also consists of the assessment of results from the algorithms, i.e., characteristics that are indispensable for automatic generalisation. The preparation of a fully automatic generalisation for spatial data requires certain standards, as well as unique and verifiable algorithms for particular groups of features. This enables cartographers to draw features from these databases to be used directly on the maps. As a result, collected data and their generalised unique counterparts at various scales should constitute standardised sets, as well as their updating procedures. This paper proposes a solution which consists in contractive self-mapping (contractor for scale s = 1) that fulfils the assumptions of the Banach fixed-point theorem. The method of generalisation of feature geometry that uses the contractive self-mapping approach is well justified due to the fact that a single update of source data can be applied to all scales simultaneously. Feature data at every scale s < 1 are generalised through contractive mapping, which leads to a unique solution. Further generalisation of the feature is carried out on larger scale spatial data (not necessarily source data), which reduces the time and cost of the new elaboration. The main part of this article is the theoretical presentation of objectifying the complex process of the generalisation of the geometry of a feature. The use of the inherent characteristics of metric spaces, narrowing mappings, Lipschitz and Cauchy conditions, Salishchev measures, and Banach theorems ensure the uniqueness of the generalisation process. Their application to generalisation makes this process objective, as it ensures that there is a single solution for portraying the generalised features at each scale. The present study is dedicated to researchers concerned with the theory of cartography.