Geometric algorithms for delineating geographic regions

Abstract

Everyone of us is used to geographical regions like the south of Utrecht, the dutch Randstad, or the mountainous areas of Austria. Some of these regions have crisp, fixed boundaries like Utrecht or Austria. Others, like the dutch Randstad and the Austrian mountains, have no such boundaries and are used in a more loose sense. However, sometimes it is neccessary to have crisp boundaries for such regions, in order to be able to answer questions like 'How many people under 25 live in the Randstad?' or 'Are there any 4 star hotels in the south of Utrecht?'. Without reasonable boundaries for these imprecisely defined regions, it is not possible to give answers to these questions. Such questions are common in geographic information systems and retrieval (GIS and GIR), but also need to be answered by Web search engines. The main topic of this thesis is to delineate imprecise regions like the ones above to aid in answering queries in GIS and GIR as well as in visualization, by using strictly geometric, algorithmic methods. In Chapters 2, 3, and 4, we present various, efficient methods to delineate boundaries for a number of imprecise regions. Chapter 2 gives three suggestions on how to divide a country into four parts that represent its north, east, west, and south. We give efficient algorithms to compute these parts and compare the results by using ten european country outlines. Chapter 3 presents two main approaches to delineate more general imprecise regions, like the dutch Randstad. We start by using the Internet as a database to classify a number of towns as inside or outside the region. It is possible that some errors are made in the classification of towns. Both approaches aim to derive reasonable boundaries for the region, such that they include (most of) the points classified as inside, and exclude (most of) the points classified as outside. Chapter 4 presents several methods to compute scale-dependent maps for gradient and aspect. Gradient and aspect belong to the most important values in geomorphometry; they describe the steepness of a terrain at a certain point, and the cardinal direction in which the terrain faces, respectively. Together with other values used in geomorphometry, such as plan and profile curvature, and height, it is possible to delineate landforms such as hills, valleys, and mountain ranges. Giving boundaries for imprecise regions is useful in GIS and GIR to be able to answer queries for 'Aikido dojos near Amsterdam'. Assume we have submitted this query to a Web search engine, and it retrieved several documents that are relevant with respect to the textual contents and the spatial location. In Chapter 5, we propose a number of geometric methods to rank documents with a textual and a spatial score. For all presented methods we assume that documents with similar scores have similar contents, and therefore should not be ranked consecutively. Our scattered ranking procedures rank documents according to the relevance to the query and the dissimilarity to the already ranked documents.