Extended Hausdorff distance for spatial objects in GIS

Distance is a fundamental concept in spatial sciences. Spatial distance is a very important parameter to measure the relative positions between spatial objects and to indicate the degree of similarity between neighbouring objects. Indeed, spatial distance plays an important role in many areas such as neighbourhood analysis, structural similarity measure, image (or object) matching, clustering analysis, and so on. In this paper, existing computational models for the distance between spatial objects are evaluated and their problems pointed out; then, the concept of the Hausdorff distance is introduced as a metric indicator for different types of spatial objects. This distance is extended to a uniform representation by the introduction of the quantile, leading to the extended Hausdorff distance. Indeed, the so-called extended Hausdorff distance is, in fact, a kind of metric characterized by the minimum distance, the Hausdorff distance, and the median Hausdorff distance. The first two can be used for measuring the dispersion and the last one for measuring the central tendency of the distance distribution between spatial objects. A method termed the ε-buffer has been proposed for the computation of the median Hausdorff distance. Finally, potential applications are discussed.