Index spaces for 3D retrieval: toward a better understanding of their geometry and distribution

Abstract

Distance is a fundamental concept when considering the information retrieval and cluster analysis of 3D information. That is, a large number of information retrieval descriptor comparison and cluster analysis algorithms are built around the very concept of the distance, such as the Mahalanobis or Manhattan distances, between points. Although not always explicitly stated, a significant proportion of these distances are, by nature, Euclidian. This implies that it is assumed that the data distribution, from a geometrical point of view, may be associated with a Euclidian flat space. In this paper, we draw attention to the fact that this association is, in many situations, not appropriate. Rather, the data should often be characterised by a Riemannian curved space. It is shown how to construct such a curved space and how to analyse its geometry from a topological point of view. The paper also illustrates how, in curved space, the distance between two points may be calculated. In addition, the consequences for information retrieval and cluster analysis algorithms are discussed.