Hierarchical Clustering and Validity Assessment

The unsupervised classification of points into groups is commonly referred to as clustering or grouping. Clustering aims at discovering structure in a point data set by dividing it into its natural groups. There are three classical problems related to the construction of the right clusters. The first is evaluating the validity of a cluster candidate. In other words, is a group of points really a cluster, i.e. a group with a large enough density? The second problem is that meaningful clusters can contain or be contained in other meaningful clusters. A rule is needed to define locally optimal clusters by inclusion. This rule, however, is not enough to interpret correctly the data. The third problem is defining a correct merging rule between meaningful clusters, and thus being able to decide whether they should stay separate or unit. A unified a contrario method will be proposed for these problems. In continuation, some complexity issues and heuristics to find sound candidate clusters will be considered. In the next chapters, the clustering theory developed here will find a main application in shape recognition: the grouping of several local matches into a more global shape matching.