Collection space distances

This paper introduces collection space theory and proposes five collection space distances: scaled city block, angle, minimum, nearest first, and farthest first distances. A collection of set S is a discrete histogram of a finite subset of S. A collection space is set C of collections of a set S with a distance on C which satisfies certain axioms. The scaled city block distance is the proportion of c-points that two scaled collections do not share. The angle distance between two collections is proportional to the angle between them. The other three distances are applicable to collections of points of a metric space. A correspondence between two n-collections defines a one-to-one mapping between the n c-points of the two collections. The deformation of a correspondence is the average displacement in the metric space for transforming one collection into the other according to the correspondence between them. The minimum distance is the minimum deformation of any correspondence between two collections. The nearest first distance and the farthest first distance are heuristic approximation of the minimum distance which are less expensive to compute. In applications where patterns can be represented by differentiated collections, pattern dissymmetry is quantified by their collection space distance. An example is given where fabric samples are classified based on their image color collections using different collection space minimum distance classifiers. A collection space method for measuring texture difference and bilateral symmetry in images is also presented