Metric Methods in Computer Vision and Pattern Recognition

Abstract

Since its inception, the notion of metric finds numerous applications not only in various domains of mathematics but also in computer science. Some special metrics (like Hausdorff metric on sets, Fréchet metrics on the set of curves, the Kantorovich metric on the set of probability measures as well as their versions and modifications) allow for obtaining quantitative estimations of dissimilarity between objects of different nature, in particular, images or sets of data. Comparison of objects is used in computer vision for evaluation of the accuracy of classifiers, search for objects by template, handwriting recognition. Therefore, we provide a survey of literature on the subject focusing on results concerning Fréchet metrics on the set of trees (in particular, curves) in metric space obtained by the authors. These metrics have their modifications called the Gromov-Fréchet metrics and they are defined on the isometry classes of (nonrooted) trees in metric spaces. In particular, we prove that the obtained space of trees is separable and non-complete. The consideration of the paper concern the quality of segmentation of objects that are presented in the form of polygons. Comparison of objects is based on the Fréchet metric between trees.