Use of Integral Signature and Hausdorff Distance in Planar Curve Matching

Abstract

Curve matching is an important problem in computer image processing and image recognition. In particular, the problem of identifying curves that are equivalent under a geometric transformation arises in a variety of applications. Two curves in $mathbb{R}^2$ are called congruent if they are equivalent under the action of the Euclidean group, i.e. if one curve can be mapped to the other by a combination of rotations, reflections, and translations. In theory, one can identify congruent curves by using differential invariants, such as infinitesimal arc-length and curvature. The practical use of differential invariants is problematic, however, due to their high sensitivity to noise and small perturbations. Other types of invariants that are less sensitive to perturbations were proposed in literature, but are much less studied than classical differential invariants. In this thesis we provide a detailed study of matching algorithms for planar curves based on Euclidean integral invariant signatures. Several types of local and global signatures are considered. We examine numerical approximations of signatures, sensitivity to perturbation, dependence on parametrization and a choice of initial point, and effects of the symmetries of the original image on signatures. Furthermore, we use Hausdorff distance between signatures to define a distance between congruence classes of curves.