Affine invariant edge completion with affine geodesics

Abstract

Edge completion is the interpolation of gaps between edge segments which are extracted from an image. We provide a new analytic solution to this problem within equi-affine plane geometry which is the natural framework for the interpolation of pairs of line segments. The desired curves are the geodesics of equi-affine plane geometry, namely parabolic arcs, which generalize the connection of points by straight lines in Euclidean geometry. Whereas most common methods of edge completion are invariant only under the group of Euclidean motions, SE(2), this solution has the advantage of being invariant under the larger group of equi-affine transformations, SA(2), that is more relevant to computer vision. In addition to these geometric qualities, the parabola is a simple algebraic curve which renders it computationally attractive, especially in comparison to the popular elastica curves.