Area distances of Convex Plane Curves and Improper Affine Spheres

TL;DR

This paper explores the connection between area distances of convex plane curves and improper affine spheres, introducing new definitions and algorithms that enhance understanding and computation in computer vision.

Contribution

It establishes a strong link between area distances and improper affine spheres, leading to new theoretical insights and fast computational algorithms.

Findings

New definition of area distance on convex curve outer parts

Fast algorithms for computing area distances

Enhanced geometric understanding of improper affine spheres

Abstract

The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an area distance on the outer part of a convex plane arc. Also, based on the theory of discrete affine spheres, we propose fast algorithms to compute the area distances. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.