Ralph Teixeira

Area Distances of Convex Plane Curves and Improper Affine Spheres

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. Based on this link, we propose an extremely fast algorithm to compute the inner area distance. Moreover, 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 curve. On the other hand, area distances provide a good geometrical understanding of improper affine spheres.

A Fast Marching Method for the Area Based Affine Distance

Abstract In a previous paper, it was proved that the area based affine distance of a convex region in the plane satisfies a non-homogeneous Monge-Ampère differential equation. Based on this equation, in this paper we propose a fast marching method for the computation of this distance. The proposed algorithm has a lower computational complexity than the direct method and we have proved its convergence. And since the algorithm allows one to obtain a connection from any point of the region to the boundary by a path of decreasing distance, it offers a dynamic point of view for the area based affine distance.

Medial Axes and Mean Curvature Motion II: Singularities

What happens to the medial axis of a curve that evolves through MCM (Mean Curvature Motion)? We explore some theoretical results regarding properties of both medial axes and curvature motions. Specifically, using singularity theory, we present all possible topological transitions of a symmetry set (of which the medial axis is a subset) whose originating curve undergoes MCM. All calculations are presented in a clear and organized fashion and are easily generalized for other front motions. A companion article deals with non-singular points of the medial axis through direct calculations.

A Numerical Scheme for the Curvature Equation Near the Singularities

In this paper we propose a modification in the usual numerical method for computing the solutions of the curvature equation in the plane . This modification takes place near the singularities of the image. We propose to use zero as the vertical speed at a saddle point and, at an extremum, the geometric mean of the eigenvalues of the Hessian matrix. This modification is theoretically justified and the preliminary experimental results show that it makes the algorithm more reliable.

Closed cycloids in a normed plane

Given a normed plane

Polygons with Parallel Opposite Sides

\mathcal {P}

Discrete Cycloids from Convex Symmetric Polygons

P, we call