Area-Based Medial Axis of Planar Curves

Abstract

A new definition of affine invariant medial axis of planar closed curves is introduced. A point belongs to the affine medial axis if and only if it is equidistant from at least two points of the curve, with the distance being a minimum and given by the areas between the curve and its corresponding chords. The medial axis is robust, eliminating the need for curve denoising. In a dynamical interpretation of this affine medial axis, the medial axis points are the affine shock positions of the affine erosion of the curve. We propose a simple method to compute the medial axis and give examples. We also demonstrate how to use this method to detect affine skew symmetry in real images.

Area Distances of Convex Plane Curves and Improper Affine Spheres

“…Denote by D the plane region whose boundary is C and the curve(s) obtained from C by a similarity of ratio Any p ∈ D is the mean point of the extremities of the chord l(p). This important property was proved first in [6] (see also [7]). Another important property is that ∇(f )(p) is orthogonal to the chord l(p), with half of its length (see Figure 2).…”

Affine Skeletons and Monge–Ampère Equations

“…The area distance was introduced by Moisan [2], and used, with a slight modication, by Giblin and Sapiro et al [3]. The process is the following: take a point γ (s) on Γ and a point X inside Γ; consider the chord that starts at γ(s) and goes through X this chord meets the curve Γ at one other point γ(r) (remember, Γ is convex).…”

“…The structure of this paper is as follows: rst, we briey review the main properties of medial axis and the denitions of the ane skeletons as in [5], [6], [3] and [1]. Next, we show how the Ane Distance (and the old Euclidean Distance) can be thought of as solutions to homogeneous Monge-Ampère equations (which distance is obtained depends on specic boundary conditions).…”

“…One of its attractive properties is the covariance by rigid transformations. In a series of papers ( [5], [6], [3] and [1]), Giblin, Sapiro et al, introduced new skeletons inspired in the medial axis transformation, but having the nice property of being covariant by all ane transformations. They used two new denitions of distance (the Ane Distance and the Area Distance) to introduce two new ane symmetry sets: the Ane Distance Symmetry Set (ADSS) and the Area Distance Symmetry Set (AASS).…”

An Approach to the Parameterization of Structure for Fast Categorization

“…In other studies, the SAT model aims at evolving sym-axes which represent the shape with high accuracy, applied for instance to the analysis and/or representation of medical image structures, e.g. (Ogniewicz and Kubler, 1995;Zhu and Yuille, 1996;Zhu, 1999;Niethammer et al, 2004). The group by Zucker classifies the symmetric-axes into 4 orders (Kimia et al, 1995;Siddiqi et al, 1999;Pelillo et al, 1999): 1st order is a U-shaped curve (protrusion; half an ellipse), 2nd order is a neck (two half-way fused circles), 3rd order is an oval (bend) and 4th order is a circle (seed).…”