A Discrete λ-Medial Axis

Abstract

Abstract. The λ-medial axis was introduced in 2005 by Chazal and Lieutier as a new concept for computing the medial axis of a shape subject to filtering with a single parameter. These authors proved the stability of the λ-medial axis under small shape perturbations. In this paper, we introduce the definition of a discrete λ-medial axis (DLMA). We evaluate its stability and rotation invariance experimentally. The DLMA may be computed by efficient algorithms, furthermore we introduce a variant of the DLMA, denoted by DL'MA, which may be computed in linear-time. We compare the DLMA and the DL'MA with the recently introduced integer medial axis and show that both DLMA and DL'MA provide measurably better results.In the 60s, Blum [7,8] introduced the notion of medial axis or skeleton, which has since been the subject of numerous theoretical studies and has also proved its usefulness in practical applications. Although initially introduced as the outcome of a propagation process, the medial axis can also be defined in simple geometric terms. In the continuous Euclidean space, the two following definitions can be used to formalize this notion: let X be a bounded subset of R n ;a) The skeleton of X consists of the centers of the balls that are included in X but that are not included in any other ball included in X.b) The medial axis of X consists of the points x ∈ X that have several nearest points on the boundary of X.The skeleton and the medial axis differ only by a negligible set of points (see [22]), in general the skeleton is a strict subset of the medial axis.In this paper, we focus on medial axes in the discrete grid Z 2 or Z 3 , which are centered in the shape with respect to the Euclidean distance.A major difficulty when using the medial axis in applications (e.g., shape recognition), is its sensitivity to small contour perturbations, in other words, its lack of stability. A recent survey [1] summarizes selected relevant studies dealing with this topic. This difficulty can be expressed mathematically: the transformation which associates a shape to its medial axis is only semi-continuous. This fact, among others, explains why it is usually necessary to add a filtering step (or pruning step) to any method that aims at computing the medial axis.

Spectral Analysis on Medial Axis of 2D Shapes

“…It means that small variations of the object boundary may cause a large change to the medial axis, as illustrated by Figure 1. The excessive geometric complexity and pathological topology of such an unstable medial axis make it difficult to use, and there have been substantial efforts aiming to produce filtered or simplified medial axis [ACK01,SFM05,CCT09,GMPW09]. The excessive geometric complexity and pathological topology of such an unstable medial axis make it difficult to use, and there have been substantial efforts aiming to produce filtered or simplified medial axis [ACK01,SFM05,CCT09,GMPW09].…”

Surface Thinning in 3D Cubical Complexes

“…In [9], we introduced the definition of a discrete λ-medial axis (DLMA) in Z n . We evaluated experimentally its stability and rotation invariance.…”

“…From here, we will consider the complex X = Φ(S) − and the set W = Φ(DLM A(S, λ)). Notice that we have no guarantee that the complex W − be topologically equivalent to the complex X (see counter-examples in [9]). Thus our next step consists of computing, using Alg.…”

“…Then, we propose two variants of a surface-preserving thinning scheme, based on this parallel directional thinning algorithm. Finally, we propose a methodology to produce filtered surface skeletons, based on the above thinning methods and the recently introduced discrete λ-medial axis [9].…”

“…Transposing directly the definition of the λ-medial axis to the discrete grid Z n would yield unsatisfactory results (see [9]), this is why we need the following notion. Let S ⊆ Z n , and let x ∈ S. The extended projection of x on S, denoted by Π e S (x), is the union of the sets…”