A discrete λ-medial axis

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 ; 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 or Z, 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. Hence, there is a rich literature devoted to skeleton or medial axis pruning, in which different criteria were proposed in order to discard “spurious” skeleton This work has been partially supported by the “ANR BLAN07–2 184378 MicroFiss” project. points or branches: see [4, 24, 3, 21, 2, 27, 17, 5, 13], to cite only a few. However, we lack theoretical justification, that is, a formalized argument that would help to understand why a filtering criterion is better than another. In 2005, Chazal and Lieutier introduced the λ-medial axis and studied its properties, in particular those related to stability [9]. Consider a bounded subset X of R, as for example, for n = 2, the region enclosed by the solid curve depicted in Fig. 1 (left). Let x be a point in X , we denote by Π(x) the set of points of the boundary of X that are closest to x. For example in Fig. 1, we have Π(x) = {a, b}, Π(x) = {a, b} and Π(x) = {a}. Let λ be a non-negative real number, the λ-medial axis of X is the set of points x of X such that the smallest ball including Π(x) has a radius greater than or equal to λ. Notice that the 0-medial axis of X is equal to X , and that any λ-medial axis with λ > 0 is included in the medial axis according to definition b). We show in Fig. 1 (right) two λ-medial axes with different values of λ. A major outcome of [9] is the following property: informally, for “regular” values of λ, the λ-medial axis remains stable under perturbations of X that are small with regard to the Hausdorff distance. Typical non-regular values are radii of locally largest maximal balls.