Stability and Computation of Medial Axes - a State-of-the-Art Report

Attali, Dominique; Boissonnat, Jean‐Daniel; Edelsbrunner, Herbert

doi:10.1007/b106657_6

Cited by 132 publications

(144 citation statements)

References 45 publications

(57 reference statements)

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“…Chazal and Lieutier [6] defined the λ-medial axis, a subset of the medial axis, which has the desired stability, and is guaranteed to have the same homotopy type as the medial axis for suitably small values of the parameter λ. The largest topologically safe λ depends on the shape and can be very small and hard to determine [4]. Finally, the existing algorithm for computing the λ-medial axis requires a very dense uniform sample of the boundary of the shape [6].…”

Section: Introductionmentioning

confidence: 99%

Medial Axis Approximation and Unstable Flow Complex

Giesen

Ramos

Sadri

2008

Int. J. Comput. Geom. Appl.

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The medial axis of a shape is known to carry a lot of information about it. In particular a recent result of Lieutier establishes that every bounded open subset of R n has the same homotopy type as its medial axis. In this paper we provide an algorithm that, given a sufficiently dense but not necessarily uniform sample from the surface of a shape with smooth boundary, computes a core for its medial axis approximation, in form of a piecewise linear cell complex, that captures the topology of the medial axis of the shape. We also provide a natural method to freely augment this core in order to enhance it geometrically all the while maintaining its topological guarantees. The definition of the core and its extension method are based on the steepest ascent flow induced by the distance function to the sample. We also provide a geometric guarantee on the closeness of the core and the actual medial axis.

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Section: Introductionmentioning

confidence: 99%

Medial Axis Approximation and Unstable Flow Complex

Giesen

Ramos

Sadri

2008

Int. J. Comput. Geom. Appl.

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“…Still, the inverse event E −1 3 is capable of restoring the vanished facet in one of its instances; see above. A similar situation arises when a tetrahedron shrinks to a vertex, in the events E 5 or E 6 .…”

Section: Lemma 63 Let E Be a Generic Non-initial Eventmentioning

confidence: 88%

“…(For unweighted polygons, both edge events and split events always yield a unique offset boundary, at least in the classical setting. 6 ) However, edge weights are not independent in our case, which leads to the special property below that we will prove first. Let us call a polygon vertex convex, reflex, or flat, respectively, if its incident interior angle is smaller, larger, or equal to π .…”

Section: Pointed Case Revisitedmentioning

confidence: 95%

“…It represents the set of centers of all multi-tangential circles (or spheres) inscribed to the object. Defined by distances to the boundary, the medial axis keeps a strong correspondence to the shape of the object; we refer the reader to [6,8,29] for an account of properties and further literature. However, even when the object boundary is piecewise linear (like for a polygon in the plane, or a polytope in 3-space), the medial axis contains curved elements when the object is nonconvex.…”

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confidence: 99%

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Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Aurenhammer

Walzl

2016

Discrete Comput Geom

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We give a concise definition of mitered offset surfaces for nonconvex polytopes in R 3 , along with a proof of existence and a discussion of basic properties. These results imply the existence of 3D straight skeletons for general nonconvex polytopes. The geometric, topological, and algorithmic features of such skeletons are investigated, including a classification of their constructing events in the generic case. Our results extend to the weighted setting, to a larger class of polytope decompositions, and to general dimensions. For (weighted) straight skeletons of an n-facet polytope in R d , an upper bound of O(n d ) on their combinatorial complexity is derived. It relies on a novel layer partition for straight skeletons, and improves the trivial bound by an order of magnitude for d ≥ 3.

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“…shape recognition). A recent survey which summarises selected relevant studies dealing with this topic is presented in [2]. 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 and when a nonreversible but simplified description of binary objects is of interest.…”

Section: Introductionmentioning

confidence: 99%

Scale Filtered Euclidean Medial Axis

Postolski

Couprie

Janaszewski

2013

Discrete Geometry for Computer Imagery

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Abstract. We propose an Euclidean medial axis filtering method which generates subsets of Euclidean medial axis were filtering rate is controlled by one parameter. The method is inspired by Miklos', Giesen's and Pauly's scale axis method which preserves important features of an input object from shape understanding point of view even if they are at different scales. Our method overcomes the most important drawback of scale axis: scale axis is not, in general, a subset of Euclidean medial axis. It is even not necessarily a subset of the original shape. The method and its properties are presented in 2D space but it can be easily extended to any dimension. Experimental verification and comparison with a few previously introduced methods are also included.

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Stability and Computation of Medial Axes - a State-of-the-Art Report

Cited by 132 publications

References 45 publications

Medial Axis Approximation and Unstable Flow Complex

Medial Axis Approximation and Unstable Flow Complex

Straight Skeletons and Mitered Offsets of Nonconvex Polytopes

Scale Filtered Euclidean Medial Axis

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