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We study the problem of approximating a 3D solid with a union of overlapping spheres. In comparison with a state-of-the-art approach, our method offers more than an order of magnitude speedup and achieves a tighter approximation in terms of volume difference with the original solid while using fewer spheres. The spheres generated by our method are internal… (More)

Medial surfaces are popular representations of 3D objects in vision, graphics and geometric modeling. They capture relevant symmetries and part hierarchies and also allow for detailed differential geometric information to be recovered. However, exact algorithms for their computation from meshes must solve high-order polynomial equations, while approximation… (More)

We introduce a novel algorithm to compute a dense sample of points on the medial locus of a polyhedral object, with a guarantee that each medial point is within a specified tolerance ε from the medial surface. Motivated by Damon's work on the relationship between the differential geometry of the smooth boundary of an object and its medial surface… (More)

The medial axis transform is valuable for shape representation as it is complete and captures part structure. However, its exact computation for arbitrary 3D models is not feasible. We introduce a novel algorithm to approximate the medial axis of a poly-hedron with a dense set of medial points, with a guarantee that each medial point is within a specified… (More)

—A common approach to approximating the medial axis decides the presence of medial points in a region of non-zero size by analyzing the gradient of the distance transform at a finite number of locations in this region. In general, algorithms of this type do not guarantee completeness. In this paper, we consider a novel medial axis approximation algorithm of… (More)

We study connectivity relations among points, where the precise location of each input point lies in a region of uncertainty. We distinguish two fundamental scenarios under which uncertainty arises. In the favorable Best-Case Uncertainty, each input point can be chosen from a given set to yield the best possible objective value. In the unfavorable… (More)

We consider an algorithm, first presented in [13], that outputs regions intersected by the medial axis of a 3D solid. In practice, this algorithm is used to approximate the medial axis with a collection of points having a desired density. The quality of the medial axis approximation is supported by experimental results. Despite promising 2D results, the… (More)

- David Bremner, Jonathan Lenchner, Giuseppe Liotta, Christophe Paul, Marc Pouget, Svetlana Stolpner +1 other
- 2008

We study the problem of realizing a given graph as an α-complex of a set of points in the plane. We study the realizability problem for trees and 2-trees. In the case of 2-trees, we confine our attention to the realizability of graphs as the α-complex minus faces of dimension two; in other words, realizability of the graph in terms of the 1-skeleton of the… (More)

- Svetlana Stolpner, Jonathan Lenchner, Giuseppe Liotta, David Bremner, Christophe Paul, Marc Pouget +1 other
- CCCG
- 2008

This lecture introduces the Ellipsoid Method, the first polynomial-time algorithm to solve LP. We start by discussing the historical significance of its discovery by L. Khachiyan. Next, we argue that the ability to decide the feasibility of a version of the constraints of an LP is as hard as solving the LP. To gather the intuition behind the Ellipsoid… (More)

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