New techniques for topologically correct surface reconstruction

Adamy,; Giesen,; John, John

doi:10.1109/visual.2000.885718

Cited by 19 publications

(17 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Whether global minimization can also be applied to surface reconstruction remains an open question. A first step in this direction was recently taken by [1].…”

Section: Discussionmentioning

confidence: 99%

Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Althaus

Mehlhorn

2001

SIAM J. Comput.

View full text Add to dashboard Cite

Abstract. An instance of the curve reconstruction problem is a finite sample set V of an unknown collection of curves γ. The task is to connect the points in V in the order in which they lie on γ. Giesen [Proceedings of the 15th Annual ACM Symposium on Computational Geometry (SCG '99), 1999, pp. 207-216] showed recently that the traveling salesman tour of V solves the reconstruction problem for single closed curves under otherwise weak assumptions on γ and V ; γ must be a single closed curve. We extend his result along several directions:• we weaken the assumptions on the sample;• we show that traveling salesman-based reconstruction also works for single open curves (with and without specified endpoints) and for collections of closed curves; • we give alternative proofs; and • we show that in the context of curve reconstruction, the traveling salesman tour can be constructed in polynomial time.Key words. traveling salesman, polynomial time, curve reconstruction AMS subject classifications. 68Q25, 05C85PII. S0097539700366115 1.Introduction. An instance of the curve reconstruction problem is a finite sample set V of an unknown collection of curves γ. The task is to construct a graph G on V so that two points in V are connected by an edge of G iff the points are adjacent on γ. The curve reconstruction problem and the related surface reconstruction problem have received a lot of attention in the graphics and the computational geometry community. We are interested in reconstruction algorithms with guaranteed performance, i.e., algorithms which provably solve the reconstruction problem under certain assumptions on γ and V . Figure 1.1 illustrates the curve reconstruction problem.Many curve reconstruction algorithms have been proposed in the past; we restrict our discussion to algorithms that provably solve the reconstruction problem for a certain class of curves and under certain assumptions on the sample set. The algorithms differ with respect to the following aspects:• whether a collection of curves or just a single curve can be handled;• whether (collections of) open and closed curves can be handled or only (collections of) closed curves; • whether the sampling must be uniform or not. Uniform sampling with density requires that the sample set V contains at least one point from every curve segment of length . In nonuniform sampling, the sampling frequency may depend on local properties of the curve, e.g., can be lower in parts of low curvature;• whether nonsmooth curves can be handled or not. A smooth curve has a

show abstract

“…Whether global minimization can also be applied to surface reconstruction remains an open question. A first step in this direction was recently taken by [1].…”

Section: Discussionmentioning

confidence: 99%

Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Althaus

Mehlhorn

2001

SIAM J. Comput.

View full text Add to dashboard Cite

show abstract

“…The umbrella is full if p is mapped to the interior of the disk. In their surface reconstruction algorithm Adamy et al [2000] also implicitly assumed that it was possible to extract full umbrellas of Gabriel triangles around each vertex. A topological clean-up step filled holes in the extracted mesh, some of which were produced by other topological reparation steps.…”

Section: Related Work On Gabriel Meshesmentioning

confidence: 99%

Gabriel meshes and Delaunay edge flips

Dyer

Zhang

Möller

2009

2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling

View full text Add to dashboard Cite

We undertake a study of the local properties of 2-Gabriel meshes: manifold triangle meshes each of whose faces has an open Euclidean diametric ball that contains no mesh vertices. We show that, under mild constraints on the dihedral angles, such meshes are Delaunay meshes: the open geodesic circumdisk of each face contains no mesh vertex.The analysis is done by means of the Delaunay edge flipping algorithm and it reveals the details of the distinction between these two mesh structures. In particular we observe that the obstructions which prohibit the existence of Gabriel meshes as homeomorphic representatives of smooth surfaces do not hinder the construction of Delaunay meshes.

show abstract

“…(Voronoi/Delaunay filtering algorithms) If the points p i are sufficiently dense samples of a surface S then S can be reconstructed via filtering as a subset of the Delaunay triangulation of the p i [9,21,22]. As an advanced example, Amenta et al's power crust algorithm [18] proceeds as follows: First the Voronoi diagram of all sample points p i is computed.…”

Section: Voronoi Diagram and Delaunay Triangulationmentioning

confidence: 99%