Computational morphology of curves

Figueiredo, Luiz Henrique de; Gomes, Jonas

doi:10.1007/bf01889981

Cited by 55 publications

(37 citation statements)

References 8 publications

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“…In two dimensions, there are a number of recent theoretical results on various Delaunay-based approaches to reconstructing smooth curves. Attali [3], Bernardini and Bajaj [5], Figueiredo and Miranda Gomes [11] and ourselves [1] have all given guarantees for different algorithms.…”

Section: Related Workmentioning

confidence: 99%

A new Voronoi-based surface reconstruction algorithm

Amenta

Bern

Kamvysselis

1998

Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '98

722

490

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We describe our experience with a new algorithm for the reconstruction of surfaces from unorganized sample points in ¢ £ ¥ ¤. The algorithm is the first for this problem with provable guarantees. Given a "good sample" from a smooth surface, the output is guaranteed to be topologically correct and convergent to the original surface as the sampling density increases. The definition of a good sample is itself interesting: the required sampling density varies locally, rigorously capturing the intuitive notion that featureless areas can be reconstructed from fewer samples. The output mesh interpolates, rather than approximates, the input points.Our algorithm is based on the three-dimensional Voronoi diagram. Given a good program for this fundamental subroutine, the algorithm is quite easy to implement.

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Section: Related Workmentioning

confidence: 99%

A new Voronoi-based surface reconstruction algorithm

Amenta

Bern

Kamvysselis

1998

Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH '98

722

490

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show abstract

“…Like most other algorithms, dense, uniformly sampled point sets are rather easily handled by our algorithm; this is also evident from the proof given in [17]. For such dense point configurations, the EMS T already shares most edges with the desired boundary and the solution space tree remains small.…”

Section: Conforming Point Set Configurationsmentioning

confidence: 67%

“…This relationship between the minimum spanning tree and the shape has been mentioned in Figueiredo and Gomes [17]. However, they only prove reconstruction for very densely sampled point sets: an EMS T without branches.…”

Section: Construction As Global Minimization Of a Criterionmentioning

confidence: 87%

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Interpolating an unorganized 2D point cloud with a single closed shape

Ohrhallinger

Mudur

2011

Computer-Aided Design

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Given an unorganized two-dimensional point cloud, we address the problem of efficiently constructing a single aesthetically pleasing closed interpolating shape, without requiring dense or uniform spacing. Using Gestalt's laws of proximity, closure and good continuity as guidance for visual aesthetics, we require that our constructed shape be minimal perimeter, non-self intersecting and manifold. We find that this yields visually pleasing results. Our algorithm is distinct from earlier shape reconstruction approaches, in that it exploits the overlap between the desired shape and a related minimal graph, the Euclidean Minimum Spanning Tree (EMS T ). Our algorithm segments the EMS T to retain as much of it as required and then locally partitions and solves the problem efficiently. Comparison with some of the best currently known solutions shows that our algorithm yields better results.Keywords: computational geometry, reconstruction, construction, shape, curve, boundary, point cloud, point set, EMST

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“…A smooth curve has a For uniformly sampled collections of closed, smooth curves several methods are known to work ranging over minimum spanning trees [12], α-shapes [5,11], β-skeletons [16], and r-regular shapes [4]. A survey of these techniques appears in [10].…”

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confidence: 99%

Traveling Salesman-Based Curve Reconstruction in Polynomial Time

Althaus

Mehlhorn

2001

SIAM J. Comput.

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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

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Computational morphology of curves

Cited by 55 publications

References 8 publications

A new Voronoi-based surface reconstruction algorithm

A new Voronoi-based surface reconstruction algorithm

Interpolating an unorganized 2D point cloud with a single closed shape

Traveling Salesman-Based Curve Reconstruction in Polynomial Time

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