Approximations of 2D and 3D generalized Voronoi diagrams

Boada, Imma; Coll, Narcís; Madern, Narcis; Sellarès, J. Antoni

doi:10.1080/00207160701466362

Cited by 17 publications

(14 citation statements)

References 19 publications

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“…Boada et al [7] proposed an octree that approximates generalized Voronoi cells and that can be used to approximately solve the nearestneighbor problem [6]. Their work also gives insight into the construction costs of such an octree.…”

Section: Related Workmentioning

confidence: 99%

Almost constant-time 3D nearest-neighbor lookup using implicit octrees

Drost

Ilić

2017

Machine Vision and Applications

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A recurring problem in 3D applications is nearestneighbor lookups in 3D point clouds. In this work, a novel method for exact and approximate 3D nearest-neighbor lookups is proposed that allows lookup times that are, contrary to previous approaches, nearly independent of the distribution of data and query points, allowing to use the method in real-time scenarios. The lookup times of the proposed method outperform prior art sometimes by several orders of magnitude. This speedup is bought at the price of increased costs for creating the indexing structure, which, however, can typically be done in an offline phase. Additionally, an approximate variant of the method is proposed that significantly reduces the time required for data structure creation and further improves lookup times, outperforming all other methods and yielding almost constant lookup times. The method is based on a recursive spatial subdivision using an octree that uses the underlying Voronoi tessellation as splitting criteria, thus avoiding potentially expensive backtracking. The resulting octree is represented implicitly using a hash table, which allows finding the leaf node a query point belongs to with a runtime that is logarithmic in the tree depth. The method is also trivially extendable to 2D nearest neighbor lookups.

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Section: Related Workmentioning

confidence: 99%

Almost constant-time 3D nearest-neighbor lookup using implicit octrees

Drost

Ilić

2017

Machine Vision and Applications

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“…Etzion and Rappoport [ER02] represent the GVD bisector symbolically for lazy evaluation, but are limited to sites that are polyhedra. Boada et al [BCS02, BCMAS08] use an adaptive approach to GVD computation, but their algorithm is restricted to GVDs with connected regions and is inefficient for polyhedral objects with many facets. Two other works are adaptive [TT97,VO98] but are computationally expensive and are restricted to convex sites.…”

Section: Related Workmentioning

confidence: 99%

Approximating the Generalized Voronoi Diagram of Closely Spaced Objects

Edwards

Daniel

Pascucci

et al. 2015

Computer Graphics Forum

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We present an algorithm to compute an approximation of the generalized Voronoi diagram (GVD) on arbitrary collections of 2D or 3D geometric objects. In particular, we focus on datasets with closely spaced objects; GVD approximation is expensive and sometimes intractable on these datasets using previous algorithms. With our approach, the GVD can be computed using commodity hardware even on datasets with many, extremely tightly packed objects. Our approach is to subdivide the space with an octree that is represented with an adjacency structure. We then use a novel adaptive distance transform to compute the distance function on octree vertices. The computed distance field is sampled more densely in areas of close object spacing, enabling robust and parallelizable GVD surface generation. We demonstrate our method on a variety of data and show example applications of the GVD in 2D and 3D.

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“…Boada et al . [5] proposed an octree that approximates generalized Voronoi cells and that can be used to approximately solve the nearest neighbor problem [4]. Their work also gives insight into the construction costs of such an octree.…”

Section: Related Workmentioning

confidence: 99%

A Hierarchical Voxel Hash for Fast 3D Nearest Neighbor Lookup

Drost

Ilić

2013

Lecture Notes in Computer Science

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Abstract. We propose a data structure for finding the exact nearest neighbors in 3D in approximately O(log(log(N )) time. In contrast to standard approaches such as k-d-trees, the query time is independent of the location of the query point and the distribution of the data set. The method uses a hierarchical voxel approximation of the data point's Voronoi cells. This avoids backtracking during the query phase, which is a typical action for tree-based methods such as k-d-trees. In addition, voxels are stored in a hash table and a bisection on the voxel level is used to find the leaf voxel containing the query point. This is asymptotically faster than letting the query point fall down the tree. The experiments show the method's high performance compared to state-of-the-art approaches even for large point sets, independent of data and query set distributions, and illustrates its advantage in real-world applications.

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Approximations of 2D and 3D generalized Voronoi diagrams

Cited by 17 publications

References 19 publications

Almost constant-time 3D nearest-neighbor lookup using implicit octrees

Almost constant-time 3D nearest-neighbor lookup using implicit octrees

Approximating the Generalized Voronoi Diagram of Closely Spaced Objects

A Hierarchical Voxel Hash for Fast 3D Nearest Neighbor Lookup

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