Oct-trees and their use in representing three-dimensional objects

Jackins, Chris L.; Tanimoto, Steven L.

doi:10.1016/0146-664x(80)90055-6

Cited by 499 publications

(136 citation statements)

References 3 publications

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“…Later years saw the introduction of numerous other types of trees: oct-trees [29], metric trees [62], vantage-point trees [64], random projection trees [19], spill trees [43], cover trees [8], cone trees [56]-to name just a few.…”

Section: Speedups Via Treesmentioning

confidence: 99%

Dual‐tree fast exact max‐kernel search

Curtin

Ram

2014

Statistical Analysis

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The problem of max-kernel search arises everywhere: given a query point pq, a set of reference objects Sr and some kernel K, find arg max pr ∈Sr K(pq, pr). Max-kernel search is ubiquitous and appears in countless domains of science, thanks to the wide applicability of kernels. A few domains include image matching, information retrieval, bio-informatics, similarity search, and collaborative filtering (to name just a few). However, there are no generalized techniques for efficiently solving max-kernel search. This paper presents a single-tree algorithm called single-tree FastMKS which returns the max-kernel solution for a single query point in provably O(log N ) time (where N is the number of reference objects), and also a dual-tree algorithm (dual-tree FastMKS) which is useful for max-kernel search with many query points. If the set of query points is of size O(N ), this algorithm returns a solution in provably O(N ) time, which is significantly better than the O(N 2 ) linear scan solution; these bounds are dependent on the expansion constant of the data. These algorithms work for abstract objects, as they do not require explicit representation of the points in kernel space. Empirical results for a variety of datasets show up to 5 orders of magnitude speedup in some cases. In addition, we present approximate extensions of the FastMKS algorithms that can achieve further speedups. Max-kernel searchOne particularly ubiquitous problem in computer science is that of max-kernel search: for a given set S r of N objects (the reference set), a similarity function K(·, ·), and a query object p q , find the object p r ∈ R such thatOften, max-kernel search is performed for a large set of query objects S q .The most simple approach to this general problem is a linear scan over all the objects in S r . However, the computational cost of this approach scales linearly with the size of the reference set for a single query, making it computationally prohibitive for large datasets. If |S q | = |S r | = O(N ), then this approach scales as O(N 2 ); thus, the approach quickly becomes infeasible for large N .In our setting we restrict the similarity function K(·, ·) to be a Mercer kernel. As we will see, this is not very restrictive. A Mercer kernel is a positive semidefinite kernel function; these can be expressed as an inner product in some Hilbert space H:

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Section: Speedups Via Treesmentioning

confidence: 99%

Dual‐tree fast exact max‐kernel search

Curtin

Ram

2014

Statistical Analysis

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“…The octree is a space-dividing, hierarchical tree data structure (Hunter, 1978;Jackins & Tanimoto, 1980;Meagher, 1982;Samet, 1985).…”

Section: Octree Encodingmentioning

confidence: 99%

Implementing Metric Operators of a Spatial Query Language for 3D Building Models: Octree and B-Rep Approaches

Borrmann

Schraufstetter

Rank

2009

J. Comput. Civ. Eng.

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A Spatial Query Language for Building Information Models enables the spatial analysis of buildings and the extraction of partial models that fulfill certain spatial constraints. Among other features, the developed spatial query language includes metric operators, i.e. operators that reflect distance relationships between spatial objects, such as mindist, maxdist, isCloser and isFarther. The paper presents formal definitions of the semantics of these operators by using point set theory notation. It further describes two possible implementation methods: the first one is based on a discrete representation of the operands' geometry by means of the hierarchical, space-partitioning data structure octree. The octree allows for the application of recursive algorithms that successively increase the discrete resolution of the spatial objects employed and thereby enables the user to trade-off between computational effort and the required accuracy. By contrast, the second approach uses the exact boundary representation (B-Rep) of both spatial objects resulting in precise distance measurements. Here the bounding facets of each operand are indexed by a so-called axis-aligned bounding boxes tree (AABB tree). The algorithm uses the AABB-tree structure to identify candidate pairs of facets, for which an exact but expensive distance algorithm is employed.

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“…In [45], Chien and Aggarwal presented an algorithm for generating an octree [46,47] of an object from 3 orthogonal views under orthographic projection. Their work was further developed by Ahuja and Veenstra [48], who extended the algorithm to handle images from any subset of 13 standard viewing directions.…”

Section: Volumetric Approachmentioning

confidence: 99%

Structure and motion estimation from apparent contours under circular motion

Wong

Mendonça

Cipolla

2002

Image and Vision Computing

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In this paper we address the problem of recovering structure and motion from the apparent contours of a smooth surface. Fixed image features under circular motion and their relationships with the intrinsic parameters of the camera are exploited to provide a simple parameterization of the fundamental matrix relating any pair of views in the sequence. Such a parameterization allows a trivial initialization of the motion parameters, which all bear physical meanings. It also greatly reduces the dimension of the search space for the optimization problem, which can now be solved using only 2 epipolar tangents. In contrast to previous methods, the motion estimation algorithm introduced here can cope with incomplete circular motion and more widely spaced images. Existing techniques for model reconstruction from apparent contours are then reviewed and compared. Experiment on real data has been carried out and the 3D model reconstructed from the estimated motion is presented.

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Oct-trees and their use in representing three-dimensional objects

Cited by 499 publications

References 3 publications

Dual‐tree fast exact max‐kernel search

Dual‐tree fast exact max‐kernel search

Implementing Metric Operators of a Spatial Query Language for 3D Building Models: Octree and B-Rep Approaches

Structure and motion estimation from apparent contours under circular motion

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