Helly, Radon, and Carathéodory Type Theorems

Eckhoff, Jürgen

doi:10.1016/b978-0-444-89596-7.50017-1

Cited by 243 publications

(203 citation statements)

References 230 publications

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“…Note that the distance between I k and the point m + π is larger than 2π/3 and thus m + π lies in no interval. Thus we can cut the unit circle at m + π and map it to the real line R. The theorem now follows from Helly's theorem [2]. Now, if all pairwise correspondences fulfill (7) with uncertainty intervals less than 2π/3, then from the above lemma we know that there exists an angle α such that constraint (6) is fulfilled for each single correspondence.…”

Section: Definitionmentioning

confidence: 78%

Two View Geometry Estimation with Outliers

Enqvist¹,

Kahl²

2009

Procedings of the British Machine Vision Conference 2009

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We study the relative orientation problem for two calibrated cameras with outliers from the feature matching. In recent years there has been a growing interest in optimal algorithms for computer vision. Most people agree that to get accurate solutions to multiview geometry problems, an appropriate norm of the reprojection errors should be minimized. To this end local as well as global optimization methods have been employed.To handle outliers though, heuristic methods still dominate the field. In this paper we address the problem of estimating relative orientation from uncertain feature correspondences. We formulate this task as an optimization problem and propose a branchand-bound algorithm to find the optimal set of correspondences as well as the optimal relative orientation. The approach is based on geometric constraints for pairs of correspondences. The experimental results are promising, especially for omnidirectional cameras. An implementation of the algorithm is also made publicly available to facilitate further research.

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Section: Definitionmentioning

confidence: 78%

Two View Geometry Estimation with Outliers

Enqvist¹,

Kahl²

2009

Procedings of the British Machine Vision Conference 2009

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“…. , A n } of nonempty and convex subsets of R 1 has as property that A j ∩ A k = ∅ for any 1 ≤ j, k ≤ n, j = k. Hence, by Helly's theorem (Eckhoff 1993…”

Section: N} Of Functions On [A B] Be Pairwise Poorly Convex Thementioning

confidence: 94%

Poor convexity and Nash equilibria in games

Radzik

2013

Int J Game Theory

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This paper considers two-person non-zero-sum games on the unit square with payoff functions having a new property called poor convexity. This property describes "something between" the classical convexity and quasi-convexity. It is proved that various types of such games have Nash equilibria with a very simple structure, consisting of the players' mixed strategies with at most two-element supports. Since poor convexity is a basic notion in the paper, also a theory of poorly convex functions is also developed.

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“…Proof: As long as τ (p, Q) > 0, p ∈ ch(Q), and by Carathéodory Theorem [Eck93], there is a d-simplex S with its d + 1 vertices in Q such that p ∈ S. Remove the vertices of S from Q, and repeat the argument. Let S 1 , .…”

Section: Tukey Depth and Convex Hullmentioning

confidence: 99%

Convex Hulls Under Uncertainty

et al. 2016

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We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input site is described by a probability distribution over a finite number of possible locations including a null location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of β-hull that may be a useful representation of uncertain hulls.

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Helly, Radon, and Carathéodory Type Theorems

Cited by 243 publications

References 230 publications

Two View Geometry Estimation with Outliers

Two View Geometry Estimation with Outliers

Poor convexity and Nash equilibria in games

Convex Hulls Under Uncertainty

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