Separability and Tight Enclosure of Point Sets

Veelaert, Peter

doi:10.1007/978-94-007-4174-4_8

Cited by 3 publications

(7 citation statements)

References 26 publications

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“…This result has an immediate corollary [26]. It suffices to note that an elemental subset can only give rise to enclosing hyperplanes if its height is maximal.…”

Section: Exchange Algorithm For Unsigned Enclosuresmentioning

confidence: 87%

“…On the other hand, E ∪ {q} must contain at least one elemental subset that has the same height as E ∪ {q}. Thus, we have the following result, which is also known as the Stiefel exchange property [18,26].…”

Section: Exchange Algorithm For Unsigned Enclosuresmentioning

confidence: 91%

“…A proof of this particular version can be found in [25,26]. Geometrically, the theorem states that the height (E) is one half times the difference in height between the enclosing hyperplanes.…”

Section: Enclosure Of Elemental Subsetsmentioning

confidence: 96%

“…There is no signed enclosure that is tighter than the unsigned enclosure of Theorem 1 [25,26]. To formulate this more precisely, given a partition {S + , S − } of S, for each elemental subset E in S + ∪S − , we define…”

Section: Lemma 2 Let C I Be the Cofactors Of The Last Column Of The Mmentioning

confidence: 99%

“…In this section we describe a combinatorial algorithm for finding unsigned enclosures, which is based on elemental subsets [26]. Elemental is a term borrowed from robust regression [10,21].…”

Section: Elemental Subsetsmentioning

confidence: 99%

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Combinatorial properties of support vectors of separating hyperplanes

Veelaert

2014

Ann Math Artif Intell

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In this paper we study the relationship between separating hyperplanes and the Radon partitions of their support vectors. This study is relevant for maximal margin separations, which appear in Support Vector Machines (SVM), as well as for separations that optimize a Chebyshev norm. We propose a new version of the Stiefel exchange algorithm where we exploit the property that each Stiefel exchange is in fact a Radon exchange. Originally, the Stiefel exchange algorithm was developed to find Chebyshev approximations, but we show that it is also suited for finding hyperplane separations. We also show that many important properties in approximation theory are closely related to fundamental results in convex set theory, in particular to Helly's, Radon's and Caratheodory's Theorem. Within this context, we prove a new result that generalizes both Radon's and Caratheodory's Theorem.

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“…This result has an immediate corollary [26]. It suffices to note that an elemental subset can only give rise to enclosing hyperplanes if its height is maximal.…”

Section: Exchange Algorithm For Unsigned Enclosuresmentioning

confidence: 87%

Section: Exchange Algorithm For Unsigned Enclosuresmentioning

confidence: 91%

Section: Enclosure Of Elemental Subsetsmentioning

confidence: 96%

Section: Lemma 2 Let C I Be the Cofactors Of The Last Column Of The Mmentioning

confidence: 99%

Section: Elemental Subsetsmentioning

confidence: 99%

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Combinatorial properties of support vectors of separating hyperplanes

Veelaert

2014

Ann Math Artif Intell

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Fast Combinatorial Algorithm for Tightly Separating Hyperplanes

Veelaert¹

2012

Combinatorial Image Analaysis

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Curvature-based Human Body Parts Segmentation in Physiotherapy

Deboeverie

Geest

Tuytelaars

et al. 2015

Proceedings of the 10th International Conference on Computer Vision Theory and Applications

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Analysing human sports activity in computer vision requires reliable segmentation of the human body into meaningful parts, such as arms, torso and legs. Therefore, we present a novel strategy for human body segmentation. Firstly, greyscale images of human bodies are divided into smooth intensity patches with an adaptive region growing algorithm based on low-degree polynomial fitting. Then, the key idea in this paper is that human body parts are approximated by nearly cylindrical surfaces, of which the axes of minimum curvature accurately reconstruct the human body skeleton. Next, human body segmentation is qualitatively evaluated with a line segment distance between reconstructed human body skeletons and ground truth skeletons. When compared with human body parts segmentations based on mean shift, normalized cuts and watersheds, the proposed method achieves more accurate segmentations and better reconstructions of human body skeletons.

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Separability and Tight Enclosure of Point Sets

Cited by 3 publications

References 26 publications

Combinatorial properties of support vectors of separating hyperplanes

Combinatorial properties of support vectors of separating hyperplanes

Fast Combinatorial Algorithm for Tightly Separating Hyperplanes

Curvature-based Human Body Parts Segmentation in Physiotherapy

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