Binary clustering

Barthélemy, Jean‐Pierre; Brucker, François

doi:10.1016/j.dam.2007.05.024

Cited by 17 publications

(16 citation statements)

References 11 publications

(16 reference statements)

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“…We say that a set system C is identified by the transit function R if C = C R . A set system is called binary [3], if it is pre-binary and for every cluster C ∈ C there is a pair p, q ∈ X such that C = R C (p, q). The latter condition and the definition of pre-binary are equivalent to (KR) and (KC), respectively.…”

Section: Proposition 1 ([8]mentioning

confidence: 99%

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Cut Vertex Transit Functions of Hypergraphs

Changat

Nezhad

Stadler

2021

Algorithms and Discrete Applied Mathematics

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Transit functions serve not only as abstractions of betweenness and convexity but are also closely connected with clustering systems. Here, we investigate the canonical transit functions of binary clustering systems inspired by pyramids, i.e., interval hypergraphs. We provide alternative characterizations of weak hierarchies, and describe union-closed binary clustering systems as a subclass of pyramids and weakly pyramidal clustering systems as an interesting generalization.

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Section: Proposition 1 ([8]mentioning

confidence: 99%

“…This connection between transit function and clustering systems has been explored e.g. in [3,6,7,8].…”

Section: Introductionmentioning

confidence: 99%

Cut Vertex Transit Functions of Hypergraphs

Changat

Nezhad

Stadler

2021

Algorithms and Discrete Applied Mathematics

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“…• S denotes the set of objects or data to be clustered, S = 1, 2, ..., n. • A is a payoff matrix or similarity matrix. For binary clustering [24], a similarity matrix, which represents the similarities between objects to cluster, will be used as a payoff matrix in the clustering game model. In the image segmentation, an element (A i j ) is the distance in a color space between pixel i and pixel j.…”

Section: A Clustering Game Modelmentioning

confidence: 99%

A clustering game based framework for image segmentation

Shen

Blasch

Pham

et al. 2012

2012 11th International Conference on Information Science, Signal Processing and Their Applications (ISSPA)

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Image segmentation decomposes a given image into segments, i.e. regions containing "similar" pixels, that aids computer vision applications such as face, medical, and fingerprint recognition as well as scene characterization. Effective segmentation requires domain knowledge or strategies for object designation as no universal segmentation algorithm exists. In this paper, we propose a holistic framework to perform image segmentation in color space. Our approach unifies the linear smoothing filter, a similarity calculation in selected color space, and a clustering game model with various evolution dynamics. In our framework, the problem of image segmentation can be considered as a "clustering game". Within this context, the notion of a cluster turns out to be equivalent to a classical equilibrium concept from game theory, as the game equilibrium reflects both the internal and external cluster conditions. Experiments on image segmentation problems show the superiority of the proposed clustering game based image segmentation framework (CGBISF) using both the Berkeley segmentation dataset and infrared images (for which, we need to perform color fusion first) in autonomy, speed, and efficiency.

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“…They subsume a number of less general constructions such as paired hierarchies [6,7] and pyramids [20]. In [4] systems of clusters are considered that are generated by two elements. These are, as we shall see, closely related to set systems identified by transit functions.…”

Section: Introductionmentioning

confidence: 99%

“…In particular in the context hierarchies, much of the previous work focuses the relationships of transit functions and convexities [8,9], see also [4,15,25] on convexities in relation to graph-theoretical constructions. Here we take a somewhat different perspective and focus on axiom systems on transit functions giving raise to set systems such as those arising naturally in the context of clustering.…”

Section: Introductionmentioning

confidence: 99%

Axiomatic characterization of transit functions of weak hierarchies

Changat

Narasimha-Shenoi²,

Stadler

2018

Art Discrete Appl. Math.

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Transit functions provide a unified approach to study notions of intervals, convexities, and betweenness. Recently, their scope has been extended to certain set systems associated with clustering. We characterize here the class of set systems that correspond to k-ary monotonic transit functions. Convexities form a subclass and are characterized in terms of transit functions by two additional axioms. We then focus on axiom systems associated with weak hierarchies as well as other generalizations of hierarchical set systems. c b This work is licensed under http://creativecommons.org/licenses/by/3.0/ 2 Set systems identified by transit functions Throughout this contribution, V is a finite, non-empty set. Consider an arbitrary set system X ⊆ 2 V and the function R X (x, y) ={A ∈ X | x, y ∈ A}.(2.1)

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Binary clustering

Cited by 17 publications

References 11 publications

Cut Vertex Transit Functions of Hypergraphs

Cut Vertex Transit Functions of Hypergraphs

A clustering game based framework for image segmentation

Axiomatic characterization of transit functions of weak hierarchies

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