Loops in reeb graphs of 2-manifolds

Cole-McLaughlin, Kree; Edelsbrunner, Herbert; Harer, John; Natarajan, Vijay; Pascucci, Valerio

doi:10.1145/777842.777844

Cited by 41 publications

(41 citation statements)

References 9 publications

(9 reference statements)

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“…The proof will clarify the concept, and its methods will also be used in Section 5 for the construction of the pair of pants decomposition. Similar methods have been used in [9]. We can now state Py's second result [18].…”

Section: Essential Critical Pointsmentioning

confidence: 94%

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Rosenberg

2010

Isr. J. Math.

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Section: Essential Critical Pointsmentioning

confidence: 94%

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Rosenberg

2010

Isr. J. Math.

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“…This means two points (α, f (α)) and (γ , f (γ )) are represented as the same node in the Reeb graph if values of f are the same and they belong to the same connected component of the inverse image of f (α) (or, equivalently f (γ )). The Reeb quotient space is coded in a Reeb graph such that the vertices represent critical points [5,7,8,14,28] for details. The right of Fig.…”

Section: Methodsmentioning

confidence: 99%

Graph-based shape indexing

Demirci

2010

Machine Vision and Applications

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Graphs have become growingly important in representing shapes in computer vision. Given a query graph, it is essential to retrieve similar database graphs efficiently from a large database. In this paper, we present a graph-based indexing technique which overcomes significant drawbacks of the previous work (Demirci et al. in Comput Vis Image Underst 110 (3):312-325, 2008) using a recently developed theorem from the domain of matrix analysis. Our technique starts by representing the topological structure of a graph in a vector space. As done in the previous work, the topological structure of a graph is constructed using its Laplacian spectra. However, unlike the previous approach, which represents all sugraphs of a database graph in the vector space to account for local similarity, a database graph in the proposed framework is represented as a single vector. By performing a range search around the query, the proposed indexing technique returns a set with both partial and global similarity. Empirical evaluation of the algorithm on an extensive set of retrieval trials including a comparison with the previous approach in both 2D and 3D demonstrates the effectiveness, efficiency, and robustness of the overall approach.

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“…This notion was first considered by Reeb [22] in the framework of Morse theory (see e.g. [9] for a detailed discussion). It is illustrated on Figs.…”

Section: A Non-linear Quasi-state On Smentioning

confidence: 99%

An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics

2007

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We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a nonlinear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories-symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.

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Loops in reeb graphs of 2-manifolds

Cited by 41 publications

References 9 publications

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

Graph-based shape indexing

An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics

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