
A Generalized Framework for Edgepreserving and Structurepreserving Image Smoothing
Image smoothing is a fundamental procedure in applications of both compu...
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nmetrics for multiple graph alignment
The work of Ioannidis et al. 2018 introduces a family of distances betwe...
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A Family of Tractable Graph Distances
Important data mining problems such as nearestneighbor search and clust...
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On constant multicommodity flowcut gaps for directed minorfree graphs
The multicommodity flowcut gap is a fundamental parameter that affects...
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Realizable piecewise linear paths of persistence diagrams with Reeb graphs
Reeb graphs are widely used in a range of fields for the purposes of ana...
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A generalization of zerodivisor graphs
In this paper, we introduce a family of graphs which is a generalization...
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Finitesize effects in response functions of molecular systems
We consider an electron in a localized potential submitted to a weak ext...
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A family of metrics from the truncated smoothing of Reeb graphs
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter τ. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ≤τ≤ 2ε, where ε is the smoothing parameter. Then, for the restriction of τ∈ [0,ε], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ∈ [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ∈ [0,1], which is a generalization of the original interleaving distance, which is the case m=0. While the resulting metrics are not stable, we show that any pair of these for m,m' ∈ [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.
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