Categorified Reeb Graphs
Categorified Reeb Graphs
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DOI：
10.1007/s00454-016-9763-9
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发表时间：
2015-01
影响因子：
0.8
通讯作者：
V. Silva;E. Munch;A. Patel
中科院分区：
文献类型：
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作者：
V. Silva;E. Munch;A. Patel
Reeb图是起源于莫尔斯理论的一种构造，用于研究定义在拓扑空间上的实值函数。最近，它已被用于各种应用中，以研究噪声数据，这产生了一个愿望，以定义这些结构之间的相似性的措施。在这里，我们利用的事实，一类Reeb图是等价的一类特定的cosheaf。利用这个等价性，我们可以定义一个在函数扰动下稳定的Reeb图之间的“交织”距离。沿着的方式，我们得到了一个自然的构造光滑的Reeb图，以减少其拓扑复杂性。光滑Reeb图可以在多项式时间内构造。
The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time.