Optimal Topological Simplification of Discrete Functions on Surfaces
Optimal Topological Simplification of Discrete Functions on Surfaces
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曲面上离散函数的最优拓扑简化
DOI：
10.1007/s00454-011-9350-z
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发表时间：
2010
影响因子：
0.8
通讯作者：
M. Wardetzky
中科院分区：
文献类型：
--
作者：
Ulrich Bauer;C. Lange;M. Wardetzky
给定曲面上的函数f和公差δ>0，我们构造一个函数fδ满足<$fδ−f <$∞≤δ，使得fδ有最少临界点。我们的构造依赖于离散莫尔斯理论和持久同调之间的联系，并从输入函数f中完全去除持久性≤2δ的同调噪声。由此得到的简化函数fδ的临界点的数量达到了持久同调稳定性定理所规定的下限。我们表明，简化的功能，可以在线性时间计算持久性对计算后。
Given a function f on a surface and a tolerance δ>0, we construct a function fδ subject to ‖fδ−f‖∞≤δ such that fδ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the resulting simplified function fδ achieves the lower bound dictated by the stability theorem of persistent homology. We show that the simplified function can be computed in linear time after persistence pairs have been computed.