Reeb Graphs: Approximation and Persistence

Given a continuous function f:X→ℝ on a topological space X, its level setf−1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph, which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H1-homology of the Reeb graph from P. It takes O(nlogn) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H1(M) from point data. The best-known previous algorithm for this problem takes O(n3) time for point data. The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H1-homology for the Reeb graphs in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n n_{e}^{3})$\end{document} time, where n is the size of the 2-skeleton and ne is the number of edges in K.

在拓扑空间x上，连续函数f：x→selef -1（a）随着实际值的变化而不断变化。促进其形状和数据分析的更广泛的应用。以o（nlogn）为预期的时间，其中n是RIPS复合物的2个骨骼的大小，当M是一个有效的2个manifold时，我们还可以获得一个有效的接近线性时间（预期）算法，以计算以前的algorith section time op poins op time op-n3。 EB图同源性的REEB图序列在过滤空间上定义了在简单复合物K的过滤下定义的分段线性功能，我们的算法计算\ documentclass [12pt] {minimal} ymb} \ usepackage {amsbsy} \ usepackage {mathrsfs} \ usepackage {upgreek} \ setLength {\ oddSidemargin} { - 69pt} { - 69pt} \ begin n_ {e}^{3}）$ \ end {document}时间，其中n是2-skeleton的大小，ne是k中的边数。