Efficient Output-Sensitive Construction of Reeb Graphs
نویسندگان
چکیده
The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. This paper describes a near-optimal two-step algorithm that constructs the Reeb graph of a Morse function defined over manifolds in any dimension. The algorithm first identifies the critical points of the input manifold, and then connects these critical points in the second step to obtain the Reeb graph. A simplification mechanism based on topological persistence aids in the removal of noise and unimportant features. A radial layout scheme results in a feature-directed drawing of the Reeb graph. Experimental results demonstrate the efficiency of the Reeb graph construction in practice and its applications.
منابع مشابه
Categorified Reeb Graphs
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 cla...
متن کاملThe Reeb Graph Edit Distance is Universal
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specif...
متن کاملStrong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs
The Reeb graph is a construction that studies a topological space through the lens of a real valued function. It has been commonly used in applications, however its use on real data means that it is desirable and increasingly necessary to have methods for comparison of Reeb graphs. Recently, several metrics on the set of Reeb graphs have been proposed. In this paper, we focus on two: the functi...
متن کاملComputation of Layered Reeb Graphs ∗
Reeb graphs represent the topological structure of a manifold based on a scalar-valued, sufficiently smooth function defined on it. The use of more than one function leads to Reeb spaces, which are thus able to capture more features of an object. The structure of the Reeb space of a 3-manifold with boundary with respect to two scalarvalued functions is captured by the layered Reeb graph. We pre...
متن کاملLayered Reeb graphs for three-dimensional manifolds in boundary representation
Reeb graphs are topological graphs originating in Morse theory, which represent the topological structure of a manifold by contracting the level set components of a scalar-valued function defined on it. The generalization to several functions leads to Reeb spaces, which are thus able to capture more features of an object. We introduce the layered Reeb graph as a discrete representation for Reeb...
متن کامل