Local Equivalence and Intrinsic Metrics between Reeb Graphs

As graphical summaries for topological spaces and maps, Reeb graphs are common objects in the computer graphics or topological data analysis literature. Defining good metrics between these objects has become an important question for applications, where it matters to quantify the extent by which two given Reeb graphs differ. Recent contributions emphasize this aspect, proposing novel distances such as {\em functional distortion} or {\em interleaving} that are provably more discriminative than the so-called {\em bottleneck distance}, being true metrics whereas the latter is only a pseudo-metric. Their main drawback compared to the bottleneck distance is to be comparatively hard (if at all possible) to evaluate. Here we take the opposite view on the problem and show that the bottleneck distance is in fact good enough {\em locally}, in the sense that it is able to discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do. This suggests considering the {\em intrinsic metrics} induced by these distances, which turn out to be all {\em globally} equivalent. This novel viewpoint on the study of Reeb graphs has a potential impact on applications, where one may not only be interested in discriminating between data but also in interpolating between them

Parallel Mapper

The construction of Mapper has emerged in the last decade as a powerful and effective topological data analysis tool that approximates and generalizes other topological summaries, such as the Reeb graph, the contour tree, split, and joint trees. In this paper, we study the parallel analysis of the construction of Mapper. We give a provably correct parallel algorithm to execute Mapper on multiple processors and discuss the performance results that compare our approach to a reference sequential Mapper implementation. We report the performance experiments that demonstrate the efficiency of our method

Stability of Reeb graphs under function perturbations: the case of closed curves

Reeb graphs provide a method for studying the shape of a manifold by encoding the evolution and arrangement of level sets of a simple Morse function defined on the manifold. Since their introduction in computer graphics they have been gaining popularity as an effective tool for shape analysis and matching. In this context one question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another. Our main result is that changes in Morse functions induce smaller changes in the editing distance between Reeb graphs of curves, implying stability of Reeb graphs under function perturbations.Comment: 23 pages, 12 figure

A topological approach for segmenting human body shape

Segmentation of a 3D human body, is a very challenging problem in applications exploiting human scan data. To tackle this problem, the paper proposes a topological approach based on the discrete Reeb graph (DRG) which is an extension of the classical Reeb graph to handle unorganized clouds of 3D points. The essence of the approach concerns detecting critical nodes in the DRG, thereby permitting the extraction of branches that represent parts of the body. Because the human body shape representation is built upon global topological features that are preserved so long as the whole structure of the human body does not change, our approach is quite robust against noise, holes, irregular sampling, frame change and posture variation. Experimental results performed on real scan data demonstrate the validity of our method

Book embeddings of Reeb graphs

Let $X$ be a simplicial complex with a piecewise linear function $f:X\to\mathbb{R}$. The Reeb graph $Reeb(f,X)$ is the quotient of $X$, where we collapse each connected component of $f^{-1}(t)$ to a single point. Let the nodes of $Reeb(f,X)$ be all homologically critical points where any homology of the corresponding component of the level set $f^{-1}(t)$ changes. Then we can label every arc of $Reeb(f,X)$ with the Betti numbers $(\beta_1,\beta_2,\dots,\beta_d)$ of the corresponding $d$-dimensional component of a level set. The homology labels give more information about the original complex $X$ than the classical Reeb graph. We describe a canonical embedding of a Reeb graph into a multi-page book (a star cross a line) and give a unique linear code of this book embedding.Comment: 12 pages, 5 figures, more examples will be at http://kurlin.or

A discrete Reeb graph approach for the segmentation of human body scans

Segmentation of 3D human body (HB) scan is a very challenging problem in applications exploiting human scan data. To tackle this problem, we propose a topological approach based on discrete Reeb graph (DRG) which is an extension of the classical Reeb graph to unorganized cloud of 3D points. The essence of the approach is detecting critical nodes in the DRG thus permitting the extraction of branches that represent the body parts. Because the human body shape representation is built upon global topological features that are preserved so long as the whole structure of the human body does not change, our approach is quite robust against noise, holes, irregular sampling, moderate reference change and posture variation. Experimental results performed on real scan data demonstrate the validity of our method

Multimapper: Data Density Sensitive Topological Visualization

Mapper is an algorithm that summarizes the topological information contained in a dataset and provides an insightful visualization. It takes as input a point cloud which is possibly high-dimensional, a filter function on it and an open cover on the range of the function. It returns the nerve simplicial complex of the pullback of the cover. Mapper can be considered a discrete approximation of the topological construct called Reeb space, as analysed in the $1$-dimensional case by [Carriere et al.,2018]. Despite its success in obtaining insights in various fields such as in [Kamruzzaman et al., 2016], Mapper is an ad hoc technique requiring lots of parameter tuning. There is also no measure to quantify goodness of the resulting visualization, which often deviates from the Reeb space in practice. In this paper, we introduce a new cover selection scheme for data that reduces the obscuration of topological information at both the computation and visualisation steps. To achieve this, we replace global scale selection of cover with a scale selection scheme sensitive to local density of data points. We also propose a method to detect some deviations in Mapper from Reeb space via computation of persistence features on the Mapper graph.Comment: Accepted at ICDM