Local Equivalence and Intrinsic Metrics between Reeb Graphs

Local Equivalence and Intrinsic Metrics Between Reeb Graphs∗ Mathieu Carrière1 and Steve Oudot2 1 DataShape, Inria Saclay, France DataShape, Inria Saclay, France 2 Abstract 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 functional distortion or interleaving that are provably more discriminative than the so-called bottleneck distance, being true metrics whereas the latter is only a pseudometric. 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 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 intrinsic metrics induced by these distances, which turn out to be all 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. 1998 ACM Subject Classification F.2.2 [Nonnumerical Algorithms and Problems:]Geometrical Problems and Computations Keywords and phrases Reeb Graphs, Extended Persistence, Induced Metrics, Topological Data Analysis Digital Object Identifier 10.4230/LIPIcs.SoCG.2017.25 1 Introduction In the context of shape analysis, the Reeb graph [26] provides a meaningful summary of a topological space and a real-valued function defined on that space. Intuitively, it continuously collapses the connected components of the level sets of the function into single points, thus tracking the values of the functions at which the connected components merge or split. Reeb graphs have been widely used in computer graphics and visualization – see [7] for a survey, and their discrete versions, including the so-called Mappers [27], have become emblematic tools of topological data analysis due to their success in applications [2, 3, 20, 23]. Finding relevant dissimilarity measures for comparing Reeb graphs has become an important question in the recent years. The quality of a dissimilarity measure is usually ∗ This work was partially supported by ERC Grant Agreement No. 339025 GUDHI (Algorithmic Foundations of Geometry Understanding in Higher Dimensions) and was carried out while the second author was visiting the ICERM at Brown University. © Mathieu Carrière and Steve Oudot; licensed under Creative Commons License CC-BY 33rd International Symposium on Computational Geometry (SoCG 2017). Editors: Boris Aronov and Matthew J. Katz; Article No. 25; pp. 25:1–25:15 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 25:2 Local Equivalence and Intrinsic Metrics Between Reeb Graphs assessed through three criteria: its ability to satisfy the axioms of a metric, its discriminative power, and its computational efficiency. The most natural choice to begin with is to use the Gromov-Hausdorff distance dGH [10] for Reeb graphs seen as metric spaces. The main drawback of this distance is to quickly become intractable to compute in practice, even for graphs that are metric trees [1]. Among recent contributions, the functional distortion distance dFD [4] and the interleaving distance dI [15] share the same advantages and drawbacks as dGH , in particular they enjoy good stability and discriminativity properties but they lack efficient algorithms for their computation, moreover they can be difficult to interpret. By contrast, the bottleneck distance dB comes with a signature for Reeb graphs, called the extended persistence diagram [14], which acts as a stable bag-of-feature descriptor. Furthermore, dB can be computed efficiently in practice. Its main drawback though is to be only a pseudo-metric, so distinct graphs can have the same signature and therefore be deemed equal in dB . Another desired property for dissimilarity measures is to be intrinsic, i.e. realized as the lengths of shortest continuous paths in the space of Reeb graphs [10]. This is particularly useful when one actually needs to interpolate between data, and not just discriminate between them, which happens in applications such as image or 3-d shape morphing, skeletonization, and matching [18, 21, 22, 28]. At this time, it is unclear whether the metrics proposed so far for Reeb graphs are intrinsic or not. Using intrinsic metrics would not only open the door to the use of Reeb graphs in the aforementioned applications, but it would also provide a better understanding of the intrinsic structure of the space of Reeb graphs, and give a deeper meaning to the distance values. Our contributions. In the first part of the paper we show that the bottleneck distance can discriminate a Reeb graph from any other Reeb graph in a small enough neighborhood, as efficiently as the other metrics do, even though it is only a pseudo-metric globally. More precisely, we show that, given any constant K ∈ (0, 1/22], in a sufficiently small neighborhood of a given Reeb graph Rf in the functional distortion distance (that is: for any Reeb graph Rg such that dFD (Rf , Rg ) < c(f, K), where c(f, K) > 0 is a positive constant depending only on f and K), one has: KdFD (Rf , Rg ) ≤ dB (Rf , Rg ) ≤ 2dFD (Rf , Rg ). (1) The second inequality is already known [4], and it asserts that the bottleneck distance between Reeb graphs is stable. The first inequality is new, and it asserts that the bottleneck distance is discriminative locally, in fact just as discriminative as the other distances mentioned above. Equation (1) can be viewed as a local equivalence between metrics although not in the usual sense: firstly, all comparisons are anchored to a fixed Reeb graph Rf , and secondly, the constants K and 2 are absolute. The second part of the paper advocates the study of intrinsic metrics on the space of Reeb graphs, for the reasons mentioned above. As a first step, we propose to study the intrinsic metrics dˆGH , dˆFD , dˆI and dˆB induced respectively by dGH , dFD , dI and dB . While the first three are obviously globally equivalent because their originating metrics are, our second contribution is to show that the last one is also globally equivalent to the other three. The paper concludes with a discussion and some directions for the study of the space of Reeb graphs as an intrinsic metric space. Related work. Interpolation between Reeb graphs is also the underlying idea of the edit distance recently proposed by Di Fabio and Landi [16]. The problem with this (...truncated)