Reeb Space Research Articles

A Topological Distance Between Multi-Fields Based on Multi-Dimensional Persistence Diagrams.

The problem of computing topological distance between two scalar fields based on Reeb graphs or contour trees has been studied and applied successfully to various problems in topological shape matching, data analysis, and visualization. However, generalizing such results for computing distance measures between two multi-fields based on their Reeb spaces is still in its infancy. Towards this, in the current article we propose a technique to compute an effective distance measure between two multi-fields by computing a novel multi-dimensional persistence diagram (MDPD) corresponding to each of the (quantized) Reeb spaces. First, we construct a multi-dimensional Reeb graph (MDRG), which is a hierarchical decomposition of the Reeb space into a collection of Reeb graphs. The MDPD corresponding to each MDRG is then computed based on the persistence diagrams of the component Reeb graphs of the MDRG. Our distance measure extends the Wasserstein distance between two persistence diagrams of Reeb graphs to MDPDs of MDRGs. We prove that the proposed measure is a pseudo-metric and satisfies a stability property. Effectiveness of the proposed distance measure has been demonstrated in (i) shape retrieval contest data - SHREC 2010 and (ii) Pt-CO bond detection data from computational chemistry. Experimental results show that the proposed distance measure based on the Reeb spaces has more discriminating power in clustering the shapes and detecting the formation of a stable Pt-CO bond as compared to the similar measures between Reeb graphs.

A (corrected) sufficient condition for the Hausdorffness of a Reeb space

A (corrected) sufficient condition for the Hausdorffness of a Reeb space

Reeb spaces of smooth functions on manifolds II

Reeb spaces of smooth functions on manifolds II

A Topological Similarity Measure Between Multi-Resolution Reeb Spaces.

Searching similarity between a pair of shapes or data is an important problem in data analysis and visualization. The problem of computing similarity measures using scalar topology has been studied extensively and proven useful in the shape and data matching. Even though multi-field or multivariate (consists of multiple scalar fields) topology reveals richer topological features, research on building tools for computing similarity measures using multi-field topology is still in its infancy. In the current article, we propose a novel similarity measure between two piecewise-linear multi-fields based on their multi-resolution Reeb spaces - a newly developed data-structure that captures the topology of a multi-field. Overall, our method consists of two steps: (i) building a multi-resolution Reeb space corresponding to each of the multi-fields and (ii) proposing a similarity measure between two multi-resolution Reeb spaces by computing a list of topologically consistent matching pairs (of nodes) and the similarity between them. We demonstrate the effectiveness of the proposed similarity measure in detecting topological features from real time-varying multi-field data in two application domains - one from computational physics and one from computational chemistry.

On the Reeb Spaces of Definable Maps

We prove that the Reeb space of a proper definable map $$f:X \rightarrow Y$$ in an arbitrary o-minimal expansion of a real closed field is realizable as a proper definable quotient. This result can be seen as an o-minimal analog of Stein factorization of proper morphisms in algebraic geometry. We also show that the Betti numbers of the Reeb space of f can be arbitrarily large compared to those of X, unlike in the special case of Reeb graphs of manifolds. Nevertheless, in the special case when $$f:X \rightarrow Y$$ is a semi-algebraic map and X is closed and bounded, we prove a singly exponential upper bound on the Betti numbers of the Reeb space of f in terms of the number and degrees of the polynomials defining X, Y, and f.

Statistical analysis of Mapper for stochastic and multivariate filters

Reeb spaces, as well as their discretized versions called Mappers, are common descriptors used in topological data analysis, with plenty of applications in various fields of science, such as computational biology and data visualization, among others. The stability and quantification of the rate of convergence of the Mapper to the Reeb space has been studied a lot in recent works (Brown et al. in CoRR. arXiv:1909.03488 , 2019; Carrière and Oudot in Found Comput Math 18(6):1333–1396, 2017; Carrière et al. in J Mach Learn Res 19(12):1–39, 2018; Munch and Wang in: 32nd international symposium on computational geometry (SoCG 2016), Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 51: 53:1–53:16, 2016), focusing on the case where a scalar-valued filter is used for the computation of Mapper. On the other hand, much less is known in the multivariate case, when the codomain of the filter is $${\mathbb {R}}^p$$ , and in the general case, when it is a general metric space $$(\mathcal {Z},d_\mathcal {Z})$$ , instead of $${\mathbb {R}}$$ . The few results that are available in this setting (Dey et al. in: 33rd international symposium on computational geometry (SoCG 2017), Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 77, 36:1–36:16, 2017; Munch and Wang, 2016) can only handle continuous topological spaces and cannot be used as is for finite metric spaces representing data, such as point clouds and distance matrices. In this article, we introduce a slight modification of the usual Mapper construction and we give risk bounds for estimating the Reeb space using this estimator. Our approach applies in particular to the setting where the filter function used to compute Mapper is also estimated from data, such as the eigenfunctions of PCA. Our results are given with respect to the Gromov-Hausdorff distance, computed with specific filter-based pseudometrics for Mappers and Reeb spaces defined in Dey et al. (2017). We finally provide examples of this setting in statistics and machine learning for different kinds of target filters, as well as numerical experiments that demonstrate the relevance of our approach.

Reeb Spaces of Smooth Functions on Manifolds

AbstractThe Reeb space of a continuous function is the space of connected components of the level sets. In this paper, we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.Dedicated to Professor Toshizumi Fukui on the occasion of his 60th birthday.

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for the topological analysis of scalar data in scientific visualization. While topological data analysis has gained in popularity over the last two decades, it has not yet been widely adopted as a standard data analysis tool for end users or developers. TTK aims at addressing this problem by providing a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependency-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website [108].

Jacobi Fiber Surfaces for Bivariate Reeb Space Computation.

This paper presents an efficient algorithm for the computation of the Reeb space of an input bivariate piecewise linear scalar function f defined on a tetrahedral mesh. By extending and generalizing algorithmic concepts from the univariate case to the bivariate one, we report the first practical, output-sensitive algorithm for the exact computation of such a Reeb space. The algorithm starts by identifying the Jacobi set of f, the bivariate analogs of critical points in the univariate case. Next, the Reeb space is computed by segmenting the input mesh along the new notion of Jacobi Fiber Surfaces, the bivariate analog of critical contours in the univariate case. We additionally present a simplification heuristic that enables the progressive coarsening of the Reeb space. Our algorithm is simple to implement and most of its computations can be trivially parallelized. We report performance numbers demonstrating orders of magnitude speedups over previous approaches, enabling for the first time the tractable computation of bivariate Reeb spaces in practice. Moreover, unlike range-based quantization approaches (such as the Joint Contour Net), our algorithm is parameter-free. We demonstrate the utility of our approach by using the Reeb space as a semi-automatic segmentation tool for bivariate data. In particular, we introduce continuous scatterplot peeling, a technique which enables the reduction of the cluttering in the continuous scatterplot, by interactively selecting the features of the Reeb space to project. We provide a VTK-based C++ implementation of our algorithm that can be used for reproduction purposes or for the development of new Reeb space based visualization techniques.

Multivariate topology simplification

Topological simplification of scalar and vector fields is well-established as an effective method for analysing and visualising complex data sets. For multivariate (alternatively, multi-field) data, topological analysis requires simultaneous advances both mathematically and computationally. We propose a robust multivariate topology simplification method based on “lip”-pruning from the Reeb space. Mathematically, we show that the projection of the Jacobi set of multivariate data into the Reeb space produces a Jacobi structure that separates the Reeb space into simple components. We also show that the dual graph of these components gives rise to a Reeb skeleton that has properties similar to the scalar contour tree and Reeb graph, for topologically simple domains. We then introduce a range measure to give a scaling-invariant total ordering of the components or features that can be used for simplification. Computationally, we show how to compute Jacobi structure, Reeb skeleton, range and geometric measures in the Joint Contour Net (an approximation of the Reeb space) and that these can be used for visualisation similar to the contour tree or Reeb graph.

Interactive Visualization for Singular Fibers of Functions f : R³ → R².

Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from scientific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers-inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualizations. This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R³ → R². This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Validation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians.

Layered 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 spaces of 3D solids (embedded three-dimensional manifolds with boundary) with respect to two scalar-valued functions. After that we present an efficient algorithm for computing the layered Reeb graph, which uses only a boundary representation of the underlying three-dimensional manifold. This leads to substantial computational advantages if the manifold is given in a boundary representation, since no volumetric representation has to be constructed. However, this algorithm is applicable only if the defining functions satisfy certain conditions.

Systoles of 2-complexes, Reeb graph, and Grushko decomposition

Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model” for X, reminiscent of the “chopping off long fingers” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex.