Reeb Graph Research Articles

Reeb Graphs of Morse–Bott Functions on a Given Surface

Reeb Graphs of Morse–Bott Functions on a Given Surface

Labeled interleaving distance for Reeb graphs

Labeled interleaving distance for Reeb graphs

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.

Universal distances for extended persistence

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen–Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg–Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.

Topological awareness towards collision-free multi-axis curved layer additive manufacturing

We present a topology-aware computational framework that enables global collision-free and support-structure-free material extrusion for curved layer multi-axis printing of a range of high-genus and simply connected models. The nature of multi-axis additive manufacturing facilitates continuous variation of build direction, which paves the way for numerous approaches of geometry-based curved layer design. However, due to unawareness of global topological features, prevention of collision between the printer nozzle and the workpiece can either remain uncertain or demand for exhaustive algorithmic approaches, depending on the topological and geometrical complexity of a given mesh model. In this paper, we present a topological analysis framework that draws inspiration from the concepts of Reeb graph and Morse theory, to distinguish between collision-prone and collision-free regions of mesh models. We provide an exact definition for global collision-inducing features in regions of risk and optimise the nozzle orientation vector field to avoid global collisions whilst adhering to the overhang angle constraints. A new shape analysis method is then proposed to find the model orientation which maximises the manufacturability of bifurcating simply connected models. To validate our algorithmic framework, several high-genus and bifurcating simply connected models are printed using a robotic multi-axis printer. The experimental results demonstrate the feasibility and effectiveness of our framework for minimising global collisions in multi-axis curved layer additive manufacturing for a range of mesh models.

On the topology of the Reeb graph

On the topology of the Reeb graph

Distributed Augmentation, Hypersweeps, and Branch Decomposition of Contour Trees for Scientific Exploration.

Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale using highperformance computing. To address this challenge, recent work has introduced distributed hierarchical contour trees for distributed computation and storage of contour trees. However, effective use of these distributed structures in analysis and visualization requires subsequent computation of geometric properties and branch decomposition to support contour extraction and exploration. In this work, we introduce distributed algorithms for augmentation, hypersweeps, and branch decomposition that enable parallel computation of geometric properties, and support the use of distributed contour trees as query structures for scientific exploration. We evaluate the parallel performance of these algorithms and apply them to identify and extract important contours for scientific visualization.

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

Relations between Reeb graphs, systems of hypersurfaces and epimorphisms onto free groups

A Comparative Study of the Perceptual Sensitivity of Topological Visualizations to Feature Variations.

Color maps are a commonly used visualization technique in which data are mapped to optical properties, e.g., color or opacity. Color maps, however, do not explicitly convey structures (e.g., positions and scale of features) within data. Topology-based visualizations reveal and explicitly communicate structures underlying data. Although our understanding of what types of features are captured by topological visualizations is good, our understanding of people's perception of those features is not. This paper evaluates the sensitivity of topology-based isocontour, Reeb graph, and persistence diagram visualizations compared to a reference color map visualization for synthetically generated scalar fields on 2-manifold triangular meshes embedded in 3D. In particular, we built and ran a human-subject study that evaluated the perception of data features characterized by Gaussian signals and measured how effectively each visualization technique portrays variations of data features arising from the position and amplitude variation of a mixture of Gaussians. For positional feature variations, the results showed that only the Reeb graph visualization had high sensitivity. For amplitude feature variations, persistence diagrams and color maps demonstrated the highest sensitivity, whereas isocontours showed only weak sensitivity. These results take an important step toward understanding which topology-based tools are best for various data and task scenarios and their effectiveness in conveying topological variations as compared to conventional color mapping.

A new hierarchical analysis framework of building heights: Towards a more intuitive understanding of 3D urban structure

AbstractUnderstanding the 3D spatial structure of a city is increasingly essential for addressing various environmental and management issues, surpassing the importance of traditional 2D analysis. However, existing studies neglect the diversity of building height and still lack a clear description of 3D urban structure. This article proposes a new framework to uncover the 3D urban structure. Firstly, kernel density is employed to reveal the hierarchical spatial structure of buildings and the contour tree method is improved to quantitatively measure the spatial diversity and complexity. Then, the 3D urban structure is abstracted by spatial interpolation after feature filtration. Finally, this framework is applied to the central area of Chengdu City, revealing that: (1) Spatial structures of buildings with different heights exhibit significant diversity, location preference, and complexity; and (2) A globally “depression” 3D urban structure with low center—high periphery is obviously identified. This framework provides an effective way to reveal the 3D urban structure in a more intuitive and clearer way from various disordered urban buildings, which can be transferable to other cities and further facilitate sustainable planning and development of cities.

MORSE AND CODIMENSION 1 FUNCTIONS ON THE PROJECTIVE PLANE

The topological properties of functions on two-dimensional spaces are described using an invariant called the Reeb graph. In the case of Morse functions on closed oriented two-dimensional manifolds, this serves as a complete topological invariant for homotopy equivalence. For non-oriented two-dimensional manifolds and manifolds with boundaries, additional information is required to construct a complete topological invariant. If a linear order is imposed on the set of vertices, it becomes a complete topological invariant with respect to topological equivalence. The research explores the topological classification of Morse functions and functions of corank 1 on the projective plane. In these cases, the Reeb graph is an oriented rooted tree. An analytical expression is derived for the number of oriented Reeb graphs with a root. For functions of corank 1 and a fixed number of critical points, all possible Reeb graphs are constructed.

ReeBundle: A Method for Topological Modeling of White Matter Pathways Using Diffusion MRI.

Tractography can generate millions of complex curvilinear fibers (streamlines) in 3D that exhibit the geometry of white matter pathways in the brain. Common approaches to analyzing white matter connectivity are based on adjacency matrices that quantify connection strength but do not account for any topological information. A critical element in neurological and developmental disorders is the topological deterioration and irregularities in streamlines. In this paper, we propose a novel Reeb graph-based method "ReeBundle" that efficiently encodes the topology and geometry of white matter fibers. Given the trajectories of neuronal fiber pathways (neuroanatomical bundle), we re-bundle the streamlines by modeling their spatial evolution to capture geometrically significant events (akin to a fingerprint). ReeBundle parameters control the granularity of the model and handle the presence of improbable streamlines commonly produced by tractography. Further, we propose a new Reeb graph-based distance metric that quantifies topological differences for automated quality control and bundle comparison. We show the practical usage of our method using two datasets: (1) For International Society for Magnetic Resonance in Medicine (ISMRM) dataset, ReeBundle handles the morphology of the white matter tract configurations due to branching and local ambiguities in complicated bundle tracts like anterior and posterior commissures; (2) For the longitudinal repeated measures in the Cognitive Resilience and Sleep History (CRASH) dataset, repeated scans of a given subject acquired weeks apart lead to provably similar Reeb graphs that differ significantly from other subjects, thus highlighting ReeBundle's potential for clinical fingerprinting of brain regions.

Realization of a digraph as the Reeb graph of a Morse–Bott function on a given surface

We characterize the set of directed Reeb graphs of Morse–Bott functions on a given surface. Namely, we prove a criterion on realization of a given digraph as the directed Reeb graph of a Morse–Bott function on a given surface.

Karst landform classification considering surface flow characteristics derived from digital elevation models

AbstractKarst landforms are one of the most typical geographical units with a specific physical process on the earth's surface. The classification of karst landforms is an important aspect for understanding their landform processes and mechanisms. However, influenced by various interior and external forces, karst landforms have an extremely complex surface morphology, increasing the difficulty of their automatic classification. In this study, we considered hydrological features as an important factor in characterizing karst landforms and proposed a method that considers surface flow for karst landform classification. In this method, terrain was reversed for hydrological analysis to achieve the landform units. Then, the watershed boundary of the reversed terrain is extracted by hydrological analysis. The boundary of the karst landform unit is determined by erasing the plain area from the watershed boundary. Thereafter, the graph theory segmentation method is employed to merge the landform units belonging to the same karst landform entity. The proposed approach is validated and applied in two sample karst areas, Fenglin and Fengcong, located in Guilin, China, using digital elevation model data with 30 m spatial resolution. In addition, a comparative analysis is conducted to evaluate the accuracy of the proposed method. The results demonstrated that the typical karst landform units of Fenglin and Fengcong can be effectively classified. The overall classification accuracy is 94.44%. The proposed method produced more reasonable and accurate boundaries compared with the contour tree and terrain feature point methods. Furthermore, the classification results indicate various landform development stages of the karst landform process in the study area. The proposed method considering surface flow characteristics can be further extended to other landform types with highly complex landforms.

Exploring Divergent Patterns and Dynamics of Urban and Active Rural Developments—A Case Study of Dezhou City

Clarifying urban-rural spatial explicit structure changes is of great significance for understanding the urban-rural relationship evolution. Previous studies have mostly focused on urban internal spatial structure evolutions and less on the regional scale when it comes to exploring urban and rural evolutions. Nighttime light can timely reflect the human activities in regions and provides great potential for investigating the evolutions of urban and rural spatial explicit structures. Here, taking Dezhou City, a rapidly urbanizing city in China, as a case study, we employed the local contour tree method and nighttime light data to map urban and active rural extents from 2012 to 2020 and further explored their respective development processes. This study showed that unlike in rural regions, the internally explicit structures of urban regions were more complex, and there were often multiple hotspots inside them. The area of the urban-rural region increased significantly by 39.3% from 2012 to 2020 (p < 0.05). Populations were greatly responsible for the spatial explicit structure changes of urban and active rural regions. The urban and rural region rankings of the identified counties were basically consistent with the urban and rural population rankings. Unlike the perspectives of earlier land use (i.e., built-up land or impervious surface), this study underlined urban and active rural regions in view of the scope of active human activities. These results can likely help policymakers understand current active human activity extents and provide a data-based reference for future public services and infrastructure planning.

Estimating urban spatial structure based on remote sensing data

Understanding the spatial structure of a city is essential for formulating a spatial strategy for that city. In this study, we propose a method for analyzing the functional spatial structure of cities based on satellite remote sensing data. In this method, we first assume that urban functions consist of residential and central functions, and that these functions are measured by trip attraction by purpose. Next, we develop a model to explain trip attraction using remote sensing data, and estimate trip attraction on a grid basis. Using the estimated trip attraction, we created a contour tree to identify the spatial extent of the city and the hierarchical structure of the central functions of the city. As a result of applying this method to the Tokyo metropolitan area, we found that (1) our method reproduced 84% of urban areas and 94% of non-urban areas defined by the government, (2) our method extracted 848 urban centers, and their size distribution followed a Pareto distribution, and (3) the top-ranking urban centers were consistent with the districts defined in the master plans for the metropolitan area. Based on the results, we discussed the applicability of our method to urban structure analysis.

Poincaré–Reeb graphs of real algebraic domains

An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the geometric shape of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincaré–Reeb graph, which is naturally transversal to the foliation by vertical lines. We show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré–Reeb graph.

A Variational Loop Shrinking Analogy for Handle and Tunnel Detection and Reeb Graph Construction on Surfaces

AbstractThe humble loop shrinking property played a central role in the inception of modern topology but it has been eclipsed by more abstract algebraic formalisms. This is particularly true in the context of detecting relevant non‐contractible loops on surfaces where elaborate homological and/or graph theoretical constructs are favored in algorithmic solutions. In this work, we devise a variational analogy to the loop shrinking property and show that it yields a simple, intuitive, yet powerful solution allowing a streamlined treatment of the problem of handle and tunnel loop detection. Our formalization tracks the evolution of a diffusion front randomly initiated on a single location on the surface. Capitalizing on a diffuse interface representation combined with a set of rules for concurrent front interactions, we develop a dynamic data structure for tracking the evolution on the surface encoded as a sparse matrix which serves for performing both diffusion numerics and loop detection and acts as the workhorse of our fully parallel implementation. The substantiated results suggest our approach outperforms state of the art and robustly copes with highly detailed geometric models. As a byproduct, our approach can be used to construct Reeb graphs by diffusion thus avoiding commonly encountered issues when using Morse functions.

Using Foliation Leaves To Extract Reeb Graphs On Surfaces.

For Reeb graph extraction on surfaces, existing methods always use the isolines of a function defined on the surface to detect the surface components and the neighboring relationships between them. Since such detection is unstable, it is still a challenge for the extracted Reeb graphs to stably and concisely encode the topological information of the surface. In this paper, we address this challenge by using foliation leaves to extract Reeb graphs. In particular, we employ a method for generating measured harmonic foliations by defining loops for foliation initialization and diffusing leaves from loops over the surface. We demonstrate that when the loops are determined, the neighboring relationships between the leaves from different loops are fixed. Thus, we can use loops to represent surface components for robustly detecting the interrelationships between surface components. As a result, we are able to extract stable and concise Reeb graphs. We developed novel measures for loop determination and improved foliation generation, and our method allows the user to manually prescribe loops for generating Reeb graphs with desired structures. Therefore, the potential of Reeb graphs for representing surfaces is enhanced, including conveniently representing the symmetries of the surface and ignoring topological noise. This is verified by our experimental results which indicate that our Reeb graphs are compact and expressive, promoting shape analysis.

Reeb graphs of circle-valued functions: A survey and basic facts

The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties. However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application. In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces. We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions. In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.