Reeb Graph Research Articles

Identification and Geographic Distribution of Accommodation and Catering Centers

As the most important manifestation of the activities of the life service industry, the reasonable layout of spatial agglomeration and dispersion of the accommodation and catering industry plays an important role in guiding the spatial structure of the urban industry and population. Applying the contour tree and location quotient index methods, based on points of interest (POI) data of the accommodation and catering industry in Beijing and on the identification of the spatial structure and cluster center of the accommodation and catering industry, we investigated the distribution and agglomeration characteristics of the urban accommodation and catering industry from the perspective of industrial spatial differentiation. The results show that: (1) the accommodation and catering industry in Beijing presents a polycentric agglomeration pattern in space, mainly distributed within a radius of 20 km from the city center and on a relatively large scale; areas beyond this distance contain isolated single cluster centers. (2) From the perspective of the industry, the cluster centers close to the core area of the city are characterized by the agglomeration of multiple advantageous industries, while those in the outer suburbs of the city are more prominent in a single industry. (3) From the perspective of the location quotient of cluster centers, the leisure catering industries are mainly located close to the urban centers. On the contrary, the cluster centers in the outer suburbs and counties are relatively small and dominated by restaurants and fast food industries. Commercial accommodation businesses are mainly distributed in the transportation hub centers and in entertainment and leisure areas.

Topology of optimal flows with collective dynamics on closed orientable surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

Moduli spaces of morse functions for persistence

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse--Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in $\mathbf R^3$. Their level-set invariant, a poset generated by nested disks and annuli from levels sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.

Fuzzy Contour Trees: Alignment and Joint Layout of Multiple Contour Trees

AbstractWe describe a novel technique for the simultaneous visualization of multiple scalar fields, e.g. representing the members of an ensemble, based on their contour trees. Using tree alignments, a graph‐theoretic concept similar to edit distance mappings, we identify commonalities across multiple contour trees and leverage these to obtain a layout that can represent all trees simultaneously in an easy‐to‐interpret, minimally‐cluttered manner. We describe a heuristic algorithm to compute tree alignments for a given similarity metric, and give an algorithm to compute a joint layout of the resulting aligned contour trees. We apply our approach to the visualization of scalar field ensembles, discuss basic visualization and interaction possibilities, and demonstrate results on several analytic and real‐world examples.

Automatic 3D virtual fitting system based on skeleton driving

To facilitate the input and improve the practicality of the existing virtual fitting systems, we propose a fully automatic 3D virtual fitting system to fit garment onto human models with various shapes and poses using a 2D full-body image and a 3D garment model. The proposed method constructs the 3D human model from the input 2D full-body image of the user by adopting the SMPLify method. To automatically position garment models onto human models with arbitrary postures, we present a 3D mesh segmentation method based on the discrete Reeb graph to accurately segment the different parts of a garment model, and a skeleton driving method based on mean curvature flow, which automatically adjusts the posture of the garment model according to the skeleton structural difference between the human model and the garment model. In addition, for the purpose of obtaining a more natural dress effect, we further adopt interpenetration removal and physical simulation for the deformed garment model. Compared to existing automatic 3D virtual fitting systems, the experimental results, we obtained based on the Leeds Sports Pose dataset, reveal that the proposed virtual fitting system is stable and effective.

Tree decomposition of Reeb graphs, parametrized complexity, and applications to phylogenetics

Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, can be identified with a coproduct of at most $$2^s$$ partially ordered trees with $$(n + s)$$ leaves. Reeb graphs are therefore classified up to isomorphism by their tree-decomposition. An implication of this result, is that the isomorphism problem for Reeb graphs is fixed parameter tractable when the parameter is the first Betti number. We propose partially ordered Reeb graphs as a model for time consistent phylogenetic networks and propose a certain Hausdorff distance as a metric on these structures.

Loops in generalized Reeb graphs associated to stable circle-valued functions

Centro de Ciencias e Tecnologia Universidade Federal do Cariri - UFCA, Campus de Juazeiro do Norte-CE

A Hidden Markov Contour Tree Model for Spatial Structured Prediction

A Hidden Markov Contour Tree Model for Spatial Structured Prediction

Study on Optimization Method of Spatial Structure Characteristics of Peak Forest Landform in Wulingyuan Scenic Area

To protect the ecological environment of peak forest landform and maintain its integrity and stability, the optimization method of spatial structure characteristics of peak forest landform in Wulingyuan Scenic Area was studied. By using GIS and digital topographic analysis to study the basic features of sandstone peak forest landform, Wulingyuan peak forest landform and karst peak forest have great differences in lithological composition, weathering resistance is better than Cheltenham Badland landform; by using tree theory to analyze the features of Wulingyuan peak forest landform, according to area weight serialization of contour tree nodes, we can know the depression area. The karstification is stronger than that of the peak forest area, and the surface is relatively fragmented. Based on different landscape indices, the landscape pattern of Wulingyuan peak forest is analyzed. The fragmentation degree of vegetation is lower, and the fragmentation degree of building landscape is first increased, then decreased, and finally intensified. The proportion of artificial landscape decreases year by year and the trend of fragmentation is obvious. Based on the present situation of spatial structure characteristics of Wulingyuan peak forest landform, optimization methods such as combining centralization with decentralization and improving the quality of artificial landscape ecosystem were put forward.

Deformations in the General Position of the Optimal Functions on Oriented Surfaces with Boundary

We consider simple functions with nondegenerate singularities on smooth compact oriented surfaces with boundary. The relationship between the optimality and polarity of Morse functions, m-functions and mm-functions on smooth compact oriented connected surfaces is described. The concept of equipped Kronrod–Reeb graph is used to define deformation in the general position. Moreover, we present the entire list of deformations of simple functions of one of the classes described above on a torus, on a 2-dimensional disk with boundary, and on the connected sum of two tori.

Automorphisms of Kronrod–Reeb graphs of Morse functions on compact surfaces

Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of difeomorphisms of $M$ isotopic to the identity. Denote by $\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms "preserving" $f$, i.e. the stabilizer of $f$ with respect to the right action of $\mathcal{D}_{\mathrm{id}}(M)$ on the space $\mathcal{C}^{\infty}(M,\mathbb{R})$ of smooth functions on $M$. Let also $\mathbf{G}(f)$ be the group of automorphisms of the Kronrod-Reeb graph of $f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is an important ingredient in determining the homotopy type of the orbit of $f$ with respect to the above action of $\mathcal{D}_{\mathrm{id}}(M)$ and it is trivial if $f$ is "generic", i.e. has at most one critical point at each level set $f^{-1}(c)$, $c\in\mathbb{R}$. For the case when $M$ is distinct from $2$-sphere and $2$-torus we present a precise description of the family $\mathbf{G}(M,\mathbb{R})$ of isomorphism classes of groups $\mathbf{G}(f)$, where $f$ runs over all Morse functions on $M$, and of its subfamily $\mathbf{G}^{smp}(M,\mathbb{R}) \subset \mathbf{G}(M,\mathbb{R})$ consisting of groups corresponding to simple Morse functions, i.e. functions having at most one critical point at each connected component of each level set. In fact, $\mathbf{G}(M,\mathbb{R})$, (resp. $\mathbf{G}^{smp}(M,\mathbb{R})$), coincides with the minimal family of isomorphism classes of groups containing the trivial group and closed with respect to direct products and also with respect to wreath products "from the top" with arbitrary finite cyclic groups, (resp. with group $\mathbb{Z}_2$ only).

Reeb posets and tree approximations

A well known result in the analysis of finite metric spaces due to Gromov says that given any metric space (X,dX) there exists a tree metrictX on X such that |dX−tX|∞ is bounded above by twice hyp(X)⋅log(2|X|). Here hyp(X) is the hyperbolicity of X, a quantity that measures the treeness of 4-tuples of points in X. This bound is known to be asymptotically tight.We improve this bound by restricting ourselves to metric spaces arising from filtered posets. By doing so we are able to replace the cardinality appearing in Gromov’s bound by a certain poset theoretic invariant which can be much smaller thus significantly improving the approximation bound.The setting of metric spaces arising from posets is rich: For example, every finite metric graph can be induced from a filtered poset. Since every finite metric space can be isometrically embedded into a finite metric graph, our ideas are applicable to finite metric spaces as well.At the core of our results lies the adaptation of the Reeb graph and Reeb tree constructions and the concept of hyperbolicity to the setting of posets, which we use to formulate and prove a tree approximation result for any filtered poset.

Reckoning of Emotions Through Recognition of Posture Features

Emotions are noteworthy signs to understand the intentions of others peoples during communication with them. Similarly, in order to identify the different emotional states of individual such as joy, sadness, and anger, using facial expressions and vocal tone are less effective due to the variations in facial and vocal outputs. Thus, for recognizing emotions intensely without variation physique postures analysis can give effective outputs. Thus, physique posture analysis of the individual can be obtained by mapping the joints using Reeb graph which plots all the joints into carvative structure to learn posture, and the angles between the joints of posture are detected by law of cosines which depicts the vigorous expression along with postures. In addition to that much more reliable recognition of emotions without distractions can be obtained through the detailed features by preprocessing the input image through the fusion of Median and Wiener filter which avoids all the five types of noise occurrence so detailed features such as Invariant, Depth sequential silhouettes and Spatiotemporal body joint can be obtained to aid efficient analysis of posture that can pave a way to identify the different emotions by tactic Tree based classifier to get better performance in terms of execution time and accuracy.

A Structural Average of Labeled Merge Trees for Uncertainty Visualization.

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize topological changes in the (sub)level sets of scalar fields. One of the biggest challenges and opportunities to advance topology-based visualization is to understand and incorporate uncertainty into such topological descriptors to effectively reason about their underlying data. In this paper, we study a structural average of a set of labeled merge trees and use it to encode uncertainty in data. Specifically, we compute a 1-center tree that minimizes its maximum distance to any other tree in the set under a well-defined metric called the interleaving distance. We provide heuristic strategies that compute structural averages of merge trees whose labels do not fully agree. We further provide an interactive visualization system that resembles a numerical calculator that takes as input a set of merge trees and outputs a tree as their structural average. We also highlight structural similarities between the input and the average and incorporate uncertainty information for visual exploration. We develop a novel measure of uncertainty, referred to as consistency, via a metric-space view of the input trees. Finally, we demonstrate an application of our framework through merge trees that arise from ensembles of scalar fields. Our work is the first to employ interleaving distances and consistency to study a global, mathematically rigorous, structural average of merge trees in the context of uncertainty visualization.

Detecting Urban Polycentric Structure from POI Data

It is meaningful to analyze urban spatial structure by identifying urban subcenters, and many methods of doing so have been proposed in the published literature. Although these methods are widely applied, they exhibit obvious shortcomings that limit their further application. Therefore, it is of great value to propose a new urban subcenter identification method that can overcome these shortcomings. In this paper, we propose the density contour tree (DCT) method for detecting urban polycentric structures and their spatial distributions. Conceptually, this method is based on an analogy between urban spatial structure and terrain. The point-of-interest (POI) density is visualized as a continuous mathematical surface representing the urban terrain. Peaks represent the regions of the most frequent human activity, valleys represent regions with small population densities in the city, and slopes represent spatial changes in urban land-use intensity. Using this method, we have detected the urban “polycentric” structure of Beijing and determined the corresponding spatial relationships. In addition, several important properties of the urban centers have been identified. For example, Beijing has a typical urban polycentric structure with an urban center area accounting for 5.9% of the total urban area, and most of the urban centers in Beijing serve comprehensive functions. In general, the method and the results can serve as references for the later research on analyzing urban structure.

Characterization of lenticulostriate arteries with high resolution black-blood T1-weighted turbo spin echo with variable flip angles at 3 and 7 Tesla

ObjectivesThe lenticulostriate arteries (LSAs) with small diameters of a few hundred microns take origin directly from the high flow middle cerebral artery (MCA), making them especially susceptible to damage (e.g. by hypertension). This study aims to present high resolution (isotropic ∼0.5 mm), black blood MRI for the visualization and characterization of LSAs at both 3 T and 7 T. Materials and methodsT1-weighted 3D turbo spin-echo with variable flip angles (T1w TSE-VFA) sequences were optimized for the visualization of LSAs by performing extended phase graph (EPG) simulations. Twenty healthy volunteers (15 under 35 years old, 5 over 60 years old) were imaged with the T1w TSE-VFA sequences at both 3 T and 7 T. Contrast-to-noise ratio (CNR) was quantified, and LSAs were manually segmented using ITK-SNAP. Automated Reeb graph shape analysis was performed to extract features including vessel length and tortuosity. All quantitative metrics were compared between the two field strengths and two age groups using ANOVA. ResultsLSAs can be clearly delineated using optimized 3D T1w TSE-VFA at 3 T and 7 T, and a greater number of LSA branches can be detected compared to those by time-of-flight MR angiography (TOF MRA) at 7 T. The CNR of LSAs was comparable between 7 T and 3 T. T1w TSE-VFA showed significantly higher CNR than TOF MRA at the stem portion of the LSAs branching off the medial middle cerebral artery. The mean vessel length and tortuosity were greater on TOF MRA compared to TSE-VFA. The number of detected LSAs by both TSE-VFA and TOF MRA was significantly reduced in aged subjects, while the mean vessel length measured on 7 T TSE-VFA showed significant difference between the two age groups. ConclusionThe high-resolution black-blood 3D T1w TSE-VFA sequence offers a new method for the visualization and quantification of LSAs at both 3 T and 7 T, which may be applied for a number of pathological conditions related to the damage of LSAs.

The exact geostrophic streamfunction for neutral surfaces

McDougall (1989) proved that neutral surfaces possess an exact geostrophic streamfunction, but its form has remained unknown. The exact geostrophic streamfunction for neutral surfaces is derived here. It involves a path integral of the specific volume along a neutral trajectory. On a neutral surface, the specific volume is a multivalued function of the pressure on the surface, p∼. By decomposing the neutral surface into regions where the specific volume is a single-valued function of p∼, the path integral is simply a sum of integrals of these single-valued functions. The regions are determined by the Reeb graph of p∼, and the neutral trajectory is encoded by a walk on this graph. Islands, and other holes in the neutral surface, can create cycles in the Reeb graph, causing the exact geostrophic streamfunction on a neutral surface to be multivalued. Moreover, neutral surfaces are ill-defined in the real ocean. Hence, the topobaric geostrophic streamfunction is presented: a single-valued approximation of the exact geostrophic streamfunction for neutral surfaces, for use on any well-defined, approximately neutral surface. Numerical tests on several approximately neutral surfaces reveal that the topobaric geostrophic streamfunction estimates the geostrophic velocity with an error that is about an order of magnitude smaller than that for any previously known geostrophic streamfunction. Also, the Montgomery potential is generalized, leading to an alternate form of the exact geostrophic streamfunction for neutral surfaces. This form is used to construct the orthobaric Montgomery potential, an easily computable geostrophic streamfunction that estimates the geostrophic velocity more accurately than any previously known geostrophic streamfunction, but less so than the topobaric geostrophic streamfunction.

Neutral surface topology

Neutral surfaces, along which most of the mixing in the ocean occurs, are notoriously difficult objects: they do not exist as well-defined surfaces, and as such can only be approximated. In a hypothetical ocean where neutral surfaces are well-defined, the in-situ density on the surface is a multivalued function of the pressure on the surface, p∼. The surface is decomposed into geographic regions where there is one connected pressure contour per pressure value, making this function single-valued in each region. The regions are represented by arcs of the Reeb graph of p∼. The regions meet at saddles of p∼ which are represented by internal nodes of the Reeb graph. Leaf nodes represent extrema of p∼. Cycles in the Reeb graph are created by islands and other holes in the neutral surface. This topological theory of neutral surfaces is used to create a new class of approximately neutral surfaces in the real ocean, called topobaric surfaces, which are very close to neutral and fast to compute. Topobaric surfaces are the topologically correct extension of orthobaric density surfaces to be geographically dependent, which is fundamental to neutral surfaces. Also considered is the possibility that helical neutral trajectories might have a larger pitch around islands than in the open ocean.

Task-Based Augmented Contour Trees with Fibonacci Heaps

This paper presents a new algorithm for the fast, shared memory, multi-core computation of augmented contour trees on triangulations. In contrast to most existing parallel algorithms our technique computes augmented trees, enabling the full extent of contour tree based applications including data segmentation. Our approach completely revisits the traditional, sequential contour tree algorithm to re-formulate all the steps of the computation as a set of independent local tasks. This includes a new computation procedure based on Fibonacci heaps for the join and split trees, two intermediate data structures used to compute the contour tree, whose constructions are efficiently carried out concurrently thanks to the dynamic scheduling of task parallelism. We also introduce a new parallel algorithm for the combination of these two trees into the output global contour tree. Overall, this results in superior time performance in practice, both in sequential and in parallel thanks to the OpenMP task runtime. We report performance numbers that compare our approach to reference sequential and multi-threaded implementations for the computation of augmented merge and contour trees. These experiments demonstrate the run-time efficiency of our approach and its scalability on common workstations. We demonstrate the utility of our approach in data segmentation applications.

Partitioning of an arbitrary domain into subdomains without branching of inner boundaries

In this paper we consider an approach to the partitioning of a arbitrary domain into connected subdomains without branching of internal boundaries. The Reeb graph is a simplified representation of the topology of the required domain. Algorithms are shown in this paper is a modification of the presented earlier algorithm of the formation of the plane Reeb graph. The algorithm for constructing the volume Reeb graph allows us to determine the internal topology of the domain. A new algorithm of the formation of the partitioning into subdomains without branching of internal boundaries is presented.