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Movement pattern mining is a key focus in movement data research, with moving flock patterns holding particular significance due to their potential to reveal valuable insights across various domains. This research aimed to discover moving flock patterns in movement data. To achieve this, we first developed a more precise definition of a moving flock by refining the existing definitions. Then, we proposed a taxonomy of flock patterns, enabling the derivation of distinct types of moving flock patterns. Finally, we developed a Reeb graph-based approach to discover desired moving flock patterns. The effectiveness of the approach was validated using movement data obtained from a real football match. In the results, 72 and 94 moving flock patterns were discovered under different cases, respectively. Moreover, the proportion of desired moving flock patterns in these cases was 29.17% and 24.47%, respectively. The results show that the proposed approach effectively detects the desired moving flock patterns, and the findings provide insightful information to sports professionals. [ABSTRACT FROM AUTHOR]

Our contribution is twofold: (1) We formulate and solve a particular graph-theoretic problem: For an undirected graph G admitting a so-called MB-good orientation (acyclic with all its sources and sinks having degree at most 2), we calculate the minimum number ν (G) of “transit” vertices (neither sources nor sinks) of degree 2 over all MB-good orientations of G. Surprisingly, ν (G) ≤ 1 , and for most graphs, there exist MB-good orientations with all vertices of degree 2 being sources or sinks, i.e., with ν (G) = 0 . As an application, (2) in the context of the Reeb graph theory, we characterize the set of Reeb graphs defined by all Morse–Bott functions on a given surface; namely, we prove a criterion on realization of a graph as the Reeb graph of a Morse–Bott function on this surface. [ABSTRACT FROM AUTHOR]

Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network. We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities X of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph R which is labeled by subsets of X. By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an 'annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups — a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph R with certain merging/disbanding events in the given time-varying network. We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph R . [ABSTRACT FROM AUTHOR]

Movement pattern mining is a key focus in movement data research, with moving flock patterns holding particular significance due to their potential to reveal valuable insights across various domains. This research aimed to discover moving flock patterns in movement data. To achieve this, we first developed a more precise definition of a moving flock by refining the existing definitions. Then, we proposed a taxonomy of flock patterns, enabling the derivation of distinct types of moving flock patterns. Finally, we developed a Reeb graph-based approach to discover desired moving flock patterns. The effectiveness of the approach was validated using movement data obtained from a real football match. In the results, 72 and 94 moving flock patterns were discovered under different cases, respectively. Moreover, the proportion of desired moving flock patterns in these cases was 29.17% and 24.47%, respectively. The results show that the proposed approach effectively detects the desired moving flock patterns, and the findings provide insightful information to sports professionals.

The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties. [ABSTRACT FROM AUTHOR]

This paper proposes a new fully automatic computational framework from continuum structural topology optimization to beam structure design. Firstly, the continuum structural topology optimization is performed to find the optimal material distribution. The centers of the elements (i.e., vertices) in the final topology are considered as the original model of the skeleton extraction. Secondly, the Floyd-Warshall algorithm is used to calculate the geodesic distances between vertices. By combining the geodesic distance-based mapping function and a coarse-to-fine partition scheme, the original model is partitioned into regular components. The skeleton can be extracted by using edges to link the barycenter of the components and decomposed into branches by identified joint vertices. Each branch is normalized into a straight line. After mesh generation, a beam finite element model is established. Compared to other methods in the literature, the beam structures reconstructed by the proposed method have a desirable centeredness and keep the homotopy properties of the original models. Finally, the cross-sectional areas of members in the beam structure are considered as the design variables, and the sizing optimization is performed. Four numerical examples, both 2D and 3D, are employed to demonstrate the validity of the automatic computational framework. The proposed method extracts a parameterized beam finite element model from the topology optimization result that bridges the gap between the topology optimization of continuum structures and the subsequent optimization or design that enables a fully automatic design of beam-like structures. [ABSTRACT FROM AUTHOR]

We propose a novel and efficient algorithm to model high-level topological structures of neuronal fibers. Tractography constructs complex neuronal fibers in three dimensions that exhibit the geometry of white matter pathways in the brain. However, most tractography analysis methods are time consuming and intractable. We develop a computational geometry-based tractography representation that aims to simplify the connectivity of white matter fibers. Given the trajectories of neuronal fiber pathways, we model the evolution of trajectories that encodes geometrically significant events and calculate their point correspondence in the 3D brain space. Trajectory inter-distance is used as a parameter to control the granularity of the model that allows local or global representation of the tractogram. Using diffusion MRI data from Alzheimer's patient study, we extract tractography features from our model for distinguishing the Alzheimer's subject from the normal control. Software implementation of our algorithm is available on GitHub (https://github.com/UCSB-VRL/ReebGraph.

Build orientation has a significant impact on the surface quality, support structure, and final cost of the fabricated model. In this study, we focus our attention on the surface quality and printing cost of complex parts when fabricated under multiple build orientations. A novel method using model segmentation is proposed to search for an optimal build orientation for each sub-model decomposed by Reeb graph. The sub-model is divided into separate regions that are treated differently based on modified curvature shift strategy. Every flat facet of the regions is given a different weight factor to build the volume error function. The optimal build orientation for each sub-model is the one leading to the minimal error of the volume error function. The case study demonstrates that the proposed method obtains a smaller error without any support structure. The model surface in higher weighted regions obtains a higher surface quality and is more cost saving when printed, especially for complex models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the long-time behavior of systems close to a system with a smooth first integral. Under certain assumptions, the limiting behavior, to some extent, turns out to be universal: it is determined by the first integral, the deterministic perturbation, and the initial point. Furthermore, it is the same for a broad class of noises. In particular, the long-time behavior of a deterministic system can, in a sense, be stochastic and stochastic systems can have a reduced stochasticity. The limiting distribution is calculated explicitly. [ABSTRACT FROM AUTHOR]

This study proposes an algorithm to identify stable Kuroshio meanderings by extracting topological features from a sea surface height (SSH) gridded dataset in 1993–2020. Based on the mathematical theory of topological classifications for streamline patterns, the algorithm provides a unique symbolic representation and a discrete graph structure, which is referred to as the partially cyclically ordered rooted tree (COT) representation and the Reeb graph, respectively, to structurally stable Hamiltonian vector fields. We have confirmed that the temporal variations in the Kuroshio southernmost position south of the Tokai district captured by the algorithm are well consistent with the existing results by the Japan Meteorological Agency (JMA). The algorithm based on the topology detects five meandering periods: The three of them correspond to large meandering events detected by the JMA, while the two of them are offshore non-large meandering events. The topological data analysis reveals that a large cyclonic eddy inside of the meandering is split into two small eddies near the termination of the most meandering events. [ABSTRACT FROM AUTHOR]

Given a set S of integers, for a directed acyclic pseudograph we say that it has an S-good orientation if all its sources and sinks have degrees in S; in these terms, the existing notion of good orientation is a { 1 } -good orientation. We give a criterion for a pseudograph to admit an S-good orientation in terms of the structure of its leaf blocks. This criterion allows to compute whether a pseudograph admits an S-good orientation with optimal (linear) time and space complexity. As an application, we give a simple criterion for a graph to be realized as the Reeb graph of a Morse function in terms of leaf blocks of the graph. Similar conditions can be obtained for other classes of functions, such as (simple) Morse–Bott functions or round functions. [ABSTRACT FROM AUTHOR]

We prove criteria for a graph to be the Reeb graph of a function of a given class on a closed manifold: Morse–Bott, round, and in general smooth functions whose critical set consists of a finite number of submanifolds. The criteria are given in terms of whether the graph admits an orientation, which we call S-good orientation, with certain conditions on the degree of sources and sinks, similar to the known notion of good orientation in the context of Morse functions. We also study when such a function is the height function associated with an immersion of the manifold. The condition for a graph to admit an S-good orientation can be expressed in terms of the leaf blocks of the graph. [ABSTRACT FROM AUTHOR]

This study proposes an algorithm detecting atmospheric blocking by extracting topological features of geopotential height data at 500 hPa. The algorithm uses topological flow data analysis (TFDA) providing a unique symbolic representation, named the partially cyclically ordered rooted tree (COT) representation, and a discrete graph structure, called a Reeb graph, to each structurally stable Hamiltonian vector field based on the mathematical theory of topological classifications for streamline patterns. It recognizes blocking events more simply and effectively using fewer meteorological parameters than conventional algorithms. Furthermore, the algorithm can determine morphological types of blocking events, an Omega shape or a dipole pattern, whereas no effective algorithm has been available so far. The identified blocking events and their morphological types are consistent with synopticians' subjective judgments. [ABSTRACT FROM AUTHOR]

We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles. [ABSTRACT FROM AUTHOR]

We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n-manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ3. [ABSTRACT FROM AUTHOR]

We prove that, up to homeomorphism, any graph subject to natural necessary conditions on orientation and the cycle rank can be realized as the Reeb graph of a Morse function on a given closed manifold M. Along the way, we show that the Reeb number R (M) , i.e., the maximum cycle rank among all Reeb graphs of functions on M, is equal to the corank of fundamental group π 1 (M) , thus extending a previous result of Gelbukh to the non-orientable case. [ABSTRACT FROM AUTHOR]

One potential solution to reduce the concentration of carbon dioxide in the atmosphere is the geologic storage of captured CO2 in underground rock formations, also known as carbon sequestration. There is ongoing research to guarantee that this process is both efficient and safe. We describe tools that provide measurements of media porosity, and permeability estimates, including visualization of pore structures. Existing standard algorithms make limited use of geometric information in calculating permeability of complex microstructures. This quantity is important for the analysis of biomineralization, a subsurface process that can affect physical properties of porous media. This paper introduces geometric and topological descriptors that enhance the estimation of material permeability. Our analysis framework includes the processing of experimental data, segmentation, and feature extraction and making novel use of multiscale topological analysis to quantify maximum flow through porous networks. We illustrate our results using synchrotron-based X-ray computed microtomography of glass beads during biomineralization. We also benchmark the proposed algorithms using simulated data sets modeling jammed packed bead beds of a monodispersive material.

This paper considers two approaches to partitioning of a triangulated multiply connected domain into connected subdomains without branching of internal boundaries. A modified algorithm for constructing the Reeb graph for the topology determining of the triangulated surface of a three-dimensional domain has been proposed. On the basis of the partition of the Reeb graph, formation of subdomains of triangulation without branching of internal boundaries has been performed. Another approach is based on the formation of an ordered set of layers – subsets of 3-simplexes of triangulation using its topological properties, such as vertex and face connectivity. By construction, the layers do not contain branches of internal boundaries. At the same time, for multiply connected computational domains it is characteristic to obtain disconnected layers. The algorithm for combining layers into connected subdomains of triangulation has been developed based on the graph of sublayers, its vertices corresponding to the connected components of each layer. Thus, the union of layers reduces to the union of vertices and edges of the graph of sublayers – a problem with much smaller dimension, and the mapping of the partition of the graph of sublayers into triangulation. The proposed algorithms have been compared following the partitioning of triangulated multiply connected domains with surfaces of different types and genus. The complexity estimates of the algorithms have been given and the quality of the partitions by the number of 2-simplexes common for the obtained subregions of the triangulation has been compared.

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. [ABSTRACT FROM AUTHOR]

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. • The exact geostrophic streamfunction for neutral surfaces is presented. • Islands cause it to be ill-defined: a multivalued function in a given water column. • Using the Reeb graph, a well-defined approximation is given: the topobaric geostrophic streamfunction. • The orthobaric Montgomery potential is presented, generalizing the Montgomery potential. • Both new geostrophic streamfunctions are highly accurate and useful on any approximately neutral surface. [ABSTRACT FROM AUTHOR]

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. • Density is a multivalued function of pressure on a neutral surface. • The topology of this function is determined by the Reeb graph. • A new approximately neutral surface is built from this function: topobaric surfaces. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Reeb graph of a scalar function that is defined on a domain gives a topologically meaningful summary of that domain. Reeb graphs have been shown in the past decade to be of great importance in geometric processing, image processing, computer graphics, and computational topology. The demand for analyzing large data sets has increased in the last decade. Hence, the parallelization of topological computations needs to be more fully considered. We propose a parallel augmented Reeb graph algorithm on triangulated meshes with and without a boundary. That is, in addition to our parallel algorithm for computing a Reeb graph, we describe a method for extracting the original manifold data from the Reeb graph structure. We demonstrate the running time of our algorithm on standard datasets. As an application, we show how our algorithm can be utilized in mesh segmentation algorithms.