Discrete Vector Fields Research Articles

THEORETICAL STUDY OF VORTEX-LIKE ATOMIC STRUCTURES WITHIN A CONTINUUM FRAMEWORK

Using regression analysis through a component-wise search for regressors, a computational approach is proposed for estimating the oriented curvature of vortex-like atomic structures. The studied discrete vector fields of atomic displacements of such structural states are obtained based on the results of molecular dynamics simulations of a deformable defect-free crystallite. Within the given approach, a continuous representation of the desired discrete vector field is obtained. The redistribution of the free volume and reorientation of the crystal lattice are analyzed. It is found that the value of oriented curvature outside specific regions (vortex centers and regions of compensation of atomic displacements) is about |K| ≈ 0.286 nm-1. The relationship is shown between vortex-like structures and model concepts of a continuous disclination-type defect.

Localized Evaluation for Constructing Discrete Vector Fields.

Topological abstractions offer a method to summarize the behavior of vector fields, but computing them robustly can be challenging due to numerical precision issues. One alternative is to represent the vector field using a discrete approach, which constructs a collection of pairs of simplices in the input mesh that satisfies criteria introduced by Forman's discrete Morse theory. While numerous approaches exist to compute pairs in the restricted case of the gradient of a scalar field, state-of-the-art algorithms for the general case of vector fields require expensive optimization procedures. This paper introduces a fast, novel approach for pairing simplices of two-dimensional, triangulated vector fields that do not vary in time. The key insight of our approach is that we can employ a local evaluation, inspired by the approach used to construct a discrete gradient field, where every simplex in a mesh is considered by no more than one of its vertices. Specifically, we observe that for any edge in the input mesh, we can uniquely assign an outward direction of flow. We can further expand this consistent notion of outward flow at each vertex, which corresponds to the concept of a downhill flow in the case of scalar fields. Working with outward flow enables a linear-time algorithm that processes the (outward) neighborhoods of each vertex one-by-one, similar to the approach used for scalar fields. We couple our approach to constructing discrete vector fields with a method to extract, simplify, and visualize topological features. Empirical results on analytic and simulation data demonstrate drastic improvements in running time, produce features similar to the current state-of-the-art, and show the application of simplification to large, complex flows.

Two-dimensional vector field of the phase energy spectrum application for motion identification

In the tasks of determining the location of objects and identifying movement, it is important to use the phase information contained in the phase-frequency spatial spectrum. In these problems, the phase-energy spectrum can be applied, which combines the advantages of the energy and phase spectra. A frame of a video sequence can be associated with a discrete vector field of the phase-energy spectrum; its components are located throughout the image. No less informative, but much simpler for analysis, is the discrete vector field of phase-energy characteristics. Its components are formed by the amplitudes of the harmonics of the phase-energy spectrum of a two-dimensional frame. To use the vector field of phase-energy characteristics in algorithms for identifying dynamic objects, information about interframe changes in its components is important. In this paper, the features of the change in the components of the vector field for the cases of horizontal and vertical movement of a rectangular object of constant brightness against a homogeneous background are investigated. Graphical representations of scalar fields of phase-energy functions are given. Characteristic regions are identified and their projections onto the coordinate plane are made. It is shown that two characteristic areas are the near and far hills, by analyzing which it is possible to determine the spatial coordinates of the object before the start of movement and the value of the object's inter-frame shift. The special areas of scalar fields of phase-energy functions are determined for the case of object motion in an arbitrary direction. It has been established that the presence of extended extrema in the graphs of phase-energy functions indicates a strictly vertical or horizontal movement of the object, and the presence of point extrema indicates the movement of the object “at an angle”, and the boundaries of the moving object and the magnitude of its displacement horizontally and vertically are completely determined by the coordinates of the point extrema. A number of options for finding the parameters of objects by extremums and/or by the boundaries of characteristic regions of phase-energy functions are proposed. The "boundary" and "extreme" approaches can be applied in vision systems as both an additional and a main tool for motion identification.

A Study of the Crystal Lattice Curvature Based on the Discrete Vector Field of Atomic Displacements

A Study of the Crystal Lattice Curvature Based on the Discrete Vector Field of Atomic Displacements

A numerical method based on the Nambu bracket for the 3D vorticity equation

Abstract In this study, numerical methods for the 3D vorticity equation and Euler equation, which preserve the structure of Nambu mechanics for fluid dynamics, are investigated. Discrete vector fields of the stream function (or vector potential), velocity, and vorticity, as well as discrete counterparts of the gradient, curl, and divergence operators acting on them, are defined such that the structure of the de Rham complex in 3D Euclidean space is preserved. The inner products of the discrete vector fields are defined such that discrete counterparts of integration-by-parts formulae for the gradient, curl, and divergence operators hold. In addition, cross products of the discrete vector fields are introduced to define a skew-symmetric trilinear form. A discrete Nambu bracket, as well as the (kinetic) energy, helicity, and enstrophy of the discrete flow field, are defined straightforwardly. They are employed to derive a discrete vorticity equation in the same way as in the continuum setting. A discrete Euler equation is derived from the discrete vorticity equation based on the discrete counterpart of the Poincaré lemma, which holds under some typical conditions. It is proved that any solution to these discretized equations satisfies discrete analogues of the balances of energy, helicity, and enstrophy. Numerical experiments on a periodic array of rolls are conducted to examine the effectiveness of the method.

Inverting the discrete curl operator: A novel graph algorithm to find a vector potential of a given vector field

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of 1- and 2-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations.

Multi-task super resolution method for vector field critical points enhancement

<p>It is a challenging task to handle the vector field visualization <span style="font-family: 'Times New Roman';">at</span> local critical points. Generally, topological based methods firstly divide critical regions into different categories, and then process the different types of critical regions to improve the effect, which pipeline is complex. In the paper, a learning based multi-task super resolution (SR) method is proposed to improve the refinement of vector field, and enhance the visualization effect, especially at the critical region. In detail, the multi-task model consists of two important <span style="font-family: 'Times New Roman';">des</span>igns on task branches: one task is to simulate the interpolation of discrete vector fields based on an improved super-resolution network; and the other is a classification task to identify the types of critical vector fields. It is an efficient end-to-end architecture for both training and inferencing stages, which simplifies the pipeline of critical vector field visualization and improves the visualization effect. In experiment, we compare our method with both traditional interpolation and pure SR network on both simulation data and real data, and the reported results indicate our method lower the error and improve PSNR significantly.</p>

Multi-task super resolution method for vector field critical points enhancement

<p>It is a challenging task to handle the vector field visualization <span style="font-family: 'Times New Roman';">at</span> local critical points. Generally, topological based methods firstly divide critical regions into different categories, and then process the different types of critical regions to improve the effect, which pipeline is complex. In the paper, a learning based multi-task super resolution (SR) method is proposed to improve the refinement of vector field, and enhance the visualization effect, especially at the critical region. In detail, the multi-task model consists of two important <span style="font-family: 'Times New Roman';">des</span>igns on task branches: one task is to simulate the interpolation of discrete vector fields based on an improved super-resolution network; and the other is a classification task to identify the types of critical vector fields. It is an efficient end-to-end architecture for both training and inferencing stages, which simplifies the pipeline of critical vector field visualization and improves the visualization effect. In experiment, we compare our method with both traditional interpolation and pure SR network on both simulation data and real data, and the reported results indicate our method lower the error and improve PSNR significantly.</p>

A new numerical method for div-curl systems with low regularity assumptions

This paper presents a new numerical method for div-curl systems with the normal boundary condition by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two prominent features in that it offers not only an accurate and reliable numerical solution to the div-curl system under the low Hα-regularity (α>0) assumption for the true solution, but also an effective approximation of the normal harmonic vector fields on domains with complex topology. Seven numerical experiments are conducted and the results are presented to demonstrate the performance of the PDWG algorithm, including one example on the computation of discrete normal harmonic vector fields.

3D-print infill generation using the biological phase field of an optimized discrete material orientation vector field

A framework to produce an optimized infill structure for 3D-printed curvilinear fiber-reinforced polymer composites is proposed. The material orientation was optimized first to maximize the physical properties. Then, the phase field of the stripe pattern was developed based on the optimized material orientation vector field using the local activation and long-range inhibition system for the formation of biological patterns. Finally, the phase field was converted into a 3D print path. The proposed approach was validated both numerically and experimentally. The optimized infill structure showed a higher structural stiffness compared with commonly used infill structures prepared via concentric or rectilinear strategies, demonstrating the advantage of the proposed method.

Signal processing of simplicial complexes

The paper considered the signal processing of simplicial complexes. Hodge decomposition formula for discrete fields is given, which is similar to the Hodge decomposition formula for smooth vector fields. The construction and the estimation of the gradient, divergence and curl operators and Laplace matrices for discrete vector fields are considered.

16. DETECTING DIFFERENTIALLY METHYLATED REGIONS IN INDIVIDUAL METHYLOMES

Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are also related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a birth-death descriptor and a new invariant providing information on multifiltrations. This new invariant, in some cases, is able to provide more information than the rank invariant. We show some applications of our algorithms to spaces of infinite type via the effective homology technique, where the performance has also been improved by means of discrete vector fields.

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

Discrete line fields on surfaces

Vector fields and line fields, their counterparts without orientations on tangent lines, are familiar objects in the theory of dynamical systems. Among the techniques used in their study, the Morse–Smale decomposition of a (generic) field plays a fundamental role, relating the geometric structure of phase space to a combinatorial object consisting of critical points and separatrices. Such concepts led Forman to a satisfactory theory of discrete vector fields, in close analogy to the continuous case.In this paper, we introduce discrete line fields. Again, our definition is rich enough to provide the counterparts of the basic results in the theory of continuous line fields: an Euler–Poincaré formula, a Morse–Smale decomposition and a topologically consistent cancellation of critical elements, which allows for topological simplification of the original discrete line field.

Geometric Verification of Vector Fields for Autonomous Aircraft Navigation

New generations of small advanced aircraft may soon transform how people and freight are transported. Thus, ensuring safe and reliable operation of these systems, in all plausible scenarios, is of utmost priority. In this paper, we address this problem by proposing a method to verify the safety of an autonomous aircraft controller following a navigational policy. The policy is encoded using discrete vector fields that are meant to drive the vehicle to a goal state and away from obstacles or restricted airspace. The solution presented in this paper provides the ability to verify the policy, catching unsafe scenarios which may be missed by random Monte Carlo methods or general reachability analysis. After illustrating the main theoretical principles, a possible practical implementation is described and compared with analysis based on Monte Carlo simulations. The comparison shows an insightful example where Monte Carlo simulations fail to detect several corner cases that are uncovered by the proposed method. In the conclusion of the paper, we provide insights about possible future research to implement the proposed solution obtaining higher accuracy and efficiency.

Geometric Variational Finite Element Discretizations for Fluids

Abstract We present an overview of finite element variational integrators for compressible and incompressible fluids with variable density. The numerical schemes are derived by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart-Thomas finite element space. We illustrate the conservation properties of the scheme with the Rayleigh-Taylor instability test.

Effective homological computations on finite topological spaces

The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions of well-known results about topological spaces with combinatorial methods used on finite spaces. In the particular case of h-regular spaces, effective and reasonably efficient methods are implemented and the technique of discrete vector fields is applied in order to improve the previous algorithms.

A Variational Finite Element Discretization of Compressible Flow

We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart–Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that does not seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

Computing invariants for multipersistence via spectral systems and effective homology

Both spectral sequences and persistent homology are tools in algebraic topology defined from filtrations of objects (e.g. topological spaces or simplicial complexes) indexed over the set Z of integer numbers. A recent work has shown the details of the relation between both concepts. Moreover, generalizations of both concepts have been proposed which originate from a different choice of the set of indices of the filtration, producing the new notions of multipersistence and spectral system. In this paper, we show that these notions are also related, generalizing results valid in the case of filtrations over Z. By using this relation and some previous programs for computing spectral systems, we have developed a new module for the Kenzo system computing multipersistence. We also present a birth-death descriptor and a new invariant providing information on multifiltrations. This new invariant, in some cases, is able to provide more information than the rank invariant. We show some applications of our algorithms to spaces of infinite type via the effective homology technique, where the performance has also been improved by means of discrete vector fields.

Critical sets of PL and discrete Morse theory: A correspondence

Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical points have been given in the literature for the discrete setting, making a clear understanding of the relationships occurring between them not obvious. This paper aims at providing equivalence results about critical points of the two discretized Morse theories. First of all, we prove the equivalence of the existing notions of PL critical points. Next, under an optimality condition called relative perfectness, we show a dimension agnostic correspondence between the set of PL critical points and that of discrete critical simplices of the combinatorial approach. Finally, we show how a relatively perfect discrete gradient vector field can be algorithmically built up to dimension 3. This way, we guarantee a formal and operative connection between critical sets in the PL and discrete theories.