Original Vertex Research Articles

Laplacian2Mesh: Laplacian-Based Mesh Understanding.

Geometric deep learning has sparked a rising interest in computer graphics to perform shape understanding tasks, such as shape classification and semantic segmentation. When the input is a polygonal surface, one has to suffer from the irregular mesh structure. Motivated by the geometric spectral theory, we introduce Laplacian2Mesh, a novel and flexible convolutional neural network (CNN) framework for coping with irregular triangle meshes (vertices may have any valence). By mapping the input mesh surface to the multi-dimensional Laplacian-Beltrami space, Laplacian2Mesh enables one to perform shape analysis tasks directly using the mature CNNs, without the need to deal with the irregular connectivity of the mesh structure. We further define a mesh pooling operation such that the receptive field of the network can be expanded while retaining the original vertex set as well as the connections between them. Besides, we introduce a channel-wise self-attention block to learn the individual importance of feature ingredients. Laplacian2Mesh not only decouples the geometry from the irregular connectivity of the mesh structure but also better captures the global features that are central to shape classification and segmentation. Extensive tests on various datasets demonstrate the effectiveness and efficiency of Laplacian2Mesh, particularly in terms of the capability of being vulnerable to noise to fulfill various learning tasks.

Thick-panel Origami Cube

This paper presents a new method for constructing a novel thick-panel origami cube. By replacing the equivalent 4R-spherical linkage in a four-crease zero-thickness origami vertex with a plane-symmetric Bricard linkage, the thick-panel form of a general plane-symmetric four-crease origami vertex is identified and constructed. The proposed thick-panel vertex preserves the kinematics of the original four-crease origami vertex. Then, by utilizing the proposed thick-panel vertex, a thick-panel cube is constructed based on the zero-thickness origami cube that was proposed in our previous work. Through mechanism decomposition, geometric constraints and kinematic properties of the corresponding integrated mechanism are investigated and formulated, which reveals the kinematic equivalence between the thick-panel and zero-thickness forms of the origami cube. In addition, a prototype of the proposed thick-panel origami cube is fabricated, verifying its kinematic properties. The proposed technique can be extended to the design and construction of thick-panel polyhedrons with potential applications in the fields such as aerospace exploration, robotics and architecture.

A General Region Nesting-Based Semi-Fragile Reversible Watermarking for Authenticating 3D Mesh Models

Aiming to reduce the distortion in the existing reversible watermarking for 3D mesh models, this paper generalizes two-dimensional (2D) region nesting to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> -dimensional space, and proposes a general region nesting based semi-fragile reversible watermarking for authenticating 3D mesh models by combing watermark generation based on vertex projection and mesh topology. For any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> -dimensional space, each hypercube space is partitioned into several nested watermark sub-spaces, and the watermark is embedded by placing the original vertex and the mapped vertex on a straight line. The watermark is extracted by acquiring the position of the vertices, and its original coordinates can be restored by inverse mapping. Furthermore, a new coordinate system is built for 3D mesh models based on reference vertices to achieve semi-fragility, and the watermarks generated by vertex projection and mesh topology is utilized for integrity authentication. Experimental results and analysis show that the proposed scheme can significantly reduce the distortion, achieve semi-fragility in RST (Rotation, Scaling and Translation) and vertex reordering, identify the tampering type, and accomplish tampering location at vertex level. It has great potential to be applied for content authentication of 3D mesh models.

Approximability of the independent feedback vertex set problem for bipartite graphs

Given an undirected graph G with n vertices, the independent feedback vertex set problem is to find a vertex subset F of G with the minimum number of vertices such that F is both an independent set and a feedback vertex set of G, if it exists. This problem is known to be NP-hard for bipartite planar graphs of maximum degree four. In this paper, we study the approximability of the problem. We first show that, for any fixed ε>0, unless P=NP, there exists no polynomial-time n1−ε-approximation algorithm even for bipartite planar graphs. We then give an α(Δ−1)/2-approximation algorithm for bipartite graphs G of maximum degree Δ, which runs in t(α,G)+O(Δn) time, under the assumption that there is an α-approximation algorithm for the original feedback vertex set problem on bipartite graphs which runs in t(α,G) time. This algorithmic result also yields a polynomial-time (exact) algorithm for the independent feedback vertex set problem on bipartite graphs of maximum degree three.

A Semi-Fragile Reversible Watermarking for Authenticating 2D Engineering Graphics Based on Improved Region Nesting

To achieve high tampering localization precision and low distortion, a semi-fragile reversible watermarking for authenticating 2D engineering graphics is proposed based on an improved region nesting and a novel watermark generation. First, an intensive investigation is performed to the recently proposed region nesting (RN) partition. It is found that the original vertex and its mapped one cannot be guaranteed to be on the same line, which means that it still has room for improvement in term of distortion. Based on this, an improved region nesting partition (IRN) is proposed. Secondly, inspiring by the idea of soldiers parading, a novel watermark generation based on the short hash of the adjacent geometric features is developed. Then, a new coordinate system is constructed to achieve invariability of translation, scaling, rotation and entity re-arrangement. Based on the above techniques, a semi-fragile reversible watermarking for authenticating 2D engineering graphics is put forward. Experimental results and analysis show that IRN can obtain at least 15% distortion reduction compared with RN, the precision of tampering localization can be improved to a single vertex, and a certain semi-fragility in translation, scaling, rotation, and entity re-arrangement can be achieved. Furthermore, it has no file size expansion, and can obtain a good balance among imperceptibility, semi-fragility and tampering localization precision.

Coercing machine learning to output physically accurate results

Many machine/deep learning artificial neural networks are trained to simply be interpolation functions that map input variables to output values interpolated from the training data in a linear/nonlinear fashion. Even when the input/output pairs of the training data are physically accurate (e.g. the results of an experiment or numerical simulation), interpolated quantities can deviate quite far from being physically accurate. Although one could project the output of a network into a physically feasible region, such a postprocess is not captured by the energy function minimized when training the network; thus, the final projected result could incorrectly deviate quite far from the training data. We propose folding any such projection or postprocess directly into the network so that the final result is correctly compared to the training data by the energy function. Although we propose a general approach, we illustrate its efficacy on a specific convolutional neural network that takes in human pose parameters (joint rotations) and outputs a prediction of vertex positions representing a triangulated cloth mesh. While the original network outputs vertex positions with erroneously high stretching and compression energies, the new network trained with our physics “prior” remedies these issues producing highly improved results.

How Electroencephalogram Reference Influences the Movement Readiness Potential?

Readiness potential (RP) based on electroencephalograms (EEG) has been studied extensively in recent years, but no studies have investigated the influence of the reference electrode on RP. In order to investigate the reference effect, 10 subjects were recruited and the original vertex reference (Cz) was used to record the raw EEG signal when the subjects performed a motor preparation task. The EEG was then transformed to the common average reference (CAR) and reference electrode standardization technique (REST) reference, and we analyzed the RP waveform and voltage topographies and calculated the classification accuracy of idle and RP EEG segments. Our results showed that the RP waveform and voltage topographies were greatly influenced by the reference, but the classification accuracy was less affected if proper channels were selected as features. Since the Cz channel is near the primary motor cortex, where the source of RP is located, using the REST and CAR references is recommended to get accurate RP waveforms and voltage topographies.

A multilevel bilinear programming algorithm for the vertex separator problem

The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper, we develop a more general continuous bilinear program which incorporates vertex weights, and which applies to the coarse graphs that are generated in a multilevel compression of the original Vertex Separator Problem. We develop a method for improving upon a given vertex separator by applying a Mountain Climbing Algorithm to the bilinear program using an incidence vector for the separator as a starting guess. Sufficient conditions are developed under which the algorithm can improve upon the starting guess after at most two iterations. The refinement algorithm is augmented with a perturbation technique to enable escapes from local optima and is embedded in a multilevel framework for solving large scale instances of the problem. The multilevel algorithm is shown through computational experiments to perform particularly well on communication and collaboration networks.

A Macdonald refined topological vertex

We consider the refined topological vertex of Iqbal et al (2009 J. High Energy Phys. JHEP10(2009)069), as a function of two parameters , and deform it by introducing the Macdonald parameters , as in the work of Vuletić on plane partitions (Vuletić M 2009 Trans. Am. Math. Soc. 361 2789–804), to obtain ‘a Macdonald refined topological vertex’. In the limit , we recover the refined topological vertex of Iqbal et al and in the limit , we obtain a qt-deformation of the original topological vertex of Aganagic et al (2005 Commun. Math. Phys. 25 425–78). Copies of the vertex can be glued to obtain qt-deformed 5D instanton partition functions that have well-defined 4D limits and, for generic values of , contain infinite-towers of poles for every pole present in the limit .

Forbidden Pairs with a Common Graph Generating Almost the Same Sets

Let $\mathcal{H}$ be a family of connected graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ does not contain any members of $\mathcal{H}$ as an induced subgraph. Let $\mathcal{F}(\mathcal{H})$ be the family of connected $\mathcal{H}$-free graphs. In this context, the members of $\mathcal{H}$ are called forbidden subgraphs.In this paper, we focus on two pairs of forbidden subgraphs containing a common graph, and compare the classes of graphs satisfying each of the two forbidden subgraph conditions. Our main result is the following: Let $H_{1},H_{2},H_{3}$ be connected graphs of order at least three, and suppose that $H_{1}$ is twin-less. If the symmetric difference of $\mathcal{F}(\{H_{1},H_{2}\})$ and $\mathcal{F}(\{H_{1},H_{3}\})$ is finite and the tuple $(H_{1};H_{2},H_{3})$ is non-trivial in a sense, then $H_{2}$ and $H_{3}$ are obtained from the same vertex-transitive graph by successively replacing a vertex with a clique and joining the neighbors of the original vertex and the clique. Furthermore, we refine a result in [Combin. Probab. Comput. 22 (2013) 733–748] concerning forbidden pairs.

A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes

Starting from the recently introduced virtual element method, we construct new diffusion fluxes in two and three dimensions that give birth to symmetric, unconditionally coercive finite volume like schemes for the discretization of heterogeneous and anisotropic diffusion-reaction problems on general, possibly nonconforming meshes. Convergence of the approximate solutions is proved for general tensors and meshes. Error estimates are derived under classical regularity assumptions. Numerical results illustrate the performance of the scheme. The link with the original vertex approximate gradient scheme is emphasized.

Spectral characterizations of two families of nearly complete bipartite graphs

It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each of which is obtained from a complete bipartite graph by adding a vertex and an edge incident on the new vertex and an original vertex, which are not determined by their spectra.

Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

We consider coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We extend previously developed aggregation-based coarsening procedures applied to both sets of dofs [P. S. Vassilevski and L. T. Zikatanov, Numer. Linear Algebra Appl., 21 (2014), pp. 297--315] to now allow more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is a modification of the spectral upscaling procedure developed in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] for the mixed finite element discretization of diffu...

Basic models and questions in statistical network analysis

Extracting information from large graphs has become an important statistical problem since network data is now common in various fields. In this minicourse we will investigate the most natural statistical questions for three canonical probabilistic models of networks: (i) community detection in the stochastic block model, (ii) finding the embedding of a random geometric graph, and (iii) finding the original vertex in a preferential attachment tree. Along the way we will cover many interesting topics in probability theory such as Polya urns, large deviation theory, concentration of measure in high dimension, entropic central limit theorems, and more.

The role of non‐coaxiality in the simulation of strain localization based on classical and Cosserat continua

SummaryIt is normally accepted that materials inside the shear band undergo severe rotation of the principal stress direction, which causes non‐coaxiality between the principal stress and principal plastic strain rate. However, classical plasticity flow theory implicitly assumes that the principal stress and the principal plastic strain rate are coaxial; thus, it may not correctly predict the onset of the shear band. In addition, classical continuum does not contain any internal length scales; as a result, it cannot provide a reasonable shear band thickness. In this study, the original vertex non‐coaxial plastic model based on the classical continuum is extended to the Cosserat continuum. The corresponding codes are implemented via the interface of the user defined element subroutine in ABAQUS. Through a simple shear test, the effectiveness of the user's codes is verified. Through a uniaxial compression test, the influence of non‐coaxiality on the onset, the orientation, and the thickness of the shear band is investigated. Results show that the onset of the shear localization is delayed, and the thickness of the shear band is widened when the non‐coaxial degree increases, while the orientation of the shear band is little affected by the non‐coaxial degree. In addition, it is found that the non‐coaxiality can weaken the micro‐polar effect to some extent; nonetheless, the Cosserat non‐coaxial model still has its advantage over the classical non‐coaxial model in capturing the pre‐bifurcation as well as the post‐bifurcation behaviors of strain localization. Copyright © 2016 John Wiley &amp; Sons, Ltd.

A Triangle Mesh Standardization Method Based on Particle Swarm Optimization.

To enhance the triangle quality of a reconstructed triangle mesh, a novel triangle mesh standardization method based on particle swarm optimization (PSO) is proposed. First, each vertex of the mesh and its first order vertices are fitted to a cubic curve surface by using least square method. Additionally, based on the condition that the local fitted surface is the searching region of PSO and the best average quality of the local triangles is the goal, the vertex position of the mesh is regulated. Finally, the threshold of the normal angle between the original vertex and regulated vertex is used to determine whether the vertex needs to be adjusted to preserve the detailed features of the mesh. Compared with existing methods, experimental results show that the proposed method can effectively improve the triangle quality of the mesh while preserving the geometric features and details of the original mesh.

A progressive transmission strategy for GIS vector data under the precondition of pixel losslessness

It is well known that the final results of geographic information system (GIS) mapping are composed of rasterized pixels in most display devices. Based on this, pixel equivalence is proposed as two vector data that can produce the same raster image on a device under the same mapping processes and parameters. Moreover, if between the two data, one is considered the original data and the other is its reduced-quantity version, then the latter can be regarded as pixel losslessness relative to the former. In this case, the reduced-quantity version can be used instead of the original data to produce the same result from the perspective of visualization. The aim of this article is to explore how to produce the transmitted data, including an initial code and a series of increments that are used for progressive transmission, to retain pixel losslessness. By illustrating that the vector data can be represented by a vertex or ordered sequence of vertices, this article introduces the detailed concepts of pixel equivalence and pixel losslessness of vector data. Then, the requirements for pixel equivalence between two vertices are deduced by analyzing the coordinate transform processes under specific assumptions. The method to generate an initial code and a series of increments related to a single vertex is proposed to produce vertices that are pixel equivalence to the original vertex under different coordinate transform processes. By discussing the traversing mode of the ordered sequence of vertices based on quadtree cells, a coding algorithm is proposed to develop the progressive transmission related to the multi-vertex and to generate reduced-quantity versions to retain pixel losslessness. Finally, experiments are designed to demonstrate that, under the precondition of pixel losslessness, the proposed progressive transmission method can significantly reduce the amount of transmitted data in the network environment.

Evaluation and Comparison on the Techniques of Vertex Chain Codes

This paper firstly describes the techniques of six representative vertex chain codes, they are: original vertex chain code, extended vertex chain code, variable-length vertex chain code, variable-length compressed vertex chain code, dynamic vertex chain code, and equal-length compressed vertex chain code. The description includes the main idea and encoding method of each vertex chain code. Then the chain length, namely the code numbers, the memory occupancy, namely the general binary bits, the code average length of each code, namely bits per code of each vertex chain code were compared respectively by large numbers of experiments. In the end, the evaluation and comparison were given from the view of chain code efficiency. The goal of the paper is to provide convenience and reference for the chain code researchers and users.

Generalized Superconducting Gap in an Anisotropic Boson–Fermion Mixture with a Uniform Coulomb Field

A boson–fermion (BF) quantum-statistical binary gas mixture model of high- T c superconductors (HTSCs) recently introduced consists of resonant bosonic Cooper electron pairs (CPs) in chemical and thermal equilibrium with single unpaired electrons. Here, this model is refined and extended to include: i) the anisotropy of the original BF vertex interaction causing boson formation/disintegration and ii) momentum-independent Coulomb repulsions between electron charge carriers. It is shown that pair breakings due to Coulomb repulsion depend on the separation between boson and fermion spectra. Specifically, as such a separation shrinks, the pair-breaking ability of the Coulomb interaction weakens and disappears altogether at the Bose–Einstein condensation (BEC) T c , i.e., at the temperature at which a complete softening of bosons occurs due to boson self-energy renormalization. Simultaneous inclusion of both effects produces “islands” in momentum space of incoherent CPs above the Fermi sea as temperature is lowered. These islands grow upon further cooling and merge together just before T c is reached. The BF model thusly extended now predicts a pseudogap phase in 2D HTSCs with lines of points on the Fermi surface along which the pseudogap vanishes, hence explaining the origin of the temperature-dependent “Fermi arcs” observed in experiments.

Finite element analysis of anchor plates using non-coaxial models

The non-coaxial model simulating the non-coincidence between the principal stresses and the principal plastic strain rates is employed within the framework of finite element method (FEM) to predict the behaviors of anchors embedded in granular material. The non-coaxial model is developed based on the non-coaxial yield vertex theory, and the elastic and conventional coaxial plastic deformations are simulated by using elasto-perfectly plastic Drucker-Prager yield function according to the original yield vertex theory. Both the horizontal and vertical anchors with various embedment depths are considered. Different anchor shapes and soil friction and dilation angles are also taken into account. The predictions indicate that the use of non-coaxial models leads to softer responses, compared with those using conventional coaxial models. Besides, the predicted ultimate pulling capacities are the same for both coaxial and non-coaxial models. The non-coaxial influences increase with the increasing embedment depths, and circular anchors lead to larger non-coaxial influences than strip anchors. In view of the fact that the design of anchors is mainly determined by their displacements, ignoring the non-coaxiality in finite element numerical analysis can lead to unsafe results.