Tangent Plane Research Articles

Boundary element method for hypersingular integral equations: Implementation and applications in potential theory

The main objective of this paper is to develop effective numerical methods to solve hypersingular integral equations arising in various physical and mechanical applications. Both surface and contour integrals are considered. The novelty of the proposed approach lies in the exact formulas obtained for an arbitrary planar polygon in hypersingular integral estimations. A one-dimensional hypersingular integral equation is derived for axially symmetrical configurations, and analytical formulas are established for calculating the hypersingular parts. It is proved that the hypersingular component of the surface integral is equal to its hypersingular component along the tangent plane. These exact formulas enable the development of an effective numerical method based on boundary element implementation. Benchmark tests are considered, and the convergence of the proposed methods is demonstrated. Problems in crack analysis are formulated and solved using both surface and contour hypersingular integral equations. A comparison of the results is made between boundary element methods and finite element methods for penny-shaped cracks. Boundary value problems in fluid-structure interaction are considered, and numerical simulations are performed. An estimation of modes and frequencies of panel and blade vibrations when interacting with liquids is carried out.

Direct segmentation of cortical cytoarchitectonic domains using ultra-high-resolution whole-brain diffusion MRI.

We assess the potential of detecting cortical laminar patterns and areal borders by directly clustering voxel values of microstructural parameters derived from high-resolution mean apparent propagator (MAP) magnetic resonance imaging (MRI), as an alternative to conventional template-warping-based cortical parcellation methods. We acquired MAP-MRI data with 200μm resolution in a fixed macaque monkey brain. To improve the sensitivity to cortical layers, we processed the data with a local anisotropic Gaussian filter determined voxel-wise by the plane tangent to the cortical surface. We directly clustered all cortical voxels using only the MAP-derived microstructural imaging biomarkers, with no information regarding their relative spatial location or dominant diffusion orientations. MAP-based 3D cytoarchitectonic segmentation revealed laminar patterns similar to those observed in the corresponding histological images. Moreover, transition regions between these laminar patterns agreed more accurately with histology than the borders between cortical areas estimated using conventional atlas/template-warping cortical parcellation. By cross-tabulating all cortical labels in the atlas- and MAP-based segmentations, we automatically matched the corresponding MAP-derived clusters (i.e., cytoarchitectonic domains) across the left and right hemispheres. Our results demonstrate that high-resolution MAP-MRI biomarkers can effectively delineate three-dimensional cortical cytoarchitectonic domains in single individuals. Their intrinsic tissue microstructural contrasts enable the construction of whole-brain mesoscopic cortical atlases.

Gating-Enhanced Hierarchical Structure Learning in Hyperbolic Space and Multi-scale Neighbor Topology Learning in Euclidean Space for Prediction of Microbe-Drug Associations.

Identifying drug-related microbes may help us explore how the microbes affect the functions of drugs by promoting or inhibiting their effects. Most previous methods for the prediction of microbe-drug associations focused on integrating the attributes and topologies of microbe and drug nodes in Euclidean space. The heterogeneous network composed of microbes and drugs has a hierarchical structure, and the hyperbolic space is helpful for reflecting the structure. However, the previous methods did not fully exploit the structure. We propose a multi-space feature learning enhanced microbe-drug association prediction method, MFLP, to fuse the hierarchical structure of microbe and drug nodes in hyperbolic space and the multiscale neighbor topologies in Euclidean space. First, we project the nodes of the microbe-drug heterogeneous network on the sphere in hyperbolic space and then construct a topology which implies hierarchical structure and forms a hierarchical attribute embedding. The node information from multiple types of neighbor nodes with the new topological structure in the tangent plane space of a sphere is aggregated by the designed gating-enhanced hyperbolic graph neural network. Second, the gate at the node feature level is constructed to adaptively fuse the hierarchical features of microbe and drug nodes from two adjacent graph neural encoding layers. Third, multiple neighbor topological embeddings for each microbe and drug node are formed by neighborhood random walks on the microbe-drug heterogeneous network, and they cover neighborhood topologies with multiple scales, respectively. Finally, as each scale of topological embedding contains its specific neighborhood topology, we establish an independent graph convolutional neural network for the topology and form the topological representations of microbe and drug nodes in Euclidean space. The comparison experiments based on cross validation showed that MFLP outperformed several advanced prediction methods, and the ablation experiments verified the effectiveness of MFLP's major innovations. The case studies on three drugs further demonstrated MFLP's ability in being applied to discover potential candidate microbes for the given drugs.

IPUNet: Iterative Cross Field Guided Point Cloud Upsampling.

Point clouds acquired by 3D scanning devices are often sparse, noisy, and non-uniform, causing a loss of geometric features. To facilitate the usability of point clouds in downstream applications, given such input, we present a learning-based point upsampling method, i.e., iPUNet, which generates dense and uniform points at arbitrary ratios and better captures sharp features. To generate feature-aware points, we introduce cross fields that are aligned to sharp geometric features by self-supervision to guide point generation. Given cross field defined frames, we enable arbitrary ratio upsampling by learning at each input point a local parameterized surface. The learned surface consumes the neighboring points and 2D tangent plane coordinates as input, and maps onto a continuous surface in 3D where arbitrary ratios of output points can be sampled. To solve the non-uniformity of input points, on top of the cross field guided upsampling, we further introduce an iterative strategy that refines the point distribution by moving sparse points onto the desired continuous 3D surface in each iteration. Within only a few iterations, the sparse points are evenly distributed and their corresponding dense samples are more uniform and better capture geometric features. Through extensive evaluations on diverse scans of objects and scenes, we demonstrate that iPUNet is robust to handle noisy and non-uniformly distributed inputs, and outperforms state-of-the-art point cloud upsampling methods.

A joint lightly perturbation and synthesizing optimization attack framework for point cloud perturbation

Due to the inherent characteristic of deep learning models, prior researchers have focused on studying adversarial strategies aimed at uncovering their vulnerabilities. However, these strategies have been primarily been confined to the realm of image processing. When considering 3D vision structures, such as point clouds and meshes, the robustness of learning models also diminishes significantly in the presence of discernible outliers, and these strategies often fail to generalize across different DNN architectures. In this paper, a simple adversarial perturbation and optimization attack framework, Tangent Plane Perturbation and Synthesizing Optimization Attack (TPSOA) is presented. TPSOA provides a point cloud projection direction for point clouds, subtly inducing imperceptible perturbations on their tangent planes. Furthermore, acknowledging the geometry form of adversarial generation and its potential for 3D printing in physical scenarios, we reconstruct the adversarial mesh alongside the adversarial point cloud. A synthesized loss, incorporating both the Sobolev loss and the Chamfer loss, is utilized to optimize the distances between the adversarial generation and ground truth. Through a combination of visualization and quantitative analysis experiments, we validate the imperceptible nature of TPSOA, the necessity of the loss terms, and the magnitude of perturbations. These experiments that TPSOA enables targeted point cloud attacks that are imperceptible and of minor magnitude. In real-world adversarial attack and defense scenarios, as evidenced by both white-box and black-box experiments, the results show that TPSOA surpasses other state-of-the-art adversarial attack models, effectively manipulating a greater number of victim models while maintaining high efficiency, all without compromising its original structural integrity.

Impact of Wood Structure Variability on the Surface Roughness of Chestnut Wood

Wood constitutes a unique and valuable material that has been used from ancient times until nowadays in a wide variety of applications, in which the surface quality of wood often constitutes a critical factor. In this study, the influence of different wood areas and therefore, of different anatomical characteristic areas of chestnut wood (Castanea sativa Mill.) on the surface quality, was thoroughly studied, in terms of surface roughness. Five different chestnut tree trunks were harvested, from which five different disks were obtained corresponding to five different trunk heights. Surface roughness was measured on these disks on the transverse, radial, and tangential planes, on the areas of sapwood and heartwood, measuring the roughness in each point both vertically and in parallel to the wood grain. The results revealed that the examined roughness indexes (Ra, Rz, Rq) follow a parallel path to one another. In the case of all surfaces (transverse, radial, tangential) of the disks examined, when the measurement was implemented perpendicularly to the wood grain, a significantly higher roughness was recorded, compared to the wood grain measurements being implemented in parallel with the wood grain. Significant differences between heartwood and sapwood roughness were not demonstrated, although sapwood often appeared to exhibit a higher surface roughness than heartwood sites. Among the roughness values of the three different surfaces, the highest roughness in the vertical-to-wood-grain measurements was recorded by tangential surfaces, with slightly lower values on the transverse surfaces and the lowest roughness on radial surfaces. Meanwhile, for the measurements in parallel with the wood grain, the transverse surfaces presented significantly higher roughness values compared to the tangential and radial surfaces. Significant roughness differences were not detected among the surfaces at different trunk heights. Although, significant differences in roughness were recorded among different trees, it was observed that all the studied trees align with the identified and described within-tree trends.

Riemannian Geodesic Discriminant Analysis–Minimum Riemannian Mean Distance: A Robust and Effective Method Leveraging a Symmetric Positive Definite Manifold and Discriminant Algorithm for Image Set Classification

This study introduces a novel method for classifying sets of images, called Riemannian geodesic discriminant analysis–minimum Riemannian mean distance (RGDA-MRMD). This method first converts image data into symmetric positive definite (SPD) matrices, which capture important features related to the variability within the data. These SPD matrices are then mapped onto simpler, flat spaces (tangent spaces) using a mathematical tool called the logarithm operator, which helps to reduce their complexity and dimensionality. Subsequently, regularized local Fisher discriminant analysis (RLFDA) is employed to refine these simplified data points on the tangent plane, focusing on local data structures to optimize the distances between the points and prevent overfitting. The optimized points are then transformed back into a complex, curved space (SPD manifold) using the exponential operator to enhance robustness. Finally, classification is performed using the minimum Riemannian mean distance (MRMD) algorithm, which assigns each data point to the class with the closest mean in the Riemannian space. Through experiments on the ETH-80 (Eidgenössische Technische Hochschule Zürich-80 object category), AFEW (acted facial expressions in the wild), and FPHA (first-person hand action) datasets, the proposed method demonstrates superior performance, with accuracy scores of 97.50%, 37.27%, and 88.47%, respectively. It outperforms all the comparison methods, effectively preserving the unique topological structure of the SPD matrices and significantly boosting image set classification accuracy.

Sampling-Based Global Path Planning Using Convex Polytope Approximation for Narrow Collision-Free Space of Humanoid

In this paper, we propose a computationally fast method of sampling-based global path planning for humanoids under Manifold Constraints such as closed kinematic chains and Volume-Reducing Constraints such as collision avoidance. In multi-contact and whole-body manipulation of humanoids, narrow collision-free space causes path planning to take a long computation time (narrow corridor problem). In previous research of constrained planning, Manifold Constraints are locally approximated by tangent plane, and steering motions along Manifold Constraints are found efficiently by projection or continuation methods. In this paper, we applied constrained planning algorithms to collision avoidance. Since tangent plane cannot approximate Volume-Reducing Constraints, we adopted convex polytope instead of tangent plane. Both Manifold Constraints and Volume-Reducing Constraints are locally approximated by convex polytope with linear equality and inequality constraints, and steering motions inside the convex polytope are found efficiently by the SQP-based prioritized inverse kinematics. We developed CP-KPIECE (Convex Polytope approximation-based KPIECE) with this approach. Benchmarks using the humanoid JAXON proved the effectiveness of our approach for fast path planning in narrow collision-free space of humanoids.

A novel method to accelerate the three-phase flash calculation of the hydrocarbon-CO2 system

To achieve high computational efficiency and simulation accuracy in the equation-of-state-based compositional simulation, the fast multiphase flash calculation approach is required for the hydrocarbon-CO2 system, allowing for the simultaneous existence of vapour, CO2-lean, and CO2-rich phases. However, the conventional step-by-step method for multiphase flash calculation is associated with a high cost in identifying the three-phase equilibrium, thereby significantly impeding the computation efficiency of compositional simulation when the three-phase equilibrium appears in most grids.To address this challenge, a novel approach to multiphase flash calculation is proposed. In this study, all trial phases exhibiting negative tangent plane distance are employed to identify the three-phase equilibrium, eliminating the need for additional phase stability analysis and two-phase split calculations. Furthermore, this new method incurs minimal computational cost and requires only minor modifications to the existing stepwise procedure. Phase diagram calculations conducted on four cases demonstrate that the newly proposed method dramatically accelerates the identification of the three-phase equilibrium.

Geometric conditions to obtain Anosov geodesic flow for non-compact manifolds

Let ( M , g ) be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold with non-positive curvature, without compact quotient and geodesic flow of Anosov type.

On smoothness conditions and selection of the edge of a scattering surface in one class of 3D time-optimal problems

A class of time-optimal problems in three-dimensional space with a spherical velocity vectogram is considered. The target set is a parametrically defined smooth curve. Numerical and analytical approaches to constructing the bisector of the target set — the scattering surface — in the time-optimal problem are proposed. The algorithms are based on formulas for the edge points of the scattering surface, written in terms of curve invariants. It is shown that these points form the edge of the bisector and lie at the centers of the touching spheres to the curve. A theorem on sufficient conditions for the smoothness of a scattering surface is proven. The equations of the tangent plane to the bisector are found for those points from which exactly two optimal trajectories emerge. An example of solving a time-optimal problem in the form of a set of level surfaces of the optimal result function is given, highlighting the surface of their non-smoothness.

Phase field modeling of hyperelastic material interfaces –Theory, implementation and application to phase transformations

Interface mechanics can significantly govern the evolution of multiple phases on smaller scales, e.g., determining the properties of TWIP- and TRIP-steels, geopolymers or Li-ion batteries. The present contribution is specifically centered around the influence of interface elasticity on mechanically induced phase transformations. A geometrically exact finite element framework is developed for this purpose based on incremental energy minimization. It is set up in three-dimensional space and comprises an Allen-Cahn type phase field formulation based on a Modica-Mortola double-well functional, as well as a Ginzburg-Landau type viscosity formulation. The phase field approximation of the interface deformation gradient is derived by means of homogenization theory. A monolithic solver is employed for the resulting set of non-linear coupled equations. The example of coalescing spheres shows that larger interface energies can amplify and accelerate coalescence. Lower interface energies, in contrast, allow for distinct spreading of the energetically favorable bulk phase. Deformation-dependent interface energies particularly add stress peaks to this process that are localized at the phase transition zone. The relaxation of a stepped interface shape showed that deformation-dependent interface energies favor a plane partition of the bulk phases. Yet, they also delay breakup events compared to simple constant interface energies. This peculiarity can be beneficial for calibration purposes. Strain dependence moreover causes an additional elastic stiffness in the direction of the interface tangent plane and constitutes thus another origin for anisotropy of the overall structure.

On the Geometric and Physical Properties of Conformable Derivative

In this article, we explore the advantages geometric and physical implications of the conformable derivative. One of the key benefits of the conformable derivative is its ability to approximate the tangent at points where the classical tangent is not readily available. By employing conformable derivatives, alternative tangents can be created to overcome this limitation. Thanks to these alternative (conformable) tangents, physical interpretation can be made with alternative velocity vectors. Furthermore, the conformable derivative proves to be valuable in situations where the tangent plane cannot be defined. It enables the creation of alternative tangent planes, offering a solution in cases where the traditional approach falls short. Geometrically speaking, the conformable derivative carries significant meaning. It provides insights into the local behavior of a function and its relationship with nearby points. By understanding the conformable derivative, we gain a deeper understanding of how a function evolves and changes within its domain. A several examples are presented in the article to better understand the article and visualize the concepts discussed. These examples are accompanied by visual representations generated using the Mathematica program, aiding in a clearer understanding of the proposed ideas. By combining theoretical explanations, practical examples, and visualizations, this article aims to provide a comprehensive exploration of the advantages and geometric and physical implications of the conformable derivative.

The smallest structures in Saturn’s rings from UVIS stellar occultations

On the largest scales Saturn’s rings are often thought of as axisymmetric annuli varying only radially in optical depth with exceptions of spiral waves and eccentric ringlets. But on scales smaller than a few tens of km yet still larger than the largest common ring particles, the assumption of azimuthal symmetry breaks down. Structures such as “straw” in the troughs of density waves, partial gaps around propeller moonlets, and ephemeral particle aggregates such as self-gravity wakes are responsible for the predominant variations in optical depth. During the 13 year Cassini mission, the Ultraviolet Imaging Spectrograph (UVIS) high speed photometer (HSP) measured 80 stellar chord occultations of Saturn’s main rings. In the vicinity of the minimum ring plane radius of each chord, the occultation line-of-sight slewed across the ring plane tangent to the Keplerian motion of the ring particles. These occultations measured ultraviolet light in the wavelength range 110–190 nm with 1- or 2-ms ms integration times, providing optical depth measurements with resolution comparable to the Fresnel scale of ∼10 m in the frame co-moving with the local Keplerian speed. These high-resolution chord occultations reach their minimum ring radii over a broad range of radial locations within Saturn’s main rings and resolve non-axisymmetric structures in a wide variety of ring regions. We measure the azimuthal length scales and optical depth profiles of “ghosts”, or azimuthally limited gaps with radial scales less than 30 m (Baillié et al., 2013), in the C ring plateaus. We measure the length scale of transparent gaps and self-gravity wakes in the peaks and troughs of spiral density waves. We present optical depth profiles which resolve axisymmetric waves in the A and B rings and constrain the azimuthal length scale over which the waves remain axisymmetric to ∼1000 km. We combine autocorrelation length scales from stellar occultations which cut across the minimum ring radius of a chord occultation with line-of-sight trajectories from other occultations which are not tangent to the rings to constrain the morphology of the azimuthally and temporally averaged mesoscale structures at that ring radius. 2D autocorrelation profiles from stellar occultations show self-gravity wakes and axisymmetric waves often coexisting.

Research on the efficient EM modeling method from multilayered anisotropic medium-metal composite targets

The paper presents a high-frequency modeling approach tailored for the electromagnetic (EM) scattering characteristics from electrically large radar targets coated with multi-layered anisotropic mediums (MLAMs). The approach begins by deriving the plane-wave spectrum expression for the incident EM field within MLAMs. It then employs spectral domain full-wave analysis method (SDFWAM) to obtain an analytical representation of the scattering field, further leveraging saddle point evaluation (SPE) to derive asymptotic solutions in the spatial domain. By integrating principles of physical optics (PO) and the tangent plane approximation, the far-field scattering characteristics of target enveloped in the specified medium are efficiently delineated. Validations against standard structure and the Misty satellite model reveal the method's pronounced alignment with the method of moments – finite element method (MoM-FEM) hybrid numerical algorithm, underscoring its notable computational efficiency. Furthermore, in conjunction with scattering sources decomposition technique, the approach is applied to optimize the radar cross-section (RCS) of the Su-57 aircraft, achieving precise and intelligent RCS control at minimal material cost. In conclusion, this research offers pivotal technological and theoretical foundations for EM scattering prediction, stealth design, and performance assessment in radar target domains.

Curvature-Invariant Adversarial Attacks for 3D Point Clouds

Imperceptibility is one of the crucial requirements for adversarial examples. Previous adversarial attacks on 3D point cloud recognition suffer from noticeable outliers, resulting in low imperceptibility. We think that the drawbacks can be alleviated via taking the local curvature of the point cloud into consideration. Existing approaches introduce the local geometry distance into the attack objective function. However, their definition of the local geometry distance neglects different perceptibility of distortions along different directions. In this paper, we aim to enhance the imperceptibility of adversarial attacks on 3D point cloud recognition by better preserving the local curvature of the original 3D point clouds. To this end, we propose the Curvature-Invariant Method (CIM), which directly regularizes the back-propagated gradient during the generation of adversarial point clouds based on two assumptions. Specifically, we first decompose the back-propagated gradients into the tangent plane and the normal direction. Then we directly reduce the gradient along the large curvature direction on the tangent plane and only keep the gradient along the negative normal direction. Comprehensive experimental comparisons confirm the superiority of our approach. Notably, our strategy can achieve 7.2% and 14.5% improvements in Hausdorff distance and Gaussian curvature measurements of the imperceptibility.

A Symmetric Non-Stationary Loop Subdivision with Applications in Initial Point Interpolation

Loop subdivision is a significant surface scheme with wide applications in fields like computer graphics and wavelet. As a type of stationary scheme, Loop subdivision cannot adjust the limit surface directly. In this paper, we present a new way to solve this problem by proposing a symmetric non-stationary Loop subdivision based on a suitable iteration. This new scheme can be used to adjust the limit surfaces freely and thus can generate surfaces with different shapes. For this new scheme, we show that it is C2 convergent in the regular part of mesh and is at least tangent plane continuous at the limit positions of the extraordinary points. Additionally, we present a non-uniform generalization of this new symmetric non-stationary subdivision so as to locally control the shape of the limit surfaces. More interestingly, we present the limit positions of the initial points, both for the symmetric non-stationary Loop subdivision and its non-uniform generalization. Such limit positions can be used to interpolate the initial points with different valences, generalizing the existing result. Several numerical examples are given to illustrate the performance of the new schemes.

Automatic labeling of 3D facial acupoint landmarks

<p>As special marks on a human face, facial landmarks reflect the facial features of various parts of the face, which is crucial in biomedicine and medical imaging. In addition, facial landmarks are also important features in computer vision such as face detection, face recognition, facial pose estimation, and facial animation. In this paper, we construct a 3D facial acupoint annotated dataset by labeling 37 facial acupoints on 846 neutral face triangle mesh on the FaceScape dataset. Based on these annotated data, we use a feature template matching method to realize the automatic annotation of 37 acupoints on triangle meshes. We used 40 meshes as the training set to extract the geometric patterns of 3D acupoints and then measured the performance of the automatic labeling algorithm on 20 meshes and 806 meshes as the test sets. In the training process, we extract the tangent plane for each landmark, project the neighbor vertices of the landmark to the tangent plane, and construct the feature image with <em>R</em> × <em>R</em> resolution through the bounding box of the projected points. In the testing process, we use the pattern images extracted during training to find the average features and use them as a guide to optimize the predicted landmarks. The experimental results show that our automatic acupoint labeling method has achieved good results.<strong></strong></p>

An advanced light scattering imaging model for total internal reflection microscopy considering a stratified medium

An advanced light scattering model for Total Internal Reflection Microscopy (TIRM) is presented. The model considers the specific TIRM geometry and deals with the scattering by an axisymmetric particle of arbitrary orientation placed in a stratified medium and the imaging of the scattered field. The scattered field is computed by truncating the scattered and internal field expansions and by using spherical and plane wave expansions for the free-space dyadic Green’s function. While the first expansion is valid outside a sphere enclosing the particle, the second one is valid outside the tangent planes bounding the particle from above and below. We demonstrate that in both cases, the results are the same, and thus, that the restrictive condition according to which the interface should not intersect the particle’s circumscribed sphere is not relevant. The image of the scattered field is computed by using the Debye diffraction integral and fast Fourier transform, while for a better reconstruction of the particle orientation, an image processing step consisting in a contour extraction and ellipse fitting is considered. The numerical simulations dealing with scattering by a prolate spheroid provide evidence of the remarkably sensitivity of the geometric parameters of the image ellipse to the particle orientation angles, as well as, of the integral response of the detector to the distance between the particle and the interface.

Timelike surfaces in [formula omitted] with a canonical null direction

Let N be a Lorentz surface and let ∂t be a unit vector field on the Lorentz manifold N×R tangent to R. A timelike surface Σ in N×R is said to have a canonical null direction with respect to ∂t if the projection ∂t⊤ on the tangent space of Σ gives a lightlike vector field. We prove that these surfaces are minimal and ruled, whose rulings are lightlike geodesics and lines of curvature. Other results include the fact that these surfaces are flat if they are totally geodesic. Furthermore, let us assume that the surface Σ is the graph of a function f:N→R. Then it has a canonical null direction if and only if the gradient of f is a lightlike vector field on N. Another property is that the Gaussian curvature of Σ coincides with the Gaussian curvature of N and the sectional curvature of N×R, along tangent planes to Σ. In particular such surfaces in S12×R have constant curvature. We prove, using isothermal coordinates, that functions on N whose gradient is lightlike locally always exist. We give several examples of such surfaces.