Local Geometrical Relationship Research Articles

Tangent lines that neither touch nor cross the curves at the points of tangency

Several studies on student's understanding of the concept of the tangent line agree that the difficulties students have when dealing with lines tangent to curves at inflection points or tangent lines that have more than one point in common with the curve are due to a misconception of tangency that is limited to the geometric relationship between line and circle. Here, we report difficulties Mexican high school students had when working on tangent lines in situations like those mentioned before. For our research, we designed Activities mostly carried out with dynamic geometry. Using the zoom-in provided by the software helped strengthen the student's mental image of the local geometric relationship between a curve and a line tangent. Several of the curves we designed for the investigation surprised the students when they zoomed in on the point of tangency. From the point of view of the tangency relation that is intuitively obtained from tangents to the circle or curves at inflection points, in several curves that we presented, the tangent lines neither touch nor cross the curves at the points of tangency. These are cases of curves and tangents that would be worth analyzing in Differential Calculus courses.

Semi-supervised enhanced discriminative local constraint preserving projection for dimensionality reduction of medical hyperspectral images

Microscopic hyperspectral images has the advantage of containing rich spatial and spectral information. However, the large number of spectral bands provides a significant amount of spectral features, but also leads to data redundancy and noise, which seriously affect the recognition and classification performance of the images, as well as increasing the requirements for computation and storage. To address this issue, we propose a dimensionality reduction algorithm named enhanced discriminant local constraint preserving projection (EDLCPP). Specifically, the global spectral attention mechanism focuses on important bands, the high discriminability sample selection module measures the discriminability of samples using a modified average neighborhood margin, the graph construction module preserves the local geometric relationship and discriminant information, and the graph embedding module embeds the constructed graphs into a low-dimensional space to obtain the projection matrices. Experimental results on eight cholangiocarcinoma (CCA) hyperspectral images, Bloodcell1-3, and Bloodcell2-2 datasets have demonstrated the effectiveness of the proposed method.

GN-CNN: A Point Cloud Analysis Method for Metaverse Applications

Metaverse applications often require many new 3D point cloud models that are unlabeled and that have never been seen before; this limited information results in difficulties for data-driven model analyses. In this paper, we propose a novel data-driven 3D point cloud analysis network GN-CNN that is suitable for such scenarios. We tackle the difficulties with a few-shot learning (FSL) approach by proposing an unsupervised generative adversarial network GN-GAN to generate prior knowledge and perform warm start pre-training for GN-CNN. Furthermore, the 3D models in the Metaverse are mostly acquired with a focus on the models’ visual appearances instead of the exact positions. Thus, conceptually, we also propose to augment the information by unleashing and incorporating local variance information, which conveys the appearance of the model. This is realized by introducing a graph convolution-enhanced combined multilayer perceptron operation (CMLP), namely GCMLP, to capture the local geometric relationship as well as a local normal-aware GeoConv, namely GNConv. The GN-GAN adopts an encoder–decoder structure and the GCMLP is used as the core operation of the encoder. It can perform the reconstruction task. The GNConv is used as the convolution-like operation in GN-CNN. The classification performance of GN-CNN is evaluated on ModelNet10 with an overall accuracy of 95.9%. Its few-shot learning performance is evaluated on ModelNet40, when the training set size is reduced to 30%, the overall classification accuracy can reach 91.8%, which is 2.5% higher than Geo-CNN. Experiments show that the proposed method could improve the accuracy in 3D point cloud classification tasks and under few-shot learning scenarios, compared with existing methods such as PointNet, PointNet++, DGCNN, and Geo-CNN, making it a beneficial method for Metaverse applications.

Deep locally linear embedding network

Recent years have seen great progress in many deep learning-based super-resolution methods that learn global responses through the non-local strategy. However, improper linear fitting tends to produce unsatisfactory results with notable artifacts. To relieve the improper linear fitting and reuse the hierarchical features, we propose an end-to-end locally linear embedding super-resolution network, termed as LLE-Net, based on the assumption that the local geometric relationship in the low-resolution manifold space also exists in high-level feature manifold space. The proposed LLE-Net includes two major components: 1) the local linear embedding block (LLEB) and 2) the hierarchical non-local block (HNB). In particular, LLEB searches sparse and similar feature maps and embeds the geometric relationship into the high-level feature space, allowing more attention to be paid to objects’ textural details. To promote the representation ability, the proposed HNB is able to explore layer- and pixel-level interdependencies. We conduct extensive experiments to evaluate the superiority of LLE-Net via two groups of experiments: (1) super-resolution tasks on two satellite image datasets and a satellite video image dataset, and (2) the subsequent high-level image processing tasks (i.e., satellite semantic segmentation). The proposed method achieves an 0.7 dB improvement in PSNR value on the Draper dataset and 0.07% segmentation accuracy improvement. The source code of the proposed LLE-Net is publicly available.

Pan-Sharpening via Deep Locally Linear Embedding Residual Network

The goal of pan-sharpening tasks is to fuse panchromatic (PAN) images and low-spatial-resolution (LR) multispectral (MS) images for the purpose of aggregating texture and spectral information. Although traditional embedding-based pan-sharpening methods achieve competitive results, they are limited by the shallow network and not suitable for large-scale datasets. In this study, we design a novel multiscale locally linear embedding residual network (LLERN) that consists of two phases: the spectral preservation phase and the structural preservation phase. As the pretreatment of the structural preservation network, the spectral preservation network aims to upscale the LR MS image while retaining spectral information. The proposed locally linear embedding residual block (LLERB) in the structural preservation phase can search for similar sparse patches from the PAN image space and embed the corresponding local geometric relationship into the residual space to enhance the MS image. Extensive experiments suggest that the proposed LLERN outperforms state-of-the-art methods from visual and quantitative perspectives, and confirm the assumption that LR image patches and residual image patches in a local region share a similar manifold structure, which can be used to guide deep-learning modeling with improved interpretability. The source code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/jiaming-wang/LLERN</uri> .

Geometry Embedded Multi-Task Network for Lane Detection and Classification

Lane detection and classification are key and fundamental problems in the field of autonomous driving. Deep learning based multi-task network is commonly built with convolutional neural networks (CNNs) to handle these problems. Although existing methods have demonstrated the capability to capture the pixel-level representation and global guidance, the local geometric relationship of lanes are not taken into account to deal with the lanes with weak appearance coherence. This paper proposes geometry embedded multi-task network (GEMNet) with three branch layers, including bounding box regression, lane mask and lane classification. Convolutional operations are applied to feature maps generated by the top hidden layer of bounding box regression branch to explore the semantic context via geometric relationship. Moreover, it is able to facilitate the weight optimization in the shared layers and other branch layers simultaneously. GEMNet is evaluated on Cordova1 and Washington1 datasets, and Experimental results show that the proposed method outperforms the state-of-the art approaches. F1-Score can reach 0.906 and 0.881 with respect to Cordova1 and Washington1 datasets, respectively.

A numerical model to investigate contact status for rail grinding by abrasive belt with an axial deflection

In order to calculate analytically the contact status for rail grinding by abrasive belt with an axis deflection, as well as considering limitations of using Hertz contact theory for that, a numerical contact model (NCM) based on the local geometrical relationship between contact wheel and rail surface was developed. The effectiveness and accuracy of the proposed model have been validated by the corresponding finite element model (FEM) by comparing the contact boundary and contact stress distribution. Comparative cases at two typical grinding positions with different attitudes were involved, and the effects of applying axis deflection on contact behavior were investigated in detail. The results from NCM and FEM both agreed belt grinding with an axis deflection was able to enlarge the contact width of the belt and the total grinding force under the premise of ensuring required width of grinding trace on the rail. The model provides insights into understanding contact behavior under actual working conditions and gives a theoretical basis for future researches based on contact analysis.

Resolution-Free Accurate DNA Contour Length Estimation from Atomic Force Microscopy Images.

This research presented an accurate and efficient contour length estimation method developed for DNA digital curves acquired from Atomic Force Microscopy (AFM) images. This automation method is calibrated against different AFM resolutions and ideal to be extended to all different kinds of biopolymer samples, encompassing all different sample stiffnesses. The methodology considers the digital curve local geometric relationship, as these digital shape segments and pixel connections represent the actual morphology of the biopolymer sample as it is being imaged from the AFM scanning. In order to incorporate the true local geometry relationship that is embedded in the continuous form of the original sample, one needs to find this geometry counterpart in the digitized image. This counterpart is realized by taking the skeleton backbone of the sample contour and by using these digitized pixels' connection relationship to find its local shape representation. In this research, one uses the 8-connect Freeman Chain Code (CC) to describe the directional connection between DNA image pixels, in order to account for the local shapes of four connected pixels. The result is a novel shape number (SN) system derived from CC, which is a fully automated algorithm that can be applied to DNA samples of any length for accurate estimation, with efficient computational cost. This shape-wise consideration is weighted to modify the local length with great precision, accounting for all the different morphologies of the biopolymer sample, and resulted with accurate length estimation, as the error falls below 0.07%, an order of magnitude improvement compared to previous findings.

Robust registration for remote sensing images by combining and localizing feature- and area-based methods

Highly accurate registration is one of the essential requirements for numerous applications of remote sensing images. Toward this end, we have developed a robust algorithm by combining and localizing feature- and area-based methods. A block-weighted projective (BWP) transformation model is first employed to map the local geometric relationship with weighted feature points in the feature-based stage, for which the weight is determined by an inverse distance weighted (IDW) function. Subsequently, the outlier-insensitive (OIS) model aims to further optimize the registration in the area-based stage. Considering the inevitable outliers (e.g., cloud, noise, land-cover change), OIS integrates Huber estimation with the structure tensor (ST), which is an approach that is robust to residual errors and outliers while preserving edges. Four pairs of remote sensing images with varied terrain features were tested in the experiments. Compared with the-state-of-art algorithms, the proposed algorithm is more effective, in terms of both visual quality and quantitative evaluation.

Low-Rank Projection Learning via Graph Embedding

With robustness to various corruptions, it is the local geometrical relationship among data that plays an important role in the recognition/clustering task of subspace learning (SL). However, a lot of previous SL methods cannot take into consideration both of the local neighborhood and the robustness, which results in poor performance in image classification and feature extraction. In this paper, a robust SL method is proposed to solve the feature extraction problem, named as Low-Rank Projection Learning via Graph Embedding (LRP-GE). The proposed algorithm enjoys two merits. First, it preserves the local neighborhood information among data by introducing the graph embedding (GE). Second, it alleviates the impact of noise and corruption by learning a robust subspace based on the low-rank projection. We cast the problem as a convex optimization problem and provide an iterative solution that can be solved efficiently in polynomial time. Extensive experiments performed on four benchmark data sets demonstrate that the proposed method performs favorably against other well-established SL methods in image classification.

Studies on the EPR Parameters for the Rhombic Co2+ Center in Magnesium Acetate

The electron paramagnetic resonance (EPR) g factors gi (i = x,y, z) and the hyperfine structure constants Ai for the rhombic Co2+ center in magnesium acetate are theoretically studied from the perturbation formulas of these parameters for a 3d7 ion under rhombic symmetry. In these formulas, the contributions from the admixture among different states, covalency effect and rhombic crystal-fields are taken into account. The related crystal-field parameters are determined from the superposition model and the local geometrical relationship of the impurity center. The calculated results show reasonable agreement with the observed values. In addition, the nuclear quadrupole interaction constant Q is analyzed, and its negative sign is verified theoretically.

Theoretical Studies On The Gyromagnetic Factors For Nd3+ In Scheelites-Type ABO4 Compounds

The gyromagnetic factors for Nd3+ in scheelite-type ABO4 compounds (A = Cd, Ca, Pb, Ba; B = Mo, W) are theoretically studied by the perturbation formulas of the anisotropic g factors g∥ and g⊥ for a 4f3 ion in tetragonal symmetry. In these formulas, the contributions to the g factors due to the second-order perturbation terms and the admixtures of various energy levels are taken into account. The relevant crystal-field parameters are determined by the superposition model and the local geometrical relationship of the A2+ sites occupied by the impurity Nd3+. The obtained g factors agree reasonably with the observed values. The discrepancies between theory and experiment are discussed.

Theoretical studies of the g factors for the tetragonal and orthorhombic Er3+ sites in ErBa2Cu3O7−δ superconductors

The EPR g factors for the tetragonal and orthorhombic Er3+ sites in ErBa2Cu3O7−δ high-Tc superconductors are theoretically studied by using the perturbation formulas of the g factors for a 4f11 ion in tetragonal and orthorhombic symmetries. In these formulas, the contributions to the g factors from the second-order perturbation terms and the admixture of various states are taken into account. The relevant crystal-field parameters are determined from the local geometrical relationship of the magnetic Er3+ sites and the superposition model (SPM). The theoretical g factors for Er3+ of both symmetries are in agreement with the observed values, by adjusting the related SPM parameters. The lowest Kramers doublets are found to be, respectively, (Γ7+Γ8) and (Γ5+Γ6) for the tetragonal and orthorhombic ErBa2Cu3O7−δ superconductors. The above studies may be helpful to understand the change of the Er3+ anisotropy from Ising- to XY-like as the symmetry varies from orthorhombic to tetragonal.

Theoretical investigations of the local structure and the EPR parameters for the trigonal Co 2+ center in K 4CdCl 6

The local structure and the electron paramagnetic resonance (EPR) g factors g ||, g ⊥ and the hyperfine structure constants A || and A ⊥ for the trigonal Co 2+ center in K 4CdCl 6 are theoretically investigated by using the perturbation formulas of the EPR parameters for a 3d 7 ion in trigonal symmetry based on the cluster approach. In these formulas, the contributions to the EPR parameters arising from the admixture of different energy levels, covalency effect and low symmetry distortion are included. The parameters related to the above effects are obtained from the optical spectral data and the local geometrical relationship of the studied system. By analyzing the EPR spectra, we find that the impurity–ligand bonding angle β (≈53.54°) related to the C 3-axis in the Co 2+ impurity center is smaller than the corresponding metal–ligand bonding angle β H (≈54.90°) in the host K 4CdCl 6. From the difference Δ β between the bonding angle β (or β H) and the corresponding angle β 0 (≈54.74°) in cubic symmetry, the chlorine octahedron changes from compression (Δ β H= β H− β 0>0) in the host to elongation (Δ β= β− β 0<0) in the impurity center. The calculated EPR parameters, especially the anisotropies of the g and A factors for K 4CdCl 6:Co 2+ based on the angle β show good agreement with the observed data. The theoretical studies on the local structure for Co 2+ in K 4CdCl 6 would be helpful to understand the structure properties for the equivalent octahedral transition-metal M site in the isostructural A 3M′MO 6 compounds.