Geometric Correlation Research Articles

Entropy-based geometric measures of correlations for bell diagonal states via measurement

Abstract The classification and quantification of quantum correlations are the important issues in quantum information theory. Quantum relative entropy and Rényi relative entropy are both non-negative, and can therefore be regarded as generalized distances between quantum states. We first introduced the method proposed by Brodutch and Modi for constructing measurements of classical and quantum correlations. Subsequently, we use this method to construct measurement-induced geometric classical and quantum correlations based on quantum relative entropy and Rényi− 1 2 relative entropy. For two-qubit Bell diagonal states, the analytic expressions for these geometric classical correlations and quantum correlations are obtained.

An inter-comparison between calculated solid angle for different geometrical correlations of source-detector through analytical and Monte Carlo approaches

An inter-comparison between calculated solid angle for different geometrical correlations of source-detector through analytical and Monte Carlo approaches

Geometric and dosimetric evaluation for breast and regional nodal auto-segmentation structures.

The accuracy of artificial intelligence (AI) generated contours for intact-breast and post-mastectomy radiotherapy plans was evaluated. Geometric and dosimetric comparisons were performed between auto-contours (ACs) and manual-contours (MCs) produced by physicians for target structures. Breast and regional nodal structures were manually delineated on 66 breast cancer patients. ACs were retrospectively generated. The characteristics of the breast/post-mastectomy chestwall (CW) and regional nodal structures (axillary [AxN], supraclavicular [SC], internal mammary [IM]) were geometrically evaluated by Dice similarity coefficient (DSC), mean surface distance, and Hausdorff Distance. The structures were also evaluated dosimetrically by superimposing the MC clinically delivered plans onto the ACs to assess the impact of utilizing ACs with target dose (Vx%) evaluation. Positive geometric correlations between volume and DSC for intact-breast, AxN, and CW were observed. Little or anti correlations between volume and DSC for IM and SC were shown. For intact-breast plans, insignificant dosimetric differences between ACs and MCs were observed for AxNV95% (p=0.17) and SCV95% (p=0.16), while IMNV90% ACs and MCs were significantly different. The average V95% for intact-breast MCs (98.4%) and ACs (97.1%) were comparable but statistically different (p=0.02). For post-mastectomy plans, AxNV95% (p=0.35) and SCV95% (p=0.08) were consistent between ACs and MCs, while IMNV90% was significantly different. Additionally, 94.1% of AC-breasts met ΔV95% variation<5% when DSC>0.7. However, only 62.5% AC-CWs achieved the same metrics, despite AC-CWV95% (p=0.43) being statistically insignificant. The AC intact-breast structure was dosimetrically similar to MCs. The AC AxN and SC may require manual adjustments. Careful review should be performed for AC post-mastectomy CW and IMN before treatment planning. The findings of this study may guide the clinical decision-making process for the utilization of AI-driven ACs for intact-breast and post-mastectomy plans. Before clinical implementation of this auto-segmentation software, an in-depth assessment of agreement with each local facilities MCs is needed.

A generalized geometric mechanics theory for multi-curve-fold origami: Vertex constrained universal configurations

Folding paper along curves leads to spatial structures that have curved surfaces meeting at spatial creases, defined as curve-fold origami. In this work, we provide an Eulerian framework focusing on the mechanics of arbitrary curve-fold origami, especially for multi-curve-fold origami with vertices. We start with single-curve-fold origami that has wide panels. Wide panel leads to different domains of mechanical responses induced by various generator distributions of the curved surface. The theories are then extended to multi-curve-fold origami, involving additional geometric correlations between creases. As an illustrative example, the deformation and equilibrium configuration of origami with annular creases are studied both theoretically and numerically. Afterward, single-vertex curved origami theory is studied as a special type of multi-curve-fold origami. We find that the extra periodicity at the vertex strongly constrains the configuration space, leading to a region near the vertex that has a striking universal equilibrium configuration regardless of the mechanical properties. Both theories and numerics confirm the existence of the universality in the near-field region. In addition, the far-field deformation is obtained via energy minimization and validated by finite element analysis. Our generalized multi-curve-fold origami theory, including the vertex-contained universality, is anticipated to provide a new understanding and framework for the shape programming of the curve-fold origami system.

GeoExplainer: Interpreting Graph Convolutional Networks with geometric masking

For Graph Convolutional Network (GCN), the inherent geometric topology and relationships within spatio-temporal graphs are critical to factors affect its predictive performance. Understanding the decision-making process of GCN remains a challenging task. Existing perturbation-based explanation methods are primarily tailored for Graph Neural Network (GNN) models and focus on how the information of nodes or subgraphs in static graphs impacts GNN predictions. Due to the geometric properties of spatio-temporal graphs, screw theory serves as crucial mathematical tools for describing the geometric relationships and patterns of motion within the geometric space of spatio-temporal graphs. Therefore,we propose GeoExplainer, which utilizes screw theory to geometric masking for spatio-temporal graphs with strong geometric correlations, generating explanations for GCN. Our method perturbs spatio-temporal graphs using geometric masking, and ultimately generating explanations in the form of geometric masks serve as importance mappings for model prediction results. Firstly, we design a screw motion vector model with geometric constraints to describe the spatio-temporal graph in screw system. Then, we design a geometric screw perturbation to generate smooth geometric masks. Finally, we propose a geometric masking approach to optimize and update the geometric mixture mask. We utilize pointwise mutual information (PMI) to maximize the correlation between the model predictions after geometric perturbation and the original predictions, obtaining the optimal geometric mask as an explanation for GCN. The interpretable experiments of ST-GCN, HD-GCN, GCN and GIN models on four datasets with geometric relationships demonstrate that GeoExplainer outperforms eight competitive interpretation methods in quantitative and visualization analysis.

DMRC-Net: Dual mechanism residual connection network for point cloud analysis

The purpose of point cloud analysis is to extract effective high-dimensional feature representations from point coordinates. However, existing methods only rely solely on global or local features. In this paper, we introduce the Dual Mechanism Residual Connection Network (DMRC-Net), a new end-to-end network architecture for more comprehensive representations. First, local features are encoded using graph convolution and fine-grained attention mechanisms, whereas global features are encoded using a multi-layer perceptron. Second, adaptive residual learning is designed to integrate two encoding results effectively. Furthermore, the structural information in local space is strengthened by collecting low-level geometric correlations. With these designs, DMRC-Net is able to capture both global and local details comprehensively. Experimental results show that our method performs well in point cloud classification, part segmentation, and semantic segmentation tasks.

Dosimetrical and geometrical parameters in single-fraction lattice radiotherapy for the treatment of bulky tumors: Insights from initial clinical experience

PurposeThis study aims to investigate lattice radiotherapy (LRT) for bulky tumor in 10 patients, analyzing geometrical and dosimetrical parameters and correlations among variables. MethodsPatients were prescribed a single-fraction of 18 Gy to 50 % of each spherical vertex (1.5 cm diameter). Vertices were arranged in equidistant planes forming a triangular pattern. Center-to-center distance (Dc-c) between vertices was varied from 4 to 5 cm. A new method for calculating the valley-to-peak dose ratio (VPDR) was proposed and compared to other two from existing literature. GTV volumes (VGTV), vertex number (Nvert), low-dose related parameters and vertex D99%, D50%, and D1% were recorded. Beam-on time and Monitor Units (MU) were also evaluated. Correlations were assessed using Spearman’s coefficient, with significant differences analyzed using Mann-Whitney U test. ResultsTumor volumes ranged from 417 to 3615 cm3. Median vertex number was 14.5 (IQR:11.3–17.8). VPDR ranged from 0.16 to 0.28. Median D99% spanned from 10.0 to 13.7 Gy, median D50% exceeded 18.0 Gy, and median D1% surpassed 23.3 Gy. Periphery dose remained under 4.0 Gy. Plans exhibited high modulation, with median beam-on time and MU of 8.8 min (IQR:8.2–10.1) and 13,069 MU (IQR:11574–13639). Significant correlations were found between Nvert and VGTV (p < 0.01), MU (p < 0.02) and beam-on time (p < 0.01) and between Dc-c and two VPDR definitions (p < 0.02) and periphery dose (p < 0.01). Significant differences were observed among the three valley dose definitions (p < 0.01) and the three peak dose definitions (p < 0.01). ConclusionsReporting geometrical and dosimetrical parameters in LRT is crucial, alongside the need for unified definitions of valley and peak doses.

Self-Supervised Monocular Depth Estimation via Binocular Geometric Correlation Learning

Monocular depth estimation aims to infer a depth map from a single image. Although supervised learning-based methods have achieved remarkable performance, they generally rely on a large amount of labor-intensively annotated data. Self-supervised methods, on the other hand, do not require any annotation of ground-truth depth and have recently attracted increasing attention. In this work, we propose a self-supervised monocular depth estimation network via binocular geometric correlation learning. Specifically, considering the inter-view geometric correlation, a binocular cue prediction module is presented to generate the auxiliary vision cue for the self-supervised learning of monocular depth estimation. Then, to deal with the occlusion in depth estimation, an occlusion interference attenuated constraint is developed to guide the supervision of the network by inferring the occlusion region and producing paired occlusion masks. Experimental results on two popular benchmark datasets have demonstrated that the proposed network obtains competitive results compared to state-of-the-art self-supervised methods and achieves comparable results to some popular supervised methods.

Learning Effective Geometry Representation from Videos for Self-Supervised Monocular Depth Estimation

Recent studies on self-supervised monocular depth estimation have achieved promising results, which are mainly based on the joint optimization of depth and pose estimation via high-level photometric loss. However, how to learn the latent and beneficial task-specific geometry representation from videos is still far from being explored. To tackle this issue, we propose two novel schemes to learn more effective representation from monocular videos: (i) an Inter-task Attention Model (IAM) to learn the geometric correlation representation between the depth and pose learning networks to make structure and motion information mutually beneficial; (ii) a Spatial-Temporal Memory Module (STMM) to exploit long-range geometric context representation among consecutive frames both spatially and temporally. Systematic ablation studies are conducted to demonstrate the effectiveness of each component. Evaluations on KITTI show that our method outperforms current state-of-the-art techniques.

Phase curve wavelength dependency as revealed by shape- and geometry- corrected asteroid phase curves

ABSTRACT We investigate the photometric properties of 35 well-observed asteroids using dense ground-based and sparse ATLAS survey data. Focusing on two-colour photometric phase functions, derived using inverse methods, we explore the wavelength dependence. Our study reveals distinct (G1, G2) domains for cyan and orange filters, especially among some S-complex asteroids. For other asteroids, substantial uncertainties prevail, or their distributions of phase curve parameters overlap, precluding definitive conclusions on wavelength dependence. Notably, for S-complex objects, the effect appears systematic, characterized by lower G2 values in the cyan filter. The effect can be explained by considering the known geometric albedo correlation: higher albedo corresponds to flatter, whereas lower albedo corresponds to steeper phase curves. In the case of equal albedo, asteroids with red spectral slopes have a more pronounced opposition effect in red and asteroids with blue spectral slopes in blue filters. We explore the variation of slope differences of orange and cyan phase curves with phase angle. For most asteroids, the largest nominal variation is observed at phase angles &amp;lt; 10°. This suggests that the phase colouring for the orange–cyan colour slope is more pronounced at small phase angles. Through meticulous analysis of the opposition effect amplitudes, we also pinpoint preferred rotational pole solutions. We identify inconsistencies between phase curve parameters and spectral types in specific cases. The shape- and geometry-corrected phase curves signify an important advancement in studying asteroid photometric behaviour and may offer a deeper understanding of surface and regolith properties previously obscured by these effects, such as surface roughness.

A differential-based method for correcting ultra-short baseline positioning results

The primary factors influencing the positional accuracy of an ultra-short baseline (USBL) system are errors in sound speed, delay, installation deviation, and the bending of sound rays. Premised on the USBL positioning model, an error contribution model including ranging error, delay error, sound speed error and array element installation deviation was constructed. In parallel, an indirect difference correction method with additional coefficients was proposed based on the geometric correlation of the error function model. Specifically, a reference beacon was deployed on the seabed, and differential correction was achieved by directly computing target coordinates, integrating them with the positional data from the reference beacon. The proposed method demonstrated efficacy in mitigating errors associated with gradual variations or pronounced geometric correlations, including discrepancies in sound speed, bending of sound rays, and deviations in the installation of array elements. The correction method was applied to the USBL positioning, and the simulation results show that the proposed method notably enhanced both accuracy and consistency in positioning, particularly in shallow water environments characterised by significant sound speed errors. The sea trial data from the R/V Dayang Yihao in the Western Pacific demonstrate that the proposed method can effectively mitigate the errors induced by sound ray bending in deep-sea environments. The mean absolute error (MAE) and standard deviation (STD) were reduced by 51.3% and 55.3%, respectively, compared with the results of the USBL direct solution.

DepthGAN: GAN-based depth generation from semantic layouts

AbstractExisting GAN-based generative methods are typically used for semantic image synthesis. We pose the question of whether GAN-based architectures can generate plausible depth maps and find that existing methods have difficulty in generating depth maps which reasonably represent 3D scene structure due to the lack of global geometric correlations. Thus, we propose DepthGAN, a novel method of generating a depth map using a semantic layout as input to aid construction, and manipulation of well-structured 3D scene point clouds. Specifically, we first build a feature generation model with a cascade of semantically-aware transformer blocks to obtain depth features with global structural information. For our semantically aware transformer block, we propose a mixed attention module and a semantically aware layer normalization module to better exploit semantic consistency for depth features generation. Moreover, we present a novel semantically weighted depth synthesis module, which generates adaptive depth intervals for the current scene. We generate the final depth map by using a weighted combination of semantically aware depth weights for different depth ranges. In this manner, we obtain a more accurate depth map. Extensive experiments on indoor and outdoor datasets demonstrate that DepthGAN achieves superior results both quantitatively and visually for the depth generation task.

Probing the magnetic field and dust grain properties of two dark clouds L1495 and L1498 through photopolarimetry

ABSTRACT We present the R-band polarimetry of two low-latitude Lynd’s clouds L1495 and L1498 acquired with AIMPOL at Aryabhatta Research Institute of Observational Sciences, Nainital, India. The estimated polarization across the regions of L1495 and L1498 is typically in the range of ∼1–5 per cent and the position angle is typically in the range of ∼100°–137° from which we infer the morphologies of the corresponding magnetic field in the envelope region ($\theta_{\rm B}^{\rm env}$). In both clouds, the magnetic fields are found to be aligned with the galactic field (θGP). Due to their close proximity to the galactic plane, we incorporate the values of offset ($\theta _{\rm off} = |\theta _{\rm B}^{\rm env} - \theta _{\rm GP}|$) from this study into our previously established geometrical correlation between the variation in relative orientation with the galactic longitude. We find that the offsets estimated for two clouds L1495 and L1498 in this investigation are consistent with the offsets predicted by our previous study. We also generate the extinction map for the two clouds using the NICE technique, and the extinction structure is found to be analogous to the dust structure at the denser region of the cloud as obtained from the 500 µm dust continuum emission maps of Herschel1 SPIRE. Three distinct cores with high extinction are detected in L1495, and two such cores are also detected in L1498. We also estimate the mean particle density ($\lt n_{\rm H_2}\gt $) at the peripheral region as well as the high extinction cores of the clouds.

Geometric versus Entropic Gaussian Correlations in an Open Quantum System of Two Bosonic Modes

Different types of geometric and entropic quantum correlation quantifiers are studied for a system composed of two resonant bosonic modes embedded in a thermal bath. The description of the evolution of the correlation measures is formulated in the framework of the theory of open systems, based on completely positive quantum dynamical semigroups, by using both a geometric and entropic quantification of total nonclassical correlations of Gaussian states. We consider the special case when the initial squeezed thermal state of the system preserves its form in time. We show that time evolution of the measures strongly depends on the parameters characterising the initial state of the system (squeezing parameter and average thermal photon numbers of the two modes) and of the thermal environment (temperature of the thermal bath and dissipation rate). In the limit of large times all the considered measures asymptotically tend to zero value, corresponding to an asymptotic bimodal uncorrelated product state. We make a comparison between the behaviour of the evolution in time of the Gaussian geometric quantum correlations and Gaussian entropic quantum correlations.

Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions.

Time-dependent reliability-based design optimization of main shaft bearings in wind turbines involving mixed-integer variables

Main shaft bearings are essential transmission components in wind turbines. The failure of the main shaft bearings is inevitably accompanied by catastrophic consequences. In this regard, this paper presents a time-dependent reliability-based design optimization (TRBDO) approach for main shaft bearings involving mixed-integer variables. The maximization of fatigue life and the elastohydrodynamic film thickness under the load spectrum are selected as optimization objectives. The time-dependent reliability and geometric correlation are determined as nonlinear constraints. An efficient two-stage enrichment strategy is introduced to handle the time-dependent probabilistic constraint with mixed-integer design variables. A mixed-integer nonlinear optimization method is developed based on a meta-heuristic algorithm to solve the formulated TRBDO problem. Eventually, the effectiveness and robustness of the proposed approach are demonstrated by a real application in a 5 MW wind turbine.

Reliability-based design optimization of the spiral water jacket for motorized spindle

Spiral water jackets are essential refrigeration units in motorized spindles. The design of spiral water jackets directly affects the machining accuracy and reliability of motorized spindles. In this regard, a reliability-based design optimization (RBDO) method is presented to obtain the optimal configuration of the spiral water jacket. The proposed RBDO algorithm aims to minimize the volume of the spiral water jacket and power loss of the cooling equipment while ensuring machining reliability. The geometric correlation and machining reliability are selected as constraints. The Kriging surrogate model is established to approximate the time-consuming finite element model for improving computational efficiency. An efficient metaheuristic algorithm is introduced to perform optimal exploration for the proposed RBDO problem. The risk speed of the motorized spindle is investigated by the experiment and finite element analysis. Subsequently, the effects of various cooling configurations on thermal error are investigated thoroughly. Eventually, the proposed RBDO is conducted on a motorized spindle. Through probabilistic evaluation, machining reliability is guaranteed by the proposed RBDO compared to deterministic design optimization. This paper provides a new design paradigm for cooling configurations in motorized spindles from the perspective of machining reliability.

Multi-level correlation learning for multi-view unsupervised feature selection

Recently, unsupervised multi-view feature selection has received a lot of interest. However, current methods facilitate feature selection by preserving the local consistency of multi-view data, which is defined based on similarities of data within each view, but ignore the global topological consistency in data, which is defined based on the cross-view topological similarities of data between views and is essential for revealing the distribution of multi-view data. In light of this, this paper proposes a novel multi-view unsupervised feature selection method with multi-level correlation learning, termed Multi-Level Correlation Learning for Multi-View Unsupervised Feature Selection (MLCL). It simultaneously derives the global topological correlation structure from the cross-view topological similarities of data and the local geometric correlation structure from the local similarities of data within each view, to take advantage of both global and local consistencies of multi-view data. An effective optimization algorithm is then developed to resolve the optimization problem for the proposed model. Extensive experiments on eight publicly available datasets show that the proposed MLCL outperforms several state-of-the-art unsupervised multi-view feature selection models.

BF3D: Bi-directional fusion 3D detector with semantic sampling and geometric mapping

3D object detection is a key task in environmental awareness, which plays a vital role in autonomous driving safety. Lidars and cameras are widely used sensors that provide complementary information for accurate 3D detection. However, due to the domain difference between the two modalities, it is challenging to leverage their respective strengths and fuse them perfectly. In this paper, an end-to-end detector termed BF3D is proposed, which integrates with the semantic sampling module, geometric point-image mapping module, and bi-directional attention fusion module. Specifically, the semantic sampling module incorporates a novel downsampling strategy to preserve more foreground points and pixels. Additionally, the geometric point-image mapping module is developed to find geometric correlation pixels of the point and take advantage of the high density of image features. We also introduce a bi-directional attention fusion module to combine useful information from the two modalities by attention mechanism. Extensive experiments demonstrate that BF3D outperforms both single- and multi-modal 3D detectors. Codes are available at: https://github.com/hustzyj/BF3D.

Anisotropic flow and the valence quark skeleton of hadrons

We study transverse momentum anisotropies, in particular, the elliptic flow v2 due to the interference effect sourced by valence quarks in high-energy hadron-hadron collisions. Our main formula is derived as the high-energy (eikonal) limit of the impact-parameter dependent cross section in quantum field theory, which agrees with that in terms of the impact parameter in the classical picture. As a quantitative assessment of the interference effect, we calculate v2 in the azimuthal distribution of gluons at a comprehensive coverage of the impact parameter and the transverse momentum in high-energy pion-pion collisions. In a broad range of the impact parameter, a sizable amount of v2, comparable with that produced due to saturated dense gluons or final-state interactions, is found to develop. This is in contrast with similar studies in heavy-ion collisions using classical color charge distributions in which such a contribution from geometric correlations was found to be small and has, hence, been ignored in recent studies. In our calculations, the valence sector of the pion wave function is obtained numerically from the Basis Light-Front Quantization, a non-perturbative light-front Hamiltonian approach. And our formalism is generic and can be applied to other small collision systems like proton-proton collisions.