Geometric Approximation Research Articles

A Variational Perspective on Auxetic Metamaterials of Checkerboard-Type

The main result of this work is a homogenization theorem via variational convergence for elastic materials with stiff checkerboard-type heterogeneities under the assumptions of physical growth and non-self-interpenetration. While the obtained energy estimates are rather standard, determining the effective deformation behavior, or in other words, characterizing the weak Sobolev limits of deformation maps whose gradients are locally close to rotations on the stiff components, is the challenging part. To this end, we establish an asymptotic rigidity result, showing that, under suitable scaling assumptions, the attainable macroscopic deformations are affine conformal contractions. This identifies the composite as a mechanical metamaterial with a negative Poisson’s ratio. Our proof strategy is to tackle first an idealized model with full rigidity on the stiff tiles to acquire insight into the mechanics of the model and then transfer the findings and methodology to the model with diverging elastic constants. The latter requires, in particular, a new quantitative geometric rigidity estimate for non-connected squares touching each other at their vertices and a tailored Poincaré type inequality for checkerboard structures.

Efficient Feature Matching for Large-scale Images based on Cascade Hash and Local Geometric Constraint

Abstract. Feature matching plays a crucial role in 3D reconstruction to provide correspondences between overlapped images. The accuracy and efficiency of feature matching significantly impact the performance of 3D reconstruction. The widely used framework with the exhaustive nearest neighbor searching (NNS) between descriptors and RANSAC-based geometric estimation is, however, low-efficient and unreliable for large-scale UAV images. Inspired by indexing-based NNS, this paper implements an efficient feature matching method for large-scale images based on Cascade Hashing and local geometric constraints. Our proposed method improves upon traditional feature matching approaches by introducing a combination of image retrieval, data scheduling, and GPU-accelerated Cascade Hashing. Besides, it utilizes a local geometric constraint to filter matching results within a matching framework. On the one hand, the GPU-accelerated Cascade Hashing technique generates compact and discriminative hash codes based on image features, facilitating the rapid completion of the initial matching process, and significantly reducing the search space and time complexity. On the other hand, after the initial matching is completed, the method employs a local geometric constraint to filter the initial matching results, enhancing the accuracy of the matching results. This forms a three-tier framework based on data scheduling, GPU-accelerated Cascade Hashing, and local geometric constraints. We conducted experiments using two sets of large-scale UAV image data, comparing our method with SIFTGPU to evaluate its performance in initial matching, outlier rejection, and 3D reconstruction. The results demonstrate that our method achieves a feature matching speed 2.0 times that of SIFTGPU while maintaining matching accuracy and producing comparable reconstruction results. This suggests that our method holds promise for efficiently addressing large-scale image matching.

A wave model of high frequency vibrational energy transfer in a wedge-shaped combined plate structure

Wedge shaped combined plate structure (WCPS) is broadly applied in large-sized transport equipment such as aerospace and shipping. However, with the increase of equipment operating running, it is easy to cause high-frequency vibration of structure components. Therefore, studying the energy transfer and vibration response of high-frequency vibration in WCPS is significant for the structural design of aerospace and shipping. The promising method is energy finite element analysis (EFEA). However, the energy transfer coefficient at joint of WCPS is needed in EFEA. The current studies about the energy transfer coefficients are focus on the joint between two uniform thickness plate. But it is some different that getting the energy transfer coefficient for WCPS, because the near-field solve representation for wedge-shaped plate is complicated due to the non-uniform thickness of wedge-shaped plate member in WCPS. In this research, geometric acoustic approximation and an equilibrium equation on both ends of joints were used to derive the equation of the energy transfer coefficient for the WCPS. Then by setting coupling nodes, construction of a coupling matrix to characterize discontinuity of energy density at the coupling position of the WCPS and energy density governing equation of wedge-shaped plates were integrated. In doing so, the EFEA modeling of the WCPS was developed and the energy density distribution on each node of the WCPS could be obtained. Same-sized uniformly coupled structures is analyzed, the vibration energy distribution characteristics of two types of coupling structures were compared. The results show that the energy density value of the wedge plate member attenuate more slowly than that of the uniform plate member due to the acoustic black hole aggregation effect in the wedge plate part, and there is an abrupt change in the energy density value at the joint of WCPS. As a result, the factors influencing high-frequency vibration energy transfer were investigated.

The investigation of retrieval method for aerosol and water vapor profiles based on MAX-DOAS technology

Simultaneously obtaining the vertical distribution and profile of aerosol and water vapor is crucial for in-depth understanding of the Earth's radiation balance, regional water cycle, and the causes of frequent haze weather in China. This study developed a system and inversion method based on multi-axis differential optical absorption spectroscopy (MAX-DOAS) to simultaneously retrieve tropospheric aerosol and water vapor profiles. The absorption of dioxy (O4) was obtained by multi-elevation measurement using the developed ground-based MAX-DOAS system. By minimizing a cost function combining an atmospheric radiative transfer model, aerosol extinction profiles were first inverted, followed by water vapor profiles. Optimized parameter selection in the look-up table method reduced reliance on prior profile information, enhancing pollutant distribution retrieval accuracy. In the monitoring period in Huaibei region, aerosols were mainly below 1.5 km, while water vapor concentrations decreased with altitude. Comparisons of water vapor vertical column densities (VCDs) retrieved through the look-up table method with the European Centre for Medium-Range Weather Forecasts(ECMWF) ERA5 model and geometric approximation showed good agreement, with correlation coefficients of 0.93 and 0.98, respectively. To comprehend water vapor sources at various vertical layers, a 24-h backward trajectory clustering analysis was conducted using the HYSPLIT model based on observed wind fields. Findings revealed that at 500 m altitude, water vapor primarily emanated from the southeast, whereas at 1 km and 2 km altitudes, it predominantly originated from the southwest. The study advances simultaneous tropospheric aerosol and water vapor profile retrieval, providing reliable technical support for the investigation of regional pollution.

Monitoring and estimation of mining and geometric indicators of the deposit

The article presents the results of mining and geometric modeling of indicators of mineral deposits. In order to estimate and monitor them, a wide range of geoinformation systems-based methods and techniques of geometrization is applied. The sequence and stages of mining and geometric estimation of deposits are described considering specific mining and geological conditions. The built models are used at Ukraine’s mining enterprises that develop solid minerals. Much attention is paid to processing initial mining and geological data, providing the basis for mining and geometric modeling. Geostatistical methods give high results when processing and assessing reliability of the input modeling data and processing and presenting modeling results. Different types of deposits require different approaches to geometrization. Deposits with high variability of indicators require accounting, data grouping and zoning. For this purpose, a heuristic algorithm is developed, which is an efficient independent method of modeling and enables considering the high level of heterogeneity of the spatial distribution of mining and geological indicators of the deposit. The results of the modeling can be used for deposit estimation and monitoring, reserves calculation, and mining operations planning.

Optimizing Osteoporosis Classification through Femur Bone Volumetric Estimation with Deep Belief Network in X-ray Imaging

The accurate classification of osteoporosis is a crucial requirement in the medical field for identifying patients with skeletal disorders associated with aging. Achieving improved accuracy and reduced computational complexity in classification algorithms is essential. To address this, our research proposes a novel classification method utilizing the Hybrid Gradient Particle Swarm (HSG) Optimization-based Deep Belief Network, integrating the Particle Swarm Optimization (PSO) algorithm into the Gradient Descent (GD) algorithm. The osteoporosis classification process comprises five key steps: Preprocessing, Active Shape Model-based Segmentation, Geometric Estimation employing the proposed template search method, Feature Extraction for extracting medical and image-level features, and Osteoporosis Classification using the HSG-based Deep Belief Network. The proposed template search method efficiently and automatically updates the geometric points of the femur segment. Experimental validation using a real-time database demonstrates the effectiveness of the proposed method in terms of accuracy, sensitivity, and specificity. The results indicate an accuracy of 0.9724, affirming the efficacy of the proposed algorithm in making precise decisions regarding osteoporosis classification.

Newton Geometric Iterative Method for B-Spline Curve and Surface Approximation

We introduce a progressive and iterative method for B-spline curve and surface approximation, incorporating parameter correction based on the Newton iterative method. While parameter corrections have been used in existing Geometric Approximation (GA) methods to enhance approximation quality, they suffer from low computational efficiency. Our approach unifies control point updates and parameter corrections in a progressive and iterative procedure, employing a one-step strategy for parameter correction. We provide a theoretical proof of convergence for the algorithm, demonstrating its superior computational efficiency compared to current GA methods. Furthermore, the provided convergence proof offers a methodology for proving the convergence of existing GA methods with location parameter correction.

Geometric Co-Ordinate Feature Point Estimation (Gfpe): A Novel Approach to Analyse Of Kinematic Characteristics of Yoga Asanas Based on Anybody Simulation Software

The kinematic characteristics of yoga asanas is crucial for comprehending the biomechanical demands and benefits associated with these postures. Utilizing AnyBody simulation software, researchers delve into the intricate movements and alignments involved in various yoga poses. Through advanced biomechanical modeling, AnyBody allows for the analysis of joint angles, muscle activations, and forces exerted on the body during asana practice. This paper presented an analysis of the kinematic characteristics of yoga asanas using AnyBody Simulation in conjunction with Geometric Co-ordinate Feature Point Estimation (GFPE). By integrating advanced biomechanical modelling capabilities of AnyBody with GFPE’s precision in identifying geometric features, this study aims to elucidate the intricate movements and alignments inherent in various yoga poses. Through comprehensive simulations, joint angles, muscle activations, and forces exerted on the body during asana practice are meticulously analyzed. The examination of body posture with GFPE in the downward-facing dog pose, the average joint angles of the shoulder, elbow, and wrist were found to be 130°, 150°, and 160°, respectively. Additionally, muscle activation levels were measured, with the triceps brachii exhibiting an average activation of 70% of maximum voluntary contraction during the pose.

Asymmetry in the atmosphere of the ultra-hot Jupiter WASP-76 b

Context. WASP-76 b has been a recurrent subject of study since the detection of a signature in high-resolution transit spectroscopy data indicating an asymmetry between the two limbs of the planet. The existence of this asymmetric signature has been confirmed by multiple studies, but its physical origin is still under debate. In addition, it contrasts with the absence of asymmetry reported in the infrared (IR) phase curve. Aims. We provide a more comprehensive dataset of WASP-76 b with the goal of drawing a complete view of the physical processes at work in this atmosphere. In particular, we attempt to reconcile visible high-resolution transit spectroscopy data and IR broadband phase curves. Methods. We gathered 3 phase curves, 20 occultations, and 6 transits for WASP-76 b in the visible with the CHEOPS space telescope. We also report the analysis of three unpublished sectors observed by the TESS space telescope (also in the visible), which represents 34 phase curves. Results. WASP-76 b displays an occultation of 260 ± 11 and 152 ± 10 ppm in TESS and CHEOPS bandpasses respectively. Depending on the composition assumed for the atmosphere and the data reduction used for the IR data, we derived geometric albedo estimates that range from 0.05 ± 0.023 to 0.146 ± 0.013 and from <0.13 to 0.189 ± 0.017 in the CHEOPS and TESS bandpasses, respectively. As expected from the IR phase curves, a low-order model of the phase curves does not yield any detectable asymmetry in the visible either. However, an empirical model allowing for sharper phase curve variations offers a hint of a flux excess before the occultation, with an amplitude of ~40 ppm, an orbital offset of ~ −30°, and a width of ~20º. We also constrained the orbital eccentricity of WASP-76 b to a value lower than 0.0067, with a 99.7% confidence level. This result contradicts earlier proposed scenarios aimed at explaining the asymmetry observed in high-resolution transit spectroscopy. Conclusions. In light of these findings, we hypothesise that WASP-76 b could have night-side clouds that extend predominantly towards its eastern limb. At this limb, the clouds would be associated with spherical droplets or spherically shaped aerosols of an unknown species, which would be responsible for a glory effect in the visible phase curves.

UAV-Borne Mapping Algorithms for Low-Altitude and High-Speed Drone Applications.

This article presents an analysis of current state-of-the-art sensors and how these sensors work with several mapping algorithms for UAV (Unmanned Aerial Vehicle) applications, focusing on low-altitude and high-speed scenarios. A new experimental construct is created using highly realistic environments made possible by integrating the AirSim simulator with Google 3D maps models using the Cesium Tiles plugin. Experiments are conducted in this high-realism simulated environment to evaluate the performance of three distinct mapping algorithms: (1) Direct Sparse Odometry (DSO), (2) Stereo DSO (SDSO), and (3) DSO Lite (DSOL). Experimental results evaluate algorithms based on their measured geometric accuracy and computational speed. The results provide valuable insights into the strengths and limitations of each algorithm. Findings quantify compromises in UAV algorithm selection, allowing researchers to find the mapping solution best suited to their application, which often requires a compromise between computational performance and the density and accuracy of geometric map estimates. Results indicate that for UAVs with restrictive computing resources, DSOL is the best option. For systems with payload capacity and modest compute resources, SDSO is the best option. If only one camera is available, DSO is the option to choose for applications that require dense mapping results.

Geometric Approximation of Point Interactions in Three-Dimensional Domains

In this paper, we study a three-dimensional second-order elliptic operator with a point interaction in an arbitrary domain. The operator is supposed to be self-adjoint. We cut out a small cavity around the center of the interaction and consider an operator in such perforated domain with the Robin condition on the boundary of the cavity. Our main result states that once the coefficient in this Robin condition is appropriately chosen, the operator in the perforated domain converges to that with the point interaction in the norm resolvent sense. We also succeed in establishing order-sharp estimates for the convergence rate.

Causal Walk: Debiasing Multi-Hop Fact Verification with Front-Door Adjustment

Multi-hop fact verification aims to detect the veracity of the given claim by integrating and reasoning over multiple pieces of evidence. Conventional multi-hop fact verification models are prone to rely on spurious correlations from the annotation artifacts, leading to an obvious performance decline on unbiased datasets. Among the various debiasing works, the causal inference-based methods become popular by performing theoretically guaranteed debiasing such as casual intervention or counterfactual reasoning. However, existing causal inference-based debiasing methods, which mainly formulate fact verification as a single-hop reasoning task to tackle shallow bias patterns, cannot deal with the complicated bias patterns hidden in multiple hops of evidence. To address the challenge, we propose Causal Walk, a novel method for debiasing multi-hop fact verification from a causal perspective with front-door adjustment. Specifically, in the structural causal model, the reasoning path between the treatment (the input claim-evidence graph) and the outcome (the veracity label) is introduced as the mediator to block the confounder. With the front-door adjustment, the causal effect between the treatment and the outcome is decomposed into the causal effect between the treatment and the mediator, which is estimated by applying the idea of random walk, and the causal effect between the mediator and the outcome, which is estimated with normalized weighted geometric mean approximation. To investigate the effectiveness of the proposed method, an adversarial multi-hop fact verification dataset and a symmetric multi-hop fact verification dataset are proposed with the help of the large language model. Experimental results show that Causal Walk outperforms some previous debiasing methods on both existing datasets and the newly constructed datasets. Code and data will be released at https://github.com/zcccccz/CausalWalk.

ON THE POSSIBILITY OF ESTIMATING THE RADIUS OF A CYLINDRICAL REFLECTOR BY THE TIME OF THE DELAY OF THE PULSE OF THE PASSING WAVE IN RELATION TO THE MIRROR-REFLECTED PULSE OF THE SHEAR WAVE DURING ULTRASONIC METAL INSPECTION BY AN ANGLE PROBE

Modeling of the electro-acoustical channel of an angle probe for a side drilled hole type reflector made it possible to calculate the forms of the run round pulse generated by the radiation of a surface Rayleigh wave propagating along a concave cylindrical cavity and the wave pulse when reflected in the geometric acoustics approximation (mirror) for different radii of the hole and to obtain estimates of the ratio of the amplitudes of these pulses and the difference in their registration time. Comparison of the experimental results with those obtained by calculation for holes with radii of 2 and 3 mm showed their quite good qualitative and satisfactory quantitative coincidence. Estimates of the radius of a cylindrical hole are given based on the experimental delay time of the run round pulse and the Rayleigh wave velocity for a flat surface using a simple ratio; the range of the ratio of the diameter of the hole to the length of the transverse wave in the medium in which such an assessment is advisable is indicated, and an example of the values of the lag time in this range is given.

Marching Windows: Scalable Mesh Generation for Volumetric Data With Multiple Materials.

Volumetric data abounds in medical imaging and other fields. With the improved imaging quality and the increased resolution, volumetric datasets are getting so large that the existing tools have become inadequate for processing and analyzing the data. Here we consider the problem of computing tetrahedral meshes to represent large volumetric datasets with labeled multiple materials, which are often encountered in medical imaging or microscopy optical slice tomography. Such tetrahedral meshes are a more compact and expressive geometric representation so are in demand for efficient visualization and simulation of the data, which are impossible if the original large volumetric data are used directly due to the large memory requirement. Existing methods for meshing volumetric data are not scalable for handling large datasets due to their sheer demand on excessively large run-time memory or failure to produce a tet-mesh that preserves the multi-material structure of the original volumetric data. In this article we propose a novel approach, called Marching Windows, that uses a moving window and a disk-swap strategy to reduce the run-time memory footprint, devise a new scheme that guarantees to preserve the topological structure of the original dataset, and adopt an error-guided optimization technique to improve both geometric approximation error and mesh quality. Extensive experiments show that our method is capable of processing very large volumetric datasets beyond the capability of the existing methods and producing tetrahedral meshes of high quality.

Empirical Estimation of Landslide Runout Distance Using Geometrical Approximations in the Colombian North–East Andean Region

Landslides represent geological hazards wherein a part of a slope loses its static equilibrium and initiates movement. Once this movement begins, it becomes crucial to evaluate the land-slide runout distance (LRD). Currently, there exist numerous tools for estimating LRD, among which geometrical approximations stand as one of the most popular. These empirical models are particularly useful for wide-scale studies, aiding in the scale-down of the problem by identifying the critical areas. This study examines the application of geometrical approximations in the Colombian north–east Andean region. Within this area, a sampling of 49 was conducted using photogrammetric techniques, enabling the morphometrical characterization of each study unit. The results showcase the relationship between geometrical characteristics and LRD in the studied area, considering both land use and geomorphological settings. By exploiting these relationships, the study compares the estimation of LRD using various empirical models, many of which are already employed by practitioners within the studied region. For instance, the relationships in literature display a relative error in the estimation ranging around −50% and 100%. Furthermore, this research proposes new relationships for estimating LRD, enhancing the error estimations in a range between 0% and 50%, highlighting both the advantages and limitations of such empirical estimations. Consequently, it contributes new data to enrich the field of LRD studies.

Variational Estimation for Mechanical Systems on Lie Groups based on Geometric Mechanics

Geometric mechanics analyzes mechanical systems in the framework of variational mechanics, while accounting for the geometry of the configuration space. From the late 1970s, developments in this area produced several schemes for geometric control of mechanical systems in continuous time and discrete time. In the mid to late 2000s, geometric mechanics was first applied to state estimation of mechanical systems, particularly systems evolving on Lie groups as configuration manifolds, like rigid body systems. Much of the existing work on geometric mechanics-based estimation has been in continuous time, using deterministic, semi-stochastic and stochastic approaches. While the body of existing literature on discrete-time estimation schemes on Lie groups is not as extensive, the literature on this topic is contemporaneous with continuous-time schemes. This work describes some recent and ongoing research on geometric mechanics-based estimation schemes in continuous and discrete time from the mid-2010s, which were developed using the Lagrange-d’Alembert principle applied to rigid body systems. This approach gives (deterministic) observer designs with strong stability and robustness properties. This work concludes with potential extensions of this approach to mechanical systems with principal fiber bundles as configuration manifolds.

A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting.

This work presents a novel and effective method for fitting multidimensional ellipsoids (i.e., ellipsoids embedded in [Formula: see text]) to scattered data in the contamination of noise and outliers. Unlike conventional algebraic or geometric fitting paradigms that assume each measurement point is a noisy version of its nearest point on the ellipsoid, we approach the problem as a Bayesian parameter estimate process and maximize the posterior probability of a certain ellipsoidal solution given the data. We establish a more robust correlation between these points based on the predictive distribution within the Bayesian framework, i.e., considering each model point as a potential source for generating each measurement. Concretely, we incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain, ensuring ellipsoid-specific results regardless of inputs. We then establish the connection between measurement point and model data via Bayes' rule to enhance the method's robustness against noise. Due to independent of spatial dimensions, the proposed method not only delivers high-quality fittings to challenging elongated ellipsoids but also generalizes well to multidimensional spaces. To address outlier disturbances, often overlooked by previous approaches, we further introduce a uniform distribution on top of the predictive distribution to significantly enhance the algorithm's robustness against outliers. Thanks to the uniform prior, our maximum a posterior probability coincides with a more tractable maximum likelihood estimation problem, which is subsequently solved by a numerically stable Expectation Maximization (EM) framework. Moreover, we introduce an ε-accelerated technique to expedite the convergence of EM considerably. We also investigate the relationship between our algorithm and conventional least-squares-based ones, during which we theoretically prove our method's superior robustness. To the best of our knowledge, this is the first comprehensive method capable of performing multidimensional ellipsoid-specific fitting within the Bayesian optimization paradigm under diverse disturbances. We evaluate it across lower and higher dimensional spaces in the presence of heavy noise, outliers, and substantial variations in axis ratios. Also, we apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks. In all these test contexts, our method consistently delivers flexible, robust, ellipsoid-specific performance, and achieves the state-of-the-art results.

Finite-frequency modeling of regional tropospheric infrasound using realistic atmospheres and terrain.

Infrasonic waves have been observed to propagate to regional (greater than 15 km) distances through the troposphere. Infrasound propagation in the geometric acoustics approximation has shown that realistic terrain can scatter acoustic energy from tropospheric ducts; however, ray methods cannot intrinsically capture finite-frequency behavior such as diffraction. A two-dimensional finite-difference time-domain (FDTD) method has been developed to solve linearized equations for infrasound propagation with realistic terrain. Acoustic wave propagation over 100 km with both flat terrain and a Gaussian hill was first simulated in order to compare finite-frequency propagation with ray predictions. The effects of realistic terrain and atmospheres on infrasound signals from a 2012 surface explosion at the Utah Testing and Training Range are then investigated. Propagation through the troposphere is suggested by array processing results, but eigenrays are not predicted due to weak to nonexistent ducting conditions. FDTD modeling suggests that the inclusion of terrain and finite frequency effects helps explain much of the observed signal in a realistic scenario. These results suggest that geometric acoustics may underestimate propagation through the troposphere, and that recorded waveforms at regional distances may be noticeably affected by terrain.

ПОШУК БАЗОВОЇ МНОЖИНИ ДЛЯ ЗАДАЧ МАШИННОГО НАВЧАННЯ

The coreset discovery problem is reviewed as well as the following three main methods to solve it: geometric coreset estimation, coreset discovery using the genetic algorithm, and coreset discovery using neural networks. We analyze each of these methods and find the cases where they show the best results. The focus of the paper is on neural network-based approaches and their ability to solve the coreset discovery problem. We perform a comparative analysis of several neural network-based approaches, describe their pros and cons, and determine the next steps in solving the coreset discovery problem. Keywords: coreset, dataset distillation, dataset condensation, geometry coreset, genetic algorithm.

LOG-LIO: A LiDAR-Inertial Odometry With Efficient Local Geometric Information Estimation

LOG-LIO: A LiDAR-Inertial Odometry With Efficient Local Geometric Information Estimation