Geometric Algorithm Research Articles

Stress‐Aligned Hexahedral Lattice Structures

AbstractMaintaining the maximum stiffness of components with as little material as possible is an overarching objective in computational design and engineering. It is well‐established that in stiffness‐optimal designs, material is aligned with orthogonal principal stress directions. In the limit of material volume, this alignment forms micro‐structures resembling quads or hexahedra. Achieving a globally consistent layout of such orthogonal micro‐structures presents a significant challenge, particularly in three‐dimensional settings. In this paper, we propose a novel geometric algorithm for compiling stress‐aligned hexahedral lattice structures. Our method involves deforming an input mesh under load to align the resulting stress field along an orthogonal basis. The deformed object is filled with a hexahedral grid, and the deformation is reverted to recover the original shape. The resulting stress‐aligned mesh is used as basis for a final hollowing procedure, generating a volume‐reduced stiff infill composed of hexahedral micro‐structures. We perform quantitative comparisons with structural optimization and hexahedral meshing approaches and demonstrate the superior mechanical performance of our designs with finite element simulation experiments.

A novel real-time tension distribution method for cable-driven parallel robots

Abstract The tension distribution problem of cable-driven parallel robots is inevitable in real-time control. Currently, iterative algorithms or geometric algorithms are commonly used to solve this problem. Iterative algorithms are difficult to improve in real-time performance, and the tension obtained by geometric algorithms may not be continuous. In this paper, a novel tension distribution method for four-cable, 3-DOF cable-driven parallel robots is proposed based on the wave equation. The tension calculated by this method is continuous and differentiable, without the need for iterative computation or geometric centroid calculations, thus exhibiting good real-time performance. Furthermore, the feasibility and rationality of this algorithm are theoretically proven. Finally, the real-time performance and continuity of cable tension are analyzed through a specific numerical example.

Automated vision-based concrete crack measurement system

Maintaining and ensuring the safety of ageing infrastructures is becoming of uttermost importance. This is particularly critical in Europe, where several infrastructures like tunnels and bridges are approaching their end-of-life. Such a condition justifies the need for a prompt recognition of the deterioration status of these structures as well as the identification of strategies for effective inspection and monitoring operations. This paper focuses on cracks, and proposes a vision-based measurement system aimed at precisely and accurately detecting cracks on concrete surfaces and measuring their geometric features. The system mainly leverages RGB-D cameras to make the pixel to real world measurement units conversion possible (by assessing the camera to target relative distance and pose), and exploits an image processing pipeline consisting of the following steps: (a) an artificial intelligence-based semantic segmentation model for identifying cracks on the target structure; (b) a ridge detection algorithm for identifying the centre line of the cracks; (c) a geometric feature extraction algorithm to extract key metrics of the crack, like its width and length. Variance analysis of the crack mean width measurement shows a repeatability standard deviation of 0.01 mm and a reproducibility standard deviation of 0.02 mm, with a Type A expanded uncertainty of 0.004 mm. The system performance is discussed in the paper referring both to lab testing activities, targeted to the assessment of its metrological characteristics, and to a real field application, i.e. concrete inspection in a tunnel. This latter application is of particular interests as it is targeted to demonstrate the robustness of the approach in harsh environments (i.e., strong presence of dust, total absence of natural light) and on surfaces whose roughness generates important light gradients and shadows (e.g., sprayed concrete).

Protocol for Designing De Novo Noncanonical Peptide Binders in OSPREY.

D-peptides, the mirror image of canonical L-peptides, offer numerous biological advantages that make them effective therapeutics. This article details how to use DexDesign, the newest OSPREY-based algorithm, for designing these D-peptides de novo. OSPREY physics-based models precisely mimic energy-equivariant reflection operations, enabling the generation of D-peptide scaffolds from L-peptide templates. Due to the scarcity of D-peptide:L-protein structural data, DexDesign calls a geometric hashing algorithm, Method of Accelerated Search for Tertiary Ensemble Representatives, as a subroutine to produce a synthetic structural dataset. DexDesign enables mixed-chirality designs with a new user interface and also reduces the conformation and sequence search space using three new design techniques: Minimum Flexible Set, Inverse Alanine Scanning, and K*-based Mutational Scanning.

Robotic arm, 3D vision and geometric matching for an efficient implementation of inverse Patch Transfer Function method

The inverse patch transfer function method was developed to reconstruct the acoustic fields (velocity, pressure, intensity) at the surface of a vibrating source with a complex geometry using the concept of a virtual acoustic domain. This method can be used to manage sources with complex shapes in a non-anechoic acoustic environment, even in the presence of masking objects. However, this method is quite demanding experimentally as it requires a large number of measurement positions with good spatial positioning accuracy. This article presents a dedicated experimental platform developed to simplify and automate the entire measurement process. First, a 3D camera placed on a robotic arm constructs a point cloud characterising the source and its environment. Using a geometric matching algorithm, the CAD of the object under test is detected in the cloud of points, enabling the different coordinate systems (CAD, robot and physical part) to be matched. Secondly, the measurement points required to apply iPTF can be defined directly in the CAD and reproduced in near-real time on the test rig, regardless of the type of antenna used, providing an efficient means of applying iPTF and opening the way to many advanced applications.

Strategic allocation of landmarks to reduce uncertainty in indoor navigation

Indoor navigation systems often rely on verbal, turn-based route instructions. These can, at times, be ambiguous at complex decision points with multiple paths intersecting under angles that are not well distinguished by the turn grammar used. Landmarks can be included into turn instructions to reduce this ambiguity. Here, we propose an approach to optimize landmark allocation to improve the clarity of route instructions. This study assumes that landmark locations are constrained to a pre-determined set of slots. We select a minimum-size subset of the set of all slots and allocate it with landmarks, such that the navigation ambiguity is resolved. Our methodology leverages computational geometric analysis, graph algorithms, and optimization formulations to strategically incorporate landmarks into indoor route instructions. We propose a method to optimize landmark allocation in indoor navigation guidance systems, improving the clarity of route instructions at complex decision points that are inadequately served by turn-based instructions alone.

A geometric algorithm for the factorization of rational motions in conformal three space

Rational motions in conformal three space can be parametrized by polynomials with coefficients in a suitable Clifford algebra. We call them “spinor polynomials.” In this text we present a new algorithm to decompose generic spinor polynomials into linear factors. The factorization algorithm is based on the “kinematics at infinity”. Factorizations exist generically but not generally and are typically not unique. We prove that generic multiples of non-factorizable spinor polynomials admit factorizations and we demonstrate at hand of an example how our ideas can be used to tackle the hitherto unsolved problem of “factorizing” algebraic motions.

Dynamic phase measurement based on two-step phase-shifting interferometry with geometric phase grating

Abstract This paper proposes a novel dynamic 2-step phase-shifting interferometry method with a geometric phase grating and direct aberration-free object phase recovery algorithm. By exploring the amplitude-phase modulation property of the geometric phase grating, two-step phase-shifting interferograms with π / 2 phase shift can be obtained in a single shot through two kinds of spatial-multiplexing arrangements. Then the restriction on the background uniformity of the 2-step phase-shifting algorithms and the system aberration can be avoided simultaneously, based on the presented direct and fast object phase demodulation strategy with an object-free state recorded in advance, so an accurate object measurement result can be guaranteed. Accordingly, a detailed analysis of the theoretical model of error sources is conducted by taking the depolarization error of geometric phase grating and phase measurement model error into consideration, with a Least Squares Iteration optimization strategy established to further improve the measurement accuracy. Experiments on various types of phase objects, such as a photo-etched quartz substrate, the spatial light modulator, and a silicon chip, validate the effectiveness and accuracy of our method when compared with the reference phase obtained from 4-step temporal phase-shifting method. The dynamic phase measurement capability is demonstrated by exhibiting the evaporation process of alcohol droplet.

PharmRL: Pharmacophore elucidation with Deep Geometric Reinforcement Learning.

Molecular interactions between proteins and their ligands are important for drug design. A pharmacophore consists of favorable molecular interactions in a protein binding site and can be utilized for virtual screening. Pharmacophores are easiest to identify from co-crystal structures of a bound protein-ligand complex. In this work, however, we develop a deep learning method that can identify pharmacophores in the absence of a ligand. Specifically, we train a CNN model to identify potential favorable interactions in the the binding site, and develop a deep geometric Q-learning algorithm that attempts to select an optimal subset of these interaction points to form a pharmacophore. With this algorithm, we show better prospective virtual screening performance, in terms of F1 scores, on the DUD-E dataset than random selection of ligand identified features from co-crystal structures. We also conduct experiments on the LIT-PCBA dataset and show that it provides efficient solutions for identifying active molecules. Finally, we test our method by screening the COVID moonshot dataset and show that it would be effective in identifying prospective lead molecules even in the absence of fragment screening experiments. Alongside, we provide a Google Colab notebook for ease of use of the developed method.

Monitoring of plant diseases caused by Fusarium commune and Rhizoctonia solani in bok choy using hyperspectral remote sensing and machine learning.

Local vegetable production is susceptible to various fungal pathogens, the most common and lethal of which are Fusarium commune and Rhizoctonia solani. Early detection of these pathogens is challenging, and by the time visual symptoms appear, the pathogens may have already spread extensively, causing massive damage to the production. In this study, we explored the use of hyperspectral data for early detection of diseases caused by F. commune or R. solani in bok choy Brassica rapa subsp. chinensis by collecting hyperspectral data from healthy plants and plants inoculated with either fungal pathogen in a controlled experimental set-up. Based on the collected data, we employed various tree-based, distribution-based, geometric, neural networks and ensemble learning algorithms to train detection models. Among the trained models, Multi-Layer Perceptron (MLP) models performed the best with overall accuracy reaching 95.9 ± 0.26%. MLP models could differentiate between healthy and infected plants with 99% precision after 1 day of infection, and distinguish between different fungal pathogens with 99% precision after 2 days. During this period, no visible symptoms of fungal infection could be observed. Further analysis into trained MLP models and general reflectance profiles of plants also revealed a high correlation of the spectral regions 445-460, 560-595, 606-620 and 719-728 nm with fungal infection in bok choy plants. Our findings highlight the potential of hyperspectral imaging as a highly precise early detection tool for fungal diseases in plants. © 2024 Society of Chemical Industry.

An automated framework for pediatric hip surveillance and severity assessment using radiographs.

Hip dysplasia is the second most common orthopedic condition in children with cerebral palsy (CP) and may result in disability and pain. The migration percentage (MP) is a widely used metric in hip surveillance, calculated based on an anterior-posterior pelvis radiograph. However, manual quantification of MP values using hip X-ray scans in current standard practice has challenges including being time-intensive, requiring expert knowledge, and not considering human bias. The purpose of this study is to develop a machine learning algorithm to automatically quantify MP values using a hip X-ray scan, and hence provide an assessment for severity, which then can be used for surveillance, treatment planning, and management. X-ray scans from 210 patients were curated, pre-processed, and manually annotated at our clinical center. Several machine learning models were trained using pre-trained weights from Inception ResNet-V2, VGG-16, and VGG-19, with different strategies (pre-processing, with and without region of interest (ROI) detection, with and without data augmentation) to find an optimal model for automatic hip landmarking. The predicted landmarks were then used by our geometric algorithm to quantify the MP value for the input hip X-ray scan. The pre-trained VGG-19 model, fine-tuned with additional custom layers, outputted the lowest mean squared error values for both train and test data, when ROI cropped images were used along with data augmentation for model training. The MP value calculated by the algorithm was compared to manual ground truth labels from our orthopedic fellows using the hip screen application for benchmarking. The results showed the feasibility of the machine learning model in automatic hip landmark detection for reliably quantifying MP value from hip X-ray scans. The algorithm could be used as an accurate and reliable tool in orthopedic care for diagnosing, severity assessment, and hence treatment and surgical planning for hip displacement.

On the Construction of Congruences over Generalized Fuzzy G-Acts

Group action is defined to support Cayley’s claim that every group is isomorphic to a suitable subgroup of a symmetric group. Group actions have a wide range of applications, including the analysis of symmetries of geometric objects and algorithms and cryptographic systems. In 1965, Zadeh introduced the concept of fuzzy sets and provided a mathematical formulation for various concepts in this area. Since then, the concept of fuzziness has been integrated into several branches of mathematics to address the uncertainties of real-life scenarios. This article introduces the concept of group action in a fuzzy environment, termed fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts. The study provide the concept of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-orbits and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-stabilizers and clearly outlines fuzzy permutation representations of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document} and fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-morphisms. The research findings significantly contributes to the understanding of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-congruences and fuzzy quotient G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-subacts with the help of fuzzy G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}$$\\end{document}-partitions. This approach not only refines the underlying theories but also opens up new possibilities for practical implementation. Thus, the study demonstrates how the more complex fuzzy theory can expand and enrich the mathematical structures of abstract algebra, making them highly applicable.

Forty years: Geometric numerical integration of dynamical systems in China

Kang Feng proposed the symplectic geometric algorithms for Hamiltonian systems at Beijing “International Symposium on Differential Geometry and Differential Equations” in August 1984, opening up a new field of research. Over the past 40 years, geometric numerical integration for dynamical systems has been irreplaceably applied in numerous fields, including celestial mechanics, plasma physics, fluid mechanics, biochemistry, meteorological forecasting, seismic exploration, data science and artificial intelligence. This paper mainly summarizes the achievements of geometric numerical integration for dynamical systems in China over the past 40 years.

Experimental Comparison of Two 6D Pose Estimation Algorithms in Robotic Fruit-Picking Tasks

This paper presents an experimental comparison between two existing methods representative of two categories of 6D pose estimation algorithms nowadays commonly used in the robotics community. The first category includes purely deep learning methods, while the second one includes hybrid approaches combining learning pipelines and geometric reasoning. The hybrid method considered in this paper is a pipeline of an instance-level deep neural network based on RGB data only and a geometric pose refinement algorithm based on the availability of the depth map and the CAD model of the target object. Such a method can handle objects whose dimensions differ from those of the CAD. The pure learning method considered in this comparison is DenseFusion, a consolidated state-of-the-art pose estimation algorithm selected because it uses the same input data, namely, RGB image and depth map. The comparison is carried out by testing the success rate of fresh food pick-and-place operations. The fruit-picking scenario has been selected for the comparison because it is challenging due to the high variability of object instances in appearance and dimensions. The experiments carried out with apples and limes show that the hybrid method outperforms the pure learning one in terms of accuracy, thus allowing the pick-and-place operation of fruits with a higher success rate. An extensive discussion is also presented to help the robotics community select the category of 6D pose estimation algorithms most suitable to the specific application.

Digital reconstruction of railway steep slope from UAV+TLS using geometric transformer

Accurate representation of railway slopes, especially those that are steep, is vital for real-time risk perception. Also, temporary structures also present certain safety hazards due to lack of monitoring. Traditional point cloud modeling, employing Unmanned Aerial Vehicle (UAV) or Terrestrial Laser Scanning (TLS), often struggles to simultaneously account for the precision of both surface and overhead models, leading to considerable model distortion, roughness, and deviation. Addressing these issues, A new 3D point cloud modeling algorithm for railway slopes based on a geometric transformer is presented in this paper. This involves an innovative rough point cloud denoising technique leveraging adaptive segmentation, multi-scale denoising, and deep learning point cloud registration. Our approach significantly enhances UAV point cloud accuracy and supplements missing portions of the TLS point cloud dataset occluded by objects block, using data from the UAV point cloud set. An experimental study shows that the score-based denoising algorithm improves precision from 37.44 mm to 8.11 mm for a UAV 3D point cloud. Further, by registering the UAV and TLS point cloud sets using the Geometric Transformer algorithm, the precision of the 3D point cloud was further augmented to 5.11 mm, representing a sevenfold enhancement over the initial UAV point cloud accuracy prior to denoising. Consequently, a high-fidelity 3D point cloud model of steep railway slopes has been created.

A robust circle fitting method for component fiducialization

The fiducialization for the round component is an indispensable step in the alignment. The traditional geometric and algebraic fitting algorithms are prone to be contaminated by outliers and the accuracy of existing robust algorithms can be further improved. In this study, we present a robust weighted Hyper method that combines the strengths of both the M-estimator and the Hyper method. Through Black-Rangarajan duality, outliers are assigned minimal or even zero weights, effectively enhancing robustness. We tested the proposed algorithm with several benchmarking algorithms using Monte Carlo data. In the presence of outliers, our method demonstrates the lowest root mean squared errors (RMSE). Specifically, when fitting points observed by the laser tracker in Dalian Advanced Light Source (DALS), our method outperforms existing algorithms in terms of repeat accuracy, yielding standard errors (STD) of 11.7 μm and 14.31 μm for the center and radius respectively. The robust weighted Hyper algorithm enhances fiducialization precision without requiring additional instruments and provides valuable guidance in the alignment.

Geometric Factor Correction Algorithm Based on Temperature and Humidity Profile Lidar

Due to the influence of geometric factors, the temperature and humidity profile of lidar’s near-field signal was warped when sensing the air environment. In order to perform geometric factor correction on near-field signals, this article proposes different correction solutions for the Mie and Raman scattering channels. Here, the Mie scattering channel used the Raman method to invert the aerosol backscatter coefficient and correct the extinction coefficient in the transition zone. The geometric factor was the ratio of the measured signal to the forward-computed vibration Raman scattering signal. The aerosol optical characteristics were reversed using the corrected echo signal, and the US standard atmospheric model was added to the missing signal in the blind zone, reflecting the aerosol evolution process. The stability and dependability of the proposed algorithm were validated by the consistency between the visibility provided by the Environmental Protection Agency and the visibility acquired via lidar retrieval data. The near-field humidity data were supplemented by the interpolation method in the Raman scattering channel to reflect the water vapor transfer process in the temporal dimension. The measured transmittance curve of the filter, the theoretical normalized spectrum, and the sounding data were used to compute the delay geometric factor. The temperature was retrieved and the near-field signal distortion issue was resolved by applying the corrected quotient of the temperature channel. The proposed algorithm exhibited robustness and universality, enhancing the system’s detection accuracy compared to the temperature and humidity data constantly recorded by the probes in the meteorological gradient tower, which have a high correlation with the lidar observation data. The comparison between lidar data and instrument monitoring data showed that the proposed algorithm could effectively correct distorted echo signals in the transition zone, which was of great value for promoting the application of lidar in the meteorological monitoring of the urban canopy layer.

Virtual Screening and Molecular Docking: Discovering Novel METTL3 Inhibitors.

Methyltransferase-like 3 (METTL3) is an RNA methyltransferase that catalyzes the N6 -methyladenosine (m6A) modification of mRNA in eukaryotic cells. Past studies have shown that METTL3 is highly expressed in various cancers and is closely related to tumor development. Therefore, METTL3 inhibitors have received widespread attention as effective treatments for different types of tumors. This study proposes a hybrid high-throughput virtual screening (HTVS) protocol that combines structure-based methods with geometric deep learning-based DeepDock algorithms. We identified unique skeleton inhibitors of METTL3 from our self-built internal database. Among them, compound C3 showed significant inhibitory activity on METTL3, and further molecular dynamics simulations were performed to provide more details about the binding conformation. Overall, our research demonstrates the effectiveness of hybrid virtual algorithms, which is of great significance for understanding the biological functions of METTL3 and developing treatment methods for related diseases.

Theta-regularized Kriging: Modeling and algorithms

To obtain more accurate model parameters and improve prediction accuracy, we proposed a regularized Kriging model that penalizes the hyperparameter theta in the Gaussian stochastic process, termed the Theta-regularized Kriging. We derived the optimization problem for this model from a maximum likelihood perspective. Additionally, we presented specific implementation details for the iterative process, including the regularized optimization algorithm and the geometric search cross-validation tuning algorithm. Three distinct penalty methods, Lasso, Ridge, and Elastic-net regularization, were meticulously considered. Meanwhile, the proposed Theta-regularized Kriging models were tested on nine common numerical functions and two practical engineering examples. The results demonstrate that, compared with other penalized Kriging models, the proposed model performs better in terms of accuracy and stability.

A Novel Direct Geometric Algorithm for Worst-Case Error Analysis of Six-Port Reflectometer and Its Guiding Significance in Application

A Novel Direct Geometric Algorithm for Worst-Case Error Analysis of Six-Port Reflectometer and Its Guiding Significance in Application