Discrete Curvature Research Articles (Page 1)

This work focuses on global minimum time trajectory planning for track racing considering curvature and distance based on offline high-precision map of track. Offline mapping of track, post processing and a new gridding method are firstly introduced. Then according to the requirement of quadratic programming method, the sum of discrete squared distance and the sum of discrete squared curvature are derived respectively, which are then combined together by a weight coefficient to constitute the objective function of minimum time trajectory. Drivable curbs, size of vehicle and safe distance are considered in constraint conditions in order to make full use of the width of track. With objective function and constraint conditions, quadratic programming method is used to calculate and iterate a smooth and reasonable trajectory. Petrol and electric racecars, different driver models and different trajectories are simulated in CarSim. Results show that conspicuous improvement can be obtained if using drivable curbs, and the minimum time trajectories for different vehicles and drivers display significant difference.

A high-efficiency regulation method for optimal root zone temperature under different nitrogen fertilizer using discrete curvature

Conventional point-based algorithms largely ignore the strict sequential order of rotating multi-beam Light Detection and Ranging (LiDARs). We exploit this intrinsic structure by modeling each scan ring as a discrete curve on a conical manifold, encoded as a one-dimensional complex-valued signal. This representation preserves full 3D information while enabling efficient, principled computation of geodesic curvature via discrete difference calculus. Augmented with local smoothness and range-gradient features, this curvature forms the basis of an unsupervised classification pipeline that uses adaptive ring-wise thresholds to label points as planar, edge, or corner features. A scan-topology graph then aggregates these 1D labels into coherent 3D primitives, all in linear time. Experiments on synthetic and large-scale urban datasets confirm the method's theoretical accuracy and practical utility. Our C++ implementation processes at over 80 Frames Per Second (FPS) on a single Central Processing Unit (CPU) core, significantly outperformings traditional Principal Component Analysis (PCA)-based and clustering methods in geometric accuracy, label purity, and temporal stability. By uniting first-principles differential geometry with real-time performance, our framework provides a transparent, parameter-light alternative to learning-based pipelines, offering robust landmarks for Simultaneous Localization and Mapping (SLAM) and consistent semantics for dynamic scene understanding.

Ricci-GraphDTA: A graph neural network integrating discrete Ricci curvature for drug-target affinity prediction.

An accurate map of intracellular organelle pH is crucial for comprehending cellular metabolism and organellar functions. However, a unified intracellular pH spectrum using a single probe is still lacking. Here, we developed a novel quantum entanglement-enhanced pH- sensitive probe called SITE-pHorin (single excitation and two emissions pH sensor protein), which features a wide pH-sensitive range and ratiometric quantitative measurement capabilities. We subsequently measured the pH of various organelles and their subcompartments, including mitochondrial subspaces, Golgi stacks, endoplasmic reticulum (ER), lysosomes, peroxisomes, and endosomes in COS-7 cells. For the long-standing debate on the pH of the mitochondrial compartments, we measured the pH of the mitochondrial cristae (mito-cristae) as 6.60±0.40, the pH of the mitochondrial intermembrane space (mito-IMS) as 6.95±0.30, and the pH of the two populations of the mitochondrial matrix (mito-matrix) at approximately 7.20±0.27 and 7.50±0.16, respectively. Notably, the pH of the lysosome exhibited a single, narrow Gaussian distribution centered at 4.79±0.17, which is consistent with an optimal lysosomal acidic pH between 4.5 and 5.0. Furthermore, quantum chemistry computations revealed that both the deprotonation of the residue Y182 and the discrete curvature of the deformed benzene ring in the chromophore are necessary for the quantum entanglement mechanism of SITE-pHorin. Intriguingly, our findings reveal an accurate pH gradient (0.6-0.9 pH units) between the mitochondrial cristae and the mitochondrial matrix, suggesting that prior knowledge about ΔpH (0.4-0.6) and the mitochondrial proton motive force (pmf) is underestimated.

In this article we study two discrete curvature notions, Bakry-Émery curvature and Ollivier Ricci curvature, on Cayley graphs. We introduce Right Angled Artin-Coxeter Hybrids (RAACHs) generalizing Right Angled Artin and Coxeter groups (RAAGs and RACGs) and derive the curvatures of Cayley graphs of certain RAACHs. Moreover, we show for general finitely presented groups $\Gamma = \langle S \, \mid\, R \rangle$ that addition of relators does not lead to a decrease in the weighted curvatures of their Cayley graphs with adapted weighting schemes.

Abstract This article introduces and studies a new class of graphs motivated by discrete curvature. We call a graph resistance nonnegative if there exists a distribution on its spanning trees such that every vertex has expected degree at most two in a random spanning tree; these are precisely the graphs that admit a metric with nonnegative resistance curvature, a discrete curvature introduced by Devriendt and Lambiotte. We show that this class of graphs lies between Hamiltonian and 1-tough graphs and, surprisingly, that a graph is resistance nonnegative if and only if its twice-dilated matching polytope intersects the interior of its spanning tree polytope. We study further characterizations and basic properties of resistance nonnegative graphs and pose several questions for future research.

Abstract We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, ɛ-chains are finite collections of points with distance at most ɛ between consecutive points. They play the role of discrete curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter ɛ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradients or of the local Lipschitz constant of Lipschitz functions. The proof of this fact is inspired by a recent technique developed by Eriksson-Bique in Eriksson-Bique (2023 Calc. Var. Partial Differential Equations62 23). In the possible non-complete setting, we prove that the chain Sobolev space is equal to the one defined via relaxation of the local Lipschitz constant of Lipschitz functions, while in general they are different from the one defined via upper gradients along curves. We apply the theory developed in the paper to prove equivalent formulations of the Poincaré inequality in terms of pointwise estimates involving ɛ-upper gradients, lower bounds on modulus of chains connecting points and size of separating sets measured with the Minkowski content in the non-complete setting. Along the way, we discuss the notion of weak ɛ-upper gradients and asymmetric notions of integral along chains.

Abstract We obtain many objects of discrete differential geometry as reductions of skew parallelogram nets, a system of lattice equations that may be formulated for any unital associative algebra. The Lax representation is linear in the spectral parameter, and paths in the lattice give rise to polynomial dependencies. We prove that generic polynomials in complex 2 × 2 matrices factorize, implying that skew parallelogram nets encompass all systems with such a polynomial representation. We demonstrate factorization in the context of discrete curves by constructing pairs of Bäcklund transformations that induce Euclidean motions on discrete elastic rods. More generally, we define a hierarchy of discrete curves by requiring such an invariance after an integer number of Bäcklund transformations. Moreover, we provide the factorization explicitly for discrete constant curvature surfaces and reveal that they are slices in certain 4D cross-ratio systems. Encompassing the discrete DPW method, this interpretation constructs such surfaces from given discrete holomorphic maps.

ABSTRACTFlood impact assessment is limited by a scarcity of damage curves for critical infrastructure network components. This study presents a judgement‐based methodology for developing critical infrastructure network component flood damage curves. The 12 semi‐structured workshops record responses for estimated minimum and maximum damage ratios at 0.5, 1, 2 and 3 m water depths. The 46 responses, weighted by participant expertise level, are aggregated into a discrete minimum and maximum damage curve for each component. Damage curves are presented for 34 infrastructure network components across the transportation, energy, water, and telecommunication sectors. These damage curves are benchmarked against relevant flood damage curves from previous studies, providing insight on how flood damage models compare internationally and across methods. While the synthesised flood damage curves allow for nationally consistent risk assessments, this study highlights the need for flood damage curves that represent local risk contexts for infrastructure network components to facilitate locally applicable risk assessments that inform risk management.

This paper is based on the digital image processing technology, using the undamaged image information to restore and protect the frescoes. The discrete binary wavelet change is used to decompose and denoise the image signal. And decompose and filter the high-frequency component and low-frequency component of the image, choose different components, respectively, carry out coefficient transformation, and solve the OMP least-paradigm for different random matrices. The color space is selected, and the mural color space is channel decomposed according to the grayscale mode and restored separately. Establish an assumed datum for each independent face of the mural, establish a spatial coordinate system for it, realize the transformation of spatial coordinates, and realize the super-resolution three-dimensional reconstruction of the mural based on the generative adversarial network and the self-attention mechanism. Objective evaluation indexes and subjective evaluation indexes are established to compare the protection effect of different algorithms on murals. Compared with the traditional algorithm CDD, this paper’s algorithm improves the restoration time by 9.545~15.625 s, and the peak signal-to-noise ratio index improves by 1.35~4.769 db. In the results of the image extraction and processing, the calculated values of discrete curvature of the mural segments AB, CD, and EF ranges from -0.00945 to -0.00478, and the difference of standard deviation of the curvature from the target curvature is 6.477%. The approximate target curvature is obtained, and the algorithm has strong adaptive ability.

Graph neural networks(GNNs) have been demonstrated to depend on whether the node effective information is sufficiently passing. Discrete curvature (Ricci curvature) is used to study graph connectivity and information propagation efficiency with a geometric perspective, and has been raised in recent years to explore the efficient message-passing structure of GNNs. However, most empirical studies are based on directly observed graph structures or heuristic topological assumptions, and lack in-depth exploration of underlying optimal information transport structures for downstream tasks. We suggest that graph curvature optimization is more in-depth and essential than directly rewiring or learning for graph structure with richer message-passing characterization and better information transport interpretability. From both graph geometry and information theory perspectives, we propose the novel Discrete Curvature Graph Information Bottleneck (CurvGIB) framework to optimize the information transport structure and learn better node representations simultaneously. CurvGIB advances the Variational Information Bottleneck (VIB) principle for Ricci curvature optimization to learn the optimal information transport pattern for specific downstream tasks. The learned Ricci curvature is used to refine the optimal transport structure of the graph, and the node representation is fully and efficiently learned. Moreover, for the computational complexity of Ricci curvature differentiation, we combine Ricci flow and VIB to deduce a curvature optimization approximation to form a tractable IB objective function. Extensive experiments on various datasets demonstrate the superior effectiveness and interpretability of CurvGIB.

Geometric deep learning (GDL) models have demonstrated a great potential for the analysis of non-Euclidian data. They are developed to incorporate the geometric and topological information of non-Euclidian data into the end-to-end deep learning architectures. Motivated by the recent success of discrete Ricci curvature in graph neural network (GNNs), we propose TorGNN, an analytic Torsion enhanced Graph Neural Network model. The essential idea is to characterize graph local structures with an analytic torsion based weight formula. Mathematically, analytic torsion is a topological invariant that can distinguish spaces which are homotopy equivalent but not homeomorphic. In our TorGNN, for each edge, a corresponding local simplicial complex is identified, then the analytic torsion (for this local simplicial complex) is calculated, and further used as a weight (for this edge) in message-passing process. Our TorGNN model is validated on link prediction tasks from sixteen different types of networks and node classification tasks from four types of networks. It has been found that our TorGNN can achieve superior performance on both tasks, and outperform various state-of-the-art models. This demonstrates that analytic torsion is a highly efficient topological invariant in the characterization of graph structures and can significantly boost the performance of GNNs.

Introduction. The problem of discretization of continuous geometric objects is one of the most common problems of computational geometry. Many applications in all different fields, such as computer vision, robotics, signal processing, curve simplification in computer graphics applications, geographic information systems, and digital manufacturing applications, are based on the discretization and segmentation of plane curves, which are basic geometric objects. These methods mainly aim to solve the problem of dividing the curve into segments with the same characteristics or to minimize a predetermined error. The condition of partitioning the curve into points when the lengths of the chords connecting the segments are equal is an additional factor interesting from the point of view of practical applications. It allows, for example, to simplify the reproduction of a curve on CNC machines thanks to the constancy of the tool feed speed [1] or the reproduction of the movement of an object based on a video recording [2]. The purpose of the paper is to develop new algorithms for partitioning flat parametric curves under the condition of equality of chords (chord length connecting segments of the partition) given the two outside points included in the first and last segment and given a number of segments. Results. The problem of partitioning a curve in a parametric vector form on the Euclidean plane into segments equal in chord length having the formulation of [23, 24] was considered. A method of partitioning a flat parametric curve into equal-chord segments by crossing a circle of constant radius with the subsequent movement of the circle's center to the intersection point is proposed. The problem of the multivalued solution of the intersection equation was considered, which complicates the application of this method. This circumstance limits the use of circular partitioning by the lower limit of the values ??of the number of segments. The proposed algorithm was presented in pseudocode and described. It consists of the following procedures: the procedure for the initial initialization of the radius of a circle based on a partition with a uniform distribution by a parameter, procedures for partitioning the curve by a circle for different directions of the circle`s move (direct, reverse, two-way); the procedure for obtaining an equal-chord partition with a specified tolerance of determining the chord length. For the real curve`s example, experiments were conducted on its equipartition by this algorithm, implemented in the Julia programming language. It was established that with an increase in the degree of discretization of the value of the curve, the number of iterations required to achieve the specified accuracy stabilizes. This leads to a linear dependence of the partition execution time with an increase in the number of segments. It was found that when the accuracy of the partition is increased, the number of iterations increases slightly compared to the increase in accuracy. Conclusions. As a result of the research, it was found that the proposed algorithm is quite suitable for the equal chord segmentation of flat parametric curves in a wide range of segment values. The two-way version of the algorithm was the most promising for real applications, as it is more stable and flexible. This version is suitable for parallel execution by two processes or threads. The two-threaded version showed the best performance of all the algorithm versions. The disadvantages of the presented algorithm should include, first of all, the limitation of the lower limit of number of segments because with a small number of them, the radius of the partition circle increases, which leads to the presence of several intersection options and the need to analyze these options. Another disadvantage is that obtaining the points by way of the intersection requires solving the nonlinear equation, which depends on the representation of the curve and can be pretty tricky, even for numerical methods. Keywords: pseudocode, iteration, computational complexity, segmentation, chord, intersection equation.

This paper presents a new, efficient, accurate, and unconditionally stable second-order time-stepping method for the incompressible thermal micropolar Navier–Stokes equations (TMNSE) using mixed finite elements. The method linearizes the nonlinear convective terms in the momentum equation, microrotation equation, and temperature equation, requiring the solution of a linear problem at each time step. The discrete curvature of the solution is added as a stabilizing term for linear velocity [Formula: see text], microrotation velocity [Formula: see text], pressure [Formula: see text], and temperature [Formula: see text] in the equations, respectively. Curvature stabilization ([Formula: see text]) is a new concept in computational fluid dynamics (CFD) aimed at improving the commonly used velocity stabilization ([Formula: see text]), which only has first-order time accuracy and has adverse effects on important flow quantities such as drag coefficients. We derive a priori error estimates for the fully discrete linear extrapolation curvature stabilization method. The theoretical results and effectiveness of the new method are verified through a series of numerical experiments for [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] in 2D and 3D, respectively. In particular, this work considers the thermal cavity-driven flow experiment to validate the numerical scheme and obtains good results.

The mechanism by which myopia-managing spectacle lenses slow myopia progression remains controversial. Understanding the changes these lenses introduce to peripheral imaging properties helps shed light on this controversial issue. Given the difficulty of directly measuring these changes in clinical settings, this study combined experimental and modeling approaches to evaluate changes in images at the retina induced by myopia-managing lenses. Optical characteristics that may related to the efficacy of the lenses with concentric cylindrical annular refractive elements (CARE) in myopia control were investigated. Three lenses were evaluated: MyoCare (MC), MyoCare S (MCS), and a single vision (SV) lens with a custom-built physical eye model and optical simulations for the analysis. The simulated PSFs are consistent with the measured ones. PSF analysis showed that MC and MCS lenses produce discrete curves, resulting in remarkable distortion in the simulated retina images, especially for large eccentricities. Whether they increase or decrease contrast depends on the spatial frequencies and eccentricities. These lenses also increase retinal light intensity at different eccentricities. The positive power of the CARE structure introduces myopic defocus of less than 0.25 D at only a limited range of eccentricities. The proposed approaches present relatively straightforward techniques for evaluating the optical performance of myopia-managing spectacle lenses.

Previous studies have verified the feasibility of single unmanned roller tracking paths and have effectively evaluated the performance of pavement compaction. Nevertheless, the issue of scheduling faulty rollers (e.g., insufficient oil pressure) during collaborative pavement construction with multiple rollers has not been fully investigated, despite its prevalence in engineering practice. Although there are some patents that propose solutions for several cases, there is a lack of comprehensive, detailed and process-oriented scheduling strategies for specific scenarios. This paper introduces a framework for pavement construction comprising one paver and six rollers, and proposes a scheduling strategy for idle and faulty robots with the objective of addressing the problem of scheduling faulty rollers. Specifically, the traditional path planning task requires known beginning and end positions, while the scheduling position in this paper can be designed in advance. Consequently, this paper presents a methodology that leverages the ratio of discrete Bezier curve path points to discrete idle region points at distinct scheduling positions to ascertain the start and end positions of faulty and idle robots. In the scheduling process implementation phase, the paper considers the scene constraints and the model constraints of the rollers, and proposes a novel cost function that balances path length and safety distance. Furthermore, to address the issue of driving in opposite directions in the narrow passage, this paper proposes the interleaved scheduling scheme with the objective of enhancing the performance upper bound of the algorithm by significantly increasing the probability of finding a feasible solution. Moreover, the implementation of a discrete sampling of curve and position points ensures that the algorithm runs in an acceptable time. The results of comparative simulations demonstrate that the path planning algorithm proposed in this paper is more effective than alternative algorithms in addressing the specific application scenarios of this project. Furthermore, the integration of the interleaved scheduling framework has a considerable impact on the enhancement of the quality of the generated paths. The algorithm is capable of successfully completing the scheduling task of faulty and idle rollers with a road length of more than 100 m in a road width of 10.5 m, while ensuring that the other rollers are not affected in the completion of their respective tasks. The path search time is 24 s, the generated planning path length is 178.1 m and the safety distance is approximately 1.6 m.

A variation of the external load distribution function between nodes of a discrete grid in the static-geometric method allows discretely modeling curves of different shapes and solving problems of discrete interpolation on the area. The form of the continuous analogue of the discretely presented curve directly depends on the nature of the functions specified control load, which forms the discretely presented curve (DPC). There are known studies of the aspects of the relationship between the static geometric method of forming the DPC and the analytical description of a continuous curve through the synthesis of the static geometric method of forming discrete curves and the method of modeling them with numerical sequences. Separate issues of determining the correspondence of the equations of the continuous surface to the discrete function of the distribution of the external load are also investigated. This article examines the patterns of changes in the values of the superposition coefficients of two arbitrarily specified, both adjacent and non-adjacent nodal points, under the condition of a known distribution law of the magnitude of the finite difference, which in some cases will be a prototype of the external load between the nodes of the frame, which is a discrete model of a defined geometric image. If we change the uniformly distributed value of the finite difference or the value of the distribution function of the value of the finite difference, or the ordinates of one or two (marginal) nodal points at fixed values of the superposition coefficients, we can control the shape of the curve, discretely represented by the nodal points of its numerical sequence. The research data determine a general approach to obtaining similar patterns of changes in the values of the superposition coefficients of two arbitrarily specified, both adjacent and non-adjacent nodal points for determining the coordinates of points of modeling any one-dimensional functional dependencies and arbitrary one-dimensional sets of points. The results of the study of the regularities of changes in the values of the superposition coefficients given by two nodal points of different elementary functions, under the condition of a known distribution law of the magnitude of the finite difference, will allow solving the problems of continuous discrete interpolation and extrapolation by numerical sequences of any one-dimensional functional dependencies (determine the ordinates of the sought points of discrete curves) without time-consuming operations of assembling and solving large systems of linear equations

Static meshes with texture maps have attracted considerable attention in both industrial manufacturing and academic research, leading to an urgent requirement for effective and robust objective quality evaluation. However, current model-based static mesh quality metrics (i.e., metrics that directly use the raw data of the static mesh to extract features and predict the quality) have obvious limitations: most of them only consider geometry information, while color information is ignored, and they have strict constraints for the meshes' geometrical topology. Other metrics, such as image-based and point-based metrics, are easily influenced by the prepossessing algorithms, e.g., projection and sampling, hampering their ability to perform at their best. In this paper, we propose Geodesic Patch Similarity (GeodesicPSIM), a novel model-based metric to accurately predict human perception quality for static meshes. After selecting a group keypoints, 1-hop geodesic patches are constructed based on both the reference and distorted meshes cleaned by an effective mesh cleaning algorithm. A two-step patch cropping algorithm and a patch texture mapping module refine the size of 1-hop geodesic patches and build the relationship between the mesh geometry and color information, resulting in the generation of 1-hop textured geodesic patches. Three types of features are extracted to quantify the distortion: patch color smoothness, patch discrete mean curvature, and patch pixel color average and variance. To the best of our knowledge, GeodesicPSIM is the first model-based metric especially designed for static meshes with texture maps. GeodesicPSIM provides state-of-the-art performance in comparison with image-based, point-based, and video-based metrics on a newly created and challenging database. We also prove the robustness of GeodesicPSIM by introducing different settings of hyperparameters. Ablation studies also exhibit the effectiveness of three proposed features and the patch cropping algorithm. The code is available at https://multimedia.tencent.com/resources/GeodesicPSIM.

Sobolev metrics on spaces of discrete regular curves