Search Results (25)

In order to solve the problems of low efficiency and insufficient accuracy of the traditional manual template method in the forming detection of shell plates, a digital solution based on laser scanning detection system was proposed. By introducing a six-degree-of-freedom robotic arm and a high-precision line laser sensor to build a three-dimensional detection platform, a digital template method framework including data acquisition, point cloud registration, surface reconstruction, and deviation analysis was innovatively constructed. A point cloud non-penetration registration algorithm fused with boundary geometric information was proposed. Based on the improved Delaunay triangulation algorithm, the surface is reconstructed and the digital template is extracted. Experimental verification shows that the method achieves an accuracy of less than 1 mm of error in the detection of outer plates, shortens the single detection time to less than 10 min, and improves the detection efficiency by more than 75% compared with the traditional method. Full article

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form. Full article

Contact-based biometric applications primarily use prints from a finger or a palm for a single instance in different applications. For access control, there is an enrollment process using one or more templates which are compared with verification images. In forensics applications, randomly located, partial, and often degraded prints acquired from a crime scene are compared with the images captured from suspects or existing fingerprint databases, like AFIS. In both scenarios, if we need to use handprints which include segments from the finger and palm, what would be the solution? The motivation behind this is the concept of one single algorithm for one hand. Using an algorithm that can incorporate both prints in a common processing framework can be an alternative which will have advantages like scaling to larger existing databases. This work proposes a method that uses minutiae or minutiae-like features, Delaunay triangulation and graph matching with invariant feature representation to overcome the effects of rotation and scaling. Since palm prints have a large surface area with degradation, they tend to have many false minutiae compared to fingerprints, and the existing palm print algorithms fail to tackle this. The proposed algorithm constructs Delaunay triangulated graphs (DTG) using minutiae where Delaunay triangles form from minutiae, and initiate a collection of base triangles for opening the matching process. Several matches may be observed for a single triangle match when two images are compared. Therefore, the set of initially matched triangles may not be a true set of matched triangles. Each matched triangle is then used to extend as a sub-graph, adding more nodes to it until a maximum graph size is reached. When a significant region of the template image is matched with the test image, the highest possible order of this graph will be obtained. To prove the robustness of the algorithm to geometrical variations and working ability with extremely degraded (similar to latent prints) conditions, it is demonstrated with a subset of partial-quality and extremely-low-quality images from the FVC (fingerprint) and the THUPALMLAB (palm print) databases with and without geometrical variations. The algorithm is useful when partial matches between template and test are expected, and alignment or geometrical normalization is not accurately possible in pre-processing. It will also work for cross-comparisons between images that are not known a priori. Full article

In this paper, we propose a grain volume measurement method based on binocular structured light to address the need for fast and high-precision grain volume measurement in grain stocks. Firstly, we utilize speckle structured light imaging to tackle the image matching problem caused by non-uniform illumination in the grain depot environment and the similar texture of the grain pile surface. Secondly, we employ a semi-global stereo matching algorithm with census transformation to obtain disparity maps in grain bins, which are then converted into depth maps using the triangulation principle. Subsequently, each pixel in the depth map is transformed from camera coordinates to world coordinates using the internal and external parameter information of the camera. This allows us to construct 3D cloud data of the grain pile, including the grain warehouse scene. Thirdly, the improved European clustering method is used to achieve the segmentation of the three-dimensional point cloud data of the grain pile and the scene of the grain depot, and the pass-through filtering method is used to eliminate some outliers and poor segmentation points generated by segmentation to obtain more accurate three-dimensional point cloud data of the grain pile. Finally, the improved Delaunay triangulation method was used to construct the optimal topology of the grain surface continuous triangular mesh, and the nodes of the grain surface triangular mesh were projected vertically to the bottom of the grain warehouse to form several irregular triangular prisms; then, the cut and complement method was used to convert these non-plane triangular prisms into regular triangular prisms that could directly calculate the volume. The measured volume of the pile is then obtained by calculating the volume of the triangular prism. The experimental results indicate that the measured volume has a relative error of less than 1.5% and an average relative error of less than 0.5%. By selecting an appropriate threshold, the relative standard deviation can be maintained within 0.6%. The test results obtained from the laboratory test platform meet the requirements for field inspection of the granary. Full article

Global route planning is a pivotal function of unmanned surface vehicles (USVs). For ships, the safety of navigation is the priority. This paper presents the VK-RRT* algorithm as a way of designing the planned route automatically. Different from other algorithms or studies, this study employs electronic navigation chart (ENC) vector data instead of grid maps as the basis of the search, which reduces data error when converting the vector map into the grid map. In addition, Delaunay triangulation is employed to organize vector data, in which the depth value is taken as a factor to ensure the safety of the planning route. Furthermore, the initial planned route is not suitable for ship tracking as it does not consider the ship motion characteristics. Therefore, the planned route needs to be further optimized. In the final part, we also conducted experiments to verify the effectiveness and advantages of the proposed algorithm. The results show that the proposed algorithm could reduce the lengths of paths by about 23% on average and save planning time; these are largely dependent on the environment. Full article

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones. Full article

We present an algorithm for the fully automatic generation of a class-compliant mesh for ship structural analysis. Our algorithm is implemented as an end-to-end solution. It starts from a description of a geometry and produces a class conforming surface mesh as a result. The algorithm consists of two parts, the automatic geometry refinement and the preconditioned Delaunay frontal quad dominant mesh generator. A geometry is described by a dictionary of elements and it contains points, rods, plates, and openings. A dictionary can contain modeling errors such as unintended overlaps or an unintended loss of connectivity between elements. The main contribution of the paper is the automatic geometry refinement algorithm and the virtual stiffener procedure designed to control the local mesh orientation of a marching front meshing algorithm. The geometry refinement algorithm guarantees that the output dictionary will be such that intersections of the boundary edges of plates are guaranteed to be nodes of any mesh generated by tessellating such geometry. The algorithm is implemented in Python, using the open-source Gmsh system together with the Open CASCADE kernel which is used to implement the automatic geometry refinement. We present several benchmark models from an engineering practice to illustrate our claims as well as to benchmark the efficiency of the various stages of the processing pipeline. Full article

The accurate modeling of curved graphene layers for time-domain electromagnetic simulations is discussed in the present work. Initially, the advanced properties of graphene are presented, focusing on the propagation of strongly confined surface plasmon polariton waves at the far-infrared regime. Then, the implementation of an unstructured triangular grid was examined, based on the Delaunay triangulation method. The electric-field components were placed at the edges of the triangles, while two different techniques were proposed for the sampling of the magnetic ones. Specifically, the first one suggests that the magnetic component is placed at the triangle’s circumcenter providing more accurate results, although instability may occur for nonacute triangles. On the other hand, the magnetic field was sampled at the triangle’s centroid, considering the second technique, ensuring the algorithm’s stability, but further approximations were required, leading to a slight accuracy reduction. Moreover, the updating equations in the time-domain were extracted via an appropriate approximation of Maxwell equations in their integral form. Finally, graphene was introduced in the computational domain as an equivalent surface current density, whose location matches the corresponding electric components. The validity of our methodology was successfully performed via the comparison of graphene surface wave propagation properties to their theoretical values, whereas the global error determination indicates the minimal triangle dimensions. Additionally, an instructive setup comprising a circular graphene scatterer was analyzed thoroughly, to reveal our technique’s advantages compared to the conventional staircase discretization. Full article

This paper presents an algorithm for the fully automatic mesh generation for the finite element analysis of ships and offshore structures. The quality requirements on the mesh generator are imposed by the acceptance criteria of the classification societies as well as the need to avoid shear locking when using low degree shell elements. The meshing algorithm will be generating quadrilateral dominated meshes (consisting of quads and triangles) and the mesh quality requirements mandate that quadrilaterals with internal angles close to 90° are to be preferred. The geometry is described by a dictionary containing points, rods, surfaces, and openings. The first part of the proposed method consists of an algorithm to automatically clean the geometry. The corrected geometry is then meshed by the frontal Delaunay mesh generator as implemented in the gmsh package. We present a heuristic method to precondition the cross field of the fronatal quadrilateral mesher. In addition, the influence of the order in which the plates are meshed will be explored as a preconditioning step. Full article

At present, feature-based 3D reconstruction and tracking technology is widely applied in the medical field. In minimally invasive surgery, the surgeon can achieve three-dimensional reconstruction through the images obtained by the endoscope in the human body, restore the three-dimensional scene of the area to be operated on, and track the motion of the soft tissue surface. This enables doctors to have a clearer understanding of the location depth of the surgical area, greatly reducing the negative impact of 2D image defects and ensuring smooth operation. In this study, firstly, the 3D coordinates of each feature point are calculated by using the parameters of the parallel binocular endoscope and the spatial geometric constraints. At the same time, the discrete feature points are divided into multiple triangles using the Delaunay triangulation method. Then, the 3D coordinates of feature points and the division results of each triangle are combined to complete the 3D surface reconstruction. Combined with the feature matching method based on convolutional neural network, feature tracking is realized by calculating the three-dimensional coordinate changes of the same feature point in different frames. Finally, experiments are carried out on the endoscope image to complete the 3D surface reconstruction and feature tracking. Full article

Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve $C^1$ continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with $C^1$ continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R²) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia. Full article

In this work, we performed a systematic study on the effect of the geometry of pre-patterned templates and spin-coating conditions on the self-assembling process of colloidal nanospheres. To achieve this goal, large-scale templates, with different size and shape, were generated by direct laser-writer lithography over square millimetre areas. When deposited over patterned templates, the ordering dynamics of the self-assembled nanospheres exhibits an inverse trend with respect to that observed for the maximisation of the correlation length $\xi$ on a flat surface. Furthermore, the self-assembly process was found to be strongly dependent on the height (H) of the template sidewalls. In particular, we observed that, when H is 0.6 times the nanospheres diameter and spinning speed 2500 rpm, the formation of a confined and well ordered monolayer is promoted. To unveil the defects generation inside the templates, a systematic assessment of the directed self-assembly quality was performed by a novel method based on Delaunay triangulation. As a result of this study, we found that, in the best deposition conditions, the self-assembly process leads to well-ordered monolayer that extended for tens of micrometres within the linear templates, where 96.2% of them is aligned with the template sidewalls. Full article

LiDAR (Light Detection and Ranging)-based individual tree crown reconstruction is a challenge task due to the variable canopy morphologies and the penetrating properties of LiDAR to tree crown surfaces. Traditional methods, including LiDAR-derived rasterization, low-pass filtering smooth algorithm, and original triangular irregular network (TIN) model, have difficulties in balancing morphological accuracy and model smoothness. To address this issue, a scene-based TIN was generated with three steps based on the local scene principle. First, local Delaunay triangles were formed through connecting neighboring point sets. Second, key control points within each local Delaunay triangle, including steeple, inverted tip, ridge, saddle, and horseshoe shape control points, were extracted by analyzing multiple local scenes. These key points were derived to determine the fluctuations of forest canopies. Third, the scene-based TIN model was generated using the control points as nodes. Visual analysis indicates the new model can accurately reconstruct different canopy shapes with a relatively smooth surface, and statistical analysis of individual trees confirms that the overall error of the new model is smaller than others. Especially, the scene-based TIN derived raster reduced the average error to 0.18 m, with a standard deviation of 0.41, while the average errors of LiDAR-derived raster, low-pass filtered smooth raster, and original TIN derived raster have average errors of 0.96, 2.05, and 1.00 m, respectively. The local scene-based control point extraction also reduces data storage due to the elimination of redundant points, and furthermore the different point densities on different objects are beneficial for canopy segmentation. Full article

A crevasse is an important surface feature of a glacier. This study aims to detect crevasses from high-density airborne LiDAR point clouds. However, existing methods continue to suffer from the data holes within the crevasse region and the influence of the undulating non-crevasse glacier surfaces. Therefore, a bidirectional analysis method is proposed to robustly extract the crevasses from the point clouds, which utilizes their vertical and horizontal characteristics. First, crevasse points are separated from non-crevasse points using a hybrid-entity method, where the height difference and the nearly vertical characteristic of a crevasse sidewall are considered, to better distinguish the crevasses from the undulating non-crevasse glacier surfaces. Second, the crevasse regions/edges are adaptively delineated by a local statistical analysis method that is based on a novel feature of the Delaunay triangulation mesh of non-crevasse points in the horizontal plane. Last, the pseudo-crevasse points and regions are removed by a cross-analysis method. To test the performance of the proposed method, this study selected airborne LiDAR point clouds from two sites of Alaskan glaciers (i.e., Tyndall Glacier and Seward Glacier) as the experimental datasets. The results were verified by a comparison with the ground truth that was manually delineated. The proposed method achieved acceptable results: the recall, precision, and $F_1$ scores of both sites exceeded 94.00%. Moreover, a comparative experiment was carried out and the results confirmed that the proposed method achieved superior performance. Full article