Hermite Radial Basis Function Research Articles

A local radial basis function-compact finite difference method for Sobolev equation arising from fluid dynamics

In this article, a new high-order, local meshless technique is presented for numerically solving multi-dimensional Sobolev equation arising from fluid dynamics. In the proposed method, Hermite radial basis function (RBF) interpolation technique is applied to approximate the operators of the model over local stencils. This leads to compact RBF generated finite difference (RBF-FD) formula, which provides a significant improvement in the accuracy and computational efficiency. In the first stage of the proposed method, the time discretization is performed by Crank–Nicolson finite difference scheme along with temporal Richardson extrapolation technique. In the second stage, the space dimension is discretized by applying the local radial basis function-compact finite difference (RBF-CFD) method. By performing some numerical simulations and comparing the results with existing methods, the high accuracy and computational efficiency of the proposed method are clearly demonstrated. The numerical results show that the presented method has fourth-order accuracy in both space and time dimensions. Finally, it can be concluded that the proposed method is a suitable alternative to the existing numerical techniques for the Sobolev model.

GeoPDNN 1.0: a semi-supervised deep learning neural network using pseudo-labels for three-dimensional shallow strata modelling and uncertainty analysis in urban areas from borehole data

Abstract. Borehole data are essential for conducting precise urban geological surveys and large-scale geological investigations. Traditionally, explicit modelling and implicit modelling have been the primary methods for visualizing borehole data and constructing 3D geological models. However, explicit modelling requires substantial manual labour, while implicit modelling faces problems related to uncertainty analysis. Recently, machine learning approaches have emerged as effective solutions for addressing these issues in 3D geological modelling. Nevertheless, the use of machine learning methods for constructing 3D geological models is often limited by insufficient training data. In this paper, we propose the semi-supervised deep learning using pseudo-labels (SDLP) algorithm to overcome the issue of insufficient training data. Specifically, we construct the pseudo-labels in the training dataset using the triangular irregular network (TIN) method. A 3D geological model is constructed using borehole data obtained from a real building engineering project in Shenyang, Liaoning Province, NE China. Then, we compare the results of the 3D geological model constructed based on SDLP with those constructed by a support vector machine (SVM) method and an implicit Hermite radial basis function (HRBF) modelling method. Compared to the 3D geological models constructed using the HRBF algorithm and the SVM algorithm, the 3D geological model constructed based on the SDLP algorithm better conforms to the sedimentation patterns of the region. The findings demonstrate that our proposed method effectively resolves the issues of insufficient training data when using machine learning methods and the inability to perform uncertainty analysis when using the implicit method. In conclusion, the semi-supervised deep learning method with pseudo-labelling proposed in this paper provides a solution for 3D geological modelling in engineering project areas with borehole data.

Forward inclusion functions for ray-tracing implicit surfaces

Using Lipschitz bounds as linear inclusion functions, we show that both Lipschitz-based ray-tracing and bottom-up inclusion functions can be used together in the same framework. We propose asymmetrical forward inclusion functions that are exact at the query point and can better encode the function’s behavior on a given interval; therefore well suited for iterative processing. We show how to derive the linear and quadratic versions of these inclusion functions either by bounding the derivatives or building bottom-up inclusion functions and combining these two. We show our results on density fields defined from point primitives with compactly supported kernels and Gaussian kernels as well as Hermite radial basis functions. We demonstrate that our method provides noticeable improvement for grazing rays and transparent rendering.

Implicit Surface Reconstruction via RBF Interpolation: A Review

Background: Implicit surface is a kind of surface modeling tool, which is widely used in point cloud reconstruction, deformation and fusion due to its advantages of good smoothness and Boolean operation. The most typical method is the surface reconstruction with Radial Basis Functions (RBF) under normal constraints. RBF has become one of the main methods of point cloud fitting because it has a strong mathematical foundation, an advantage of computation simplicity, and the ability of processing nonuniform points. Objective: Techniques and patents of implicit surface reconstruction interpolation with RBF are surveyed. Theory, algorithm, and application are discussed to provide a comprehensive summary for implicit surface reconstruction in RBF and Hermite Radial Basis Functions (HRBF) interpolation. Methods: RBF implicit surface reconstruction interpolation can be divided into RBF interpolation under the constraints of points and HRBF interpolation under the constraints of points and corresponding normals. Results: A total of 125 articles were reviewed, in which more than 30% were related to RBF in the last decade. The continuity properties and application fields of the popular global supported radial basis functions and compactly supported radial basis functions are analyzed. Different methods of RBF and HRBF implicit surface reconstruction are evaluated, and the challenges of these methods are discussed. Conclusion: In future work, implicit surface reconstruction via RBF and HRBF should be further studied in fitting accuracy, computation speed, and other fundamental problems. In addition, it is a more challenging but valuable research direction to construct a new RBF with both compact support and improved fitting accuracy.

HRBF-Fusion: Accurate 3D Reconstruction from RGB-D Data Using On-the-fly Implicits

Reconstruction of high-fidelity 3D objects or scenes is a fundamental research problem. Recent advances in RGB-D fusion have demonstrated the potential of producing 3D models from consumer-level RGB-D cameras. However, due to the discrete nature and limited resolution of their surface representations (e.g., point or voxel based), existing approaches suffer from the accumulation of errors in camera tracking and distortion in the reconstruction, which leads to an unsatisfactory 3D reconstruction. In this article, we present a method using on-the-fly implicits of Hermite Radial Basis Functions (HRBFs) as a continuous surface representation for camera tracking in an existing RGB-D fusion framework. Furthermore, curvature estimation and confidence evaluation are coherently derived from the inherent surface properties of the on-the-fly HRBF implicits, which are devoted to a data fusion with better quality. We argue that our continuous but on-the-fly surface representation can effectively mitigate the impact of noise with its robustness and constrain the reconstruction with inherent surface smoothness when being compared with discrete representations. Experimental results on various real-world and synthetic datasets demonstrate that our HRBF-fusion outperforms the state-of-the-art approaches in terms of tracking robustness and reconstruction accuracy.

A finite difference Hermite RBF‐WENO scheme for hyperbolic conservation laws

AbstractOne of the best ideas for controlling the Gibbs oscillations in hyperbolic conservation laws is to apply weighted essentially non‐oscillatory (WENO) schemes. The traditional WENO and Hermite WENO (HWENO) schemes are based on the polynomial interpolation. In this research, a non‐polynomial HENO and HWENO finite difference scheme in order to enhance the local accuracy and convergence is proposed. The idea of the reconstruction in HWENO schemes is derived from traditional WENO schemes, with the difference that in traditional WENO schemes only function values are evolved and used, while in HWENO schemes both the values of the function and the first derivative of the function are evolved in time and used. The Hermite radial basis functions (RBFs) interpolation offers the flexibility to control the local error by optimizing the free parameter. The numerical results demonstrate that the developed Hermite RBF‐ENO and RBF‐WENO schemes enhance the local accuracy and give sharper solution profile than the Hermite ENO and WENO schemes based on the polynomial interpolation.

Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

Three-dimensional geological modeling and spatial analysis from geotechnical borehole data using an implicit surface and marching tetrahedra algorithm

Three-dimensional (3D) geological modeling based on borehole data usually requires many manual operations, and the modeling process remains very complicated and time consuming. This paper presents an automatic implicit 3D geological modeling and visualization method that can be applied to urban geotechnical drilling data. First, by analyzing the general characteristics of these types of data, the implicit Hermite radial basis function (HRBF) surface is used to simulate geological interfaces according to lithological layers/units in boreholes. Second, the marching tetrahedra (MT) visualization method is optimized with new data structures and split operations to extract explicit mesh models from implicit geological surfaces. Then, the optimized tetrahedron used in the field of 3D urban drilling modeling is described in detail. Finally, the local geotechnical engineering data of two Chinese cities are used to verify the modeling algorithm, and several kinds of spatial analyses are carried out, including visualization, data querying, accuracy assessment and multiple cutting methods. The results show that the proposed urban subsurface 3D geological modeling method is effective in visualizing the structural shape, topological relationship and formation properties of strata that can be used to predict underground conditions and reduce construction risks during urban geotechnical engineering projects.

Explicit-implicit-integrated 3-D geological modelling approach: A case study of the Xianyan Demolition Volcano (Fujian, China)

Explicit interactive modelling and implicit interpolation modelling are the two classic three-dimensional (3-D) modelling approaches for reconstructing 3-D geological structures from sparse data. The limitations of using a single method appear when performing complex regional modelling containing various types of geological structures. However, due to the complication of spatial geometry computation, the integration and coupling of explicit and implicit modelling approaches and modelling results remain difficult and unresolved. In this paper, we propose an explicit-implicit-integrated 3-D geological modelling approach for the geometric fusion of different types of complex geological structure models. First, suitable methods were chosen to model different types of structures. For example, a Hermite Radial Basis Function (HRBF)-based implicit method was used to model general strata, faults, and folds from geological boundaries and attitudes in a regional geological map; explicit approaches, such as the skinning method and the free-form surface, were used to model local detailed structures, such as Quaternary deposits, monoclinic strata and multi-layer volcanic edifices, from section-constrained data. Finally, the explicit models were resampled and converted to implicit models and were coupled with the implicit models, leading to the integrated regional modelling of various types of geological structures. The validity of the proposed method was tested using the geological map of the Xianyan Demolition Volcano in Fujian. The results demonstrate that the proposed explicit-implicit-integrated 3-D geological modelling approach is efficient and effective, and this approach provides a probable solution for coupling and integrating multiple modelling methods and geological models for an area containing complex geological structures. The 3-D model established using this method can directly reflect the geomorphic features and volcanic passage facies of the Xianyan Demolition Volcano. Moreover, the simulation of radioactive fractures around the center of the volcano can also help to accurately determine the volcano type.

Wrinkle synthesis for cloth mesh with hermite radial basis functions

Designing virtual clothing has received much attention recently due to the increasing need for synthesizing realistically dressed digital humans for various applications. Wrinkles are an important appearance feature of the garment in virtual environments. Generating such wrinkles currently requires a computationally expensive simulation or specialized design skills. We present a new geometric method for adding believable wrinkles to existing virtual clothing. The key novelty of our work is to use Hermite Radial Based Functions (HRBF) to reconstruct an approximation of the clothing mesh. Our method takes advantage of angles between gradients of adjacent HRBF scalar fields to trace spatially and temporally coherent wrinkle curves on a cloth mesh. We generate plausible wrinkle geometry using implicit deformers following the wrinkle curves. Our method can be used as a post-processing step on any garment simulation system. The results obtained demonstrate that our approach produces believable wrinkles and it is very satisfactory in terms of performance. The method is fully automatic and provides a large set of parameters that can be modified by the user in order to control the appearance of the resulting wrinkles in real-time and thus obtain the desired result.

A radial basis function — Hermite finite difference approach to tackle cash-or-nothing and asset-or-nothing options

A novel local meshfree method is proposed for simulating cash-or-nothing and asset-or-nothing options, at which the initial condition is discontinuous. The novelty of the scheme is in the use of Hermite radial basis function (RBF) interpolation on set of local nodes. We also construct the RBF generated Hermite finite difference (HFD) formulas and subsequently obtain their associated differentiation matrices (DMs). The developed accuracy equipped with a θ-shift algorithm for smoothing the discontinuous initial conditions are illustrated with a series of financial tests.

Implicit 3D Modeling of Ore Body from Geological Boreholes Data Using Hermite Radial Basis Functions

Modeling ore body in 3D is the basis of digital intelligent mining. However, most existing three-dimensional mining software uses the contour approach that requires too much man–machine interaction and difficult partial updating. Moreover, accounting for uncertainty and low geometric quality picking is very difficult in the direct contour approach. Therefore, an implicit modeling approach to automatically build the three-dimensional model for ore body by means of spatial interpolation directly based on the geological borehole data with Hermite radial basis function (HRBF) algorithm as the core is proposed. Furthermore, in order to solve the problems of weak continuity of models due to the long-distance original boreholes as well as the boundary-point normal solution error, the densification of original borehole data with the virtual borehole as well as the calculation of point-cloud normal direction based on the adjacent hole-drilling method is proposed. The verification of two mine engineering projects and comparison with the explicit modeling results show that this approach could realize the automatic building of three-dimensional models for the ore body with high geometric quality, timely update and accurate results.

Hermite Radial Basis Collocation Method for Unsaturated Soil Water Movement Equation

Due to the nonlinear diffusion term, it is hard to use the collocation method to solve the unsaturated soil water movement equation directly. In this paper, a nonmesh Hermite collocation method with radial basis functions was proposed to solve the nonlinear unsaturated soil water movement equation with the Neumann boundary condition. By preprocessing the nonlinear diffusion term and using the Hermite radial basis function to deal with the Neumann boundary, the phenomenon that the collocation method cannot be used directly is avoided. The numerical results of unsaturated soil moisture movement with Neumann boundary conditions on the regular and nonregular regions show that the new method improved the accuracy significantly, which can be used to solve the low precision problem for the traditional collocation method when simulating the Neumann boundary condition problem. Moreover, the effectiveness and reliability of the algorithm are proved by the one-dimensional and two-dimensional engineering problem of soil water infiltration in arid area. It can be applied to engineering problems.

A 3D Multi-scale geology modeling method for tunnel engineering risk assessment

A method for 3D multi-scale geology modeling is proposed for assessing tunnel engineering risk. Risk assessments have varying requirements at various different construction stages, therefore the proposed model can be modified for different scales. The regional scale 3D geological model, based on regional field surveys of geological boundary and attitude, is proposed for conducting preliminary evaluations with limited geological sampling data and the Hermite radial basis function (HRBF). The project scale geological model, based on updating and refining the regional scale model with dense drilling and geological prediction data, couples the geological body model for surrounding rock and the tunnel model. This project scale model is appropriate for performing pre-evaluations of tunnel construction. Once construction has begun, continuous rock structure surface information from the newly exposed tunnel face is incorporated into the outcrop scale geological model. It is based on a Monte-Carlo algorithm for conducting dynamic evaluations during construction. These modeling methods are applied to a case study, railway tunnel engineering of the Yuelongmen tunnel in China.

Rapid Interactive and Intuitive Segmentation of 3D Medical Images Using Radial Basis Function Interpolation

Segmentation is one of the most important parts of medical image analysis. Manual segmentation is very cumbersome, time-consuming, and prone to inter-observer variability. Fully automatic segmentation approaches require a large amount of labeled training data and may fail in difficult or abnormal cases. In this work, we propose a new method for 2D segmentation of individual slices and 3D interpolation of the segmented slices. The Smart Brush functionality quickly segments the region of interest in a few 2D slices. Given these annotated slices, our adapted formulation of Hermite radial basis functions reconstructs the 3D surface. Effective interactions with less number of equations accelerate the performance and, therefore, a real-time and an intuitive, interactive segmentation of 3D objects can be supported effectively. The proposed method is evaluated on 12 clinical 3D magnetic resonance imaging data sets and are compared to gold standard annotations of the left ventricle from a clinical expert. The automatic evaluation of the 2D Smart Brush resulted in an average Dice coefficient of 0.88 ± 0.09 for the individual slices. For the 3D interpolation using Hermite radial basis functions, an average Dice coefficient of 0.94 ± 0.02 is achieved.

Section-constrained local geological interface dynamic updating method based on the HRBF surface

Boundaries, attitudes and sections are the most common data acquired from regional field geological surveys, and they are used for three-dimensional (3D) geological modelling. However, constructing topologically consistent 3D geological models from rapid and automatic regional modelling with convenient local modifications remains unresolved. In previous works, the Hermite radial basis function (HRBF) surface was introduced for the simulation of geological interfaces from geological boundaries and attitudes, which allows 3D geological models to be automatically extracted from the modelling area by the interfaces. However, the reasonability and accuracy of non-supervised subsurface modelling is limited without further modifications generated through explanations and analyses performed by geology experts. In this paper, we provide flexible and convenient manual interactive manipulation tools for geologists to sketch constraint lines, and these tools may help geologists transform and apply their expert knowledge to the models. In the modified modelling workflow, the geological sections were treated as auxiliary constraints to construct more reasonable 3D geological models. The geometric characteristics of section lines were abstracted to coordinates and normal vectors, and along with the transformed coordinates and vectors from boundaries and attitudes, these characteristics were adopted to co-calculate the implicit geological surface function parameters of the HRBF equations and form constrained geological interfaces from topographic (boundaries and attitudes) and subsurface data (sketched sections). Based on this new modelling method, a prototype system was developed, in which the section lines could be imported from databases or interactively sketched, and the models could be immediately updated after the new constraints were added. Experimental comparisons showed that all boundary, attitude and section data are well represented in the constrained models, which are consistent with expert explanations and help improve the quality of the models.

Estimation of differential quantities using Hermite RBF interpolation

Curvature is an important geometric property in computer graphics that provides information about the character of object surfaces. The exact curvature can only be calculated for a limited set of surface descriptions. Most of the time, we deal with triangles, point sets or some other discrete representation of the surface. For those, curvature can only be estimated. However, surfaces can be fitted by some kind of interpolation function and from it, curvature can be calculated directly. This paper proposes a method for curvature estimation and normal vector re-estimation based on surface fitting using Hermite Radial Basis Function interpolation. Hermite variation uses not only control points, but normal vectors at those points as well. This leads to a better and more robust interpolation than if only control points are used. Once the interpolant is obtained, the curvature and other possible properties can be directly computed using known approaches. The proposed algorithm was tested on several explicit and implicit functions, and it outperforms current state-of-the-art methods if exact normals are available. For normals calculated directly from a triangle mesh, the proposed algorithm works on par with existing state-of-the-art methods.

A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces

We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in $\mathbb{R}^d$. The novelty of the method is in the approximation of the Laplace--Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace--Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computa...

IMPLICIT THREE-DIMENSIONAL GEO-MODELLING BASED ON HRBF SURFACE

Abstract. Three-dimensional (3D) geological models are important representations of the results of regional geological surveys. However, the process of constructing 3D geological models from two-dimensional (2D) geological elements remains difficult and time-consuming. This paper proposes a method of migrating from 2D elements to 3D models. First, the geological interfaces were constructed using the Hermite Radial Basis Function (HRBF) to interpolate the boundaries and attitude data. Then, the subsurface geological bodies were extracted from the spatial map area using the Boolean method between the HRBF surface and the fundamental body. Finally, the top surfaces of the geological bodies were constructed by coupling the geological boundaries to digital elevation models. Based on this workflow, a prototype system was developed, and typical geological structures (e.g., folds, faults, and strata) were simulated. Geological modes were constructed through this workflow based on realistic regional geological survey data. For extended applications in 3D modelling of other kinds of geo-objects, mining ore body models and urban geotechnical engineering stratum models were constructed by this method from drill-hole data. The model construction process was rapid, and the resulting models accorded with the constraints of the original data.

A closed-form formulation of HRBF-based surface reconstruction by approximate solution

The Hermite radial basis functions (HRBFs) implicits have been used to reconstruct surfaces from scattered Hermite data points. In this work, we propose a closed-form formulation to construct HRBF-based implicits by a quasi-solution to approximate the exact one. A scheme is developed to automatically adjust the support sizes of basis functions to hold the error bound of a quasi-solution. Our method can generate an implicit function from positions and normals of scattered points without taking any global operation. Robust and efficient reconstructions are observed in our experimental tests on real data captured from a variety of scenes.