Compactly Supported Radial Basis Functions Research Articles

LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions

Implicit 3D surface reconstruction of an object from its partial and noisy 3D point cloud scan is the classical geometry processing and 3D computer vision problem. In the literature, various 3D shape representations have been developed, differing in memory efficiency and shape retrieval effectiveness, such as volumetric, parametric, and implicit surfaces. Radial basis functions provide memory-efficient parameterization of the implicit surface. However, we show that training a neural network using the mean squared error between the ground-truth implicit surface and the linear basis-based implicit surfaces does not converge to the global solution. In this work, we propose locally supported compact radial basis functions for a linear representation of the implicit surface. This representation enables us to generate 3D shapes with arbitrary topologies at any resolution due to their continuous nature. We then propose a neural network architecture for learning the linear implicit shape representation of the 3D surface of an object. We learn linear implicit shapes within a supervised learning framework using ground truth Signed-Distance Field (SDF) data for guidance. The classical strategies face difficulties in finding linear implicit shapes from a given 3D point cloud due to numerical issues (requires solving inverse of a large matrix) in basis and query point selection. The proposed approach achieves better Chamfer distance and comparable F-score than the state-of-the-art approach on the benchmark dataset. We also show the effectiveness of the proposed approach by using it for the 3D shape completion task.

Probabilistic Modeling for Image Registration Using Radial Basis Functions: Application to Cardiac Motion Estimation.

Cardiovascular diseases (CVDs) are the leading cause of death, affecting the cardiac dynamics over the cardiac cycle. Estimation of cardiac motion plays an essential role in many medical clinical tasks. This article proposes a probabilistic framework for image registration using compact support radial basis functions (CSRBFs) to estimate cardiac motion. A variational inference-based generative model with convolutional neural networks (CNNs) is proposed to learn the probabilistic coefficients of CSRBFs used in image deformation. We designed two networks to estimate the deformation coefficients of CSRBFs: the first one solves the spatial transformation using given control points, and the second one models the transformation using drifting control points. The given-point-based network estimates the probabilistic coefficients of control points. In contrast, the drifting-point-based model predicts the probabilistic coefficients and spatial distribution of control points simultaneously. To regularize these coefficients, we derive the bending energy (BE) in the variational bound by defining the covariance of coefficients. The proposed framework has been evaluated on the cardiac motion estimation and the calculation of the myocardial strain. In the experiments, 1409 slice pairs of end-diastolic (ED) and end-systolic (ES) phase in 4-D cardiac magnetic resonance (MR) images selected from three public datasets are employed to evaluate our networks. The experimental results show that our framework outperforms the state-of-the-art registration methods concerning the deformation smoothness and registration accuracy.

Shape and Generalized Topology Optimization of Curved Grid Stiffeners Through the Level Set-Based Density Method

Abstract A shape and generalized topology optimization method based on the level set-based density method is proposed to design the curved grid stiffeners. The overall layout of the stiffeners is described by combining many single stiffeners, and each single stiffener is described by a level set function parameterized by using the compactly supported radial basis functions (CS-RBFs). The curvilinear path of each stiffener is described by the zero iso-contour of each level set function, and the width of each stiffener is described by applying an interval projection to each level set function. The combination operation that is similar to the Boolean operation “union” is achieved by using the p-norm method. The expansion coefficients of CS-RBFs are taken as part of the design variables of the optimization, and they are responsible for changing the shape of curved stiffeners. A topology design variable is assigned to each single stiffener, and it is responsible for changing the existence of single stiffeners. The proposed method is validated through several numerical examples, and the results demonstrate that the shape and topology of stiffeners can be effectively changed during the optimization.

Optimization with manufacturing constraints for composite laminates reinforced by curvilinear fibers through a parametric level set method

This paper presents the design optimization with manufacturing constraints for composite laminates reinforced by curvilinear fibers by using a parametric level set method. In each layer of the composite laminate, a level set function is defined by using the compactly supported radial basis functions (CS-RBFs), and the iso-contours of each level set function represent the fiber paths in one layer of the laminate. By this means, the fiber orientation at an arbitrary point in a layer can be computed by the orientation of the tangent vector of the iso-contour passing through this point. To avoid structure defects (such as gap/overlap, wrinkling and delamination) that would arise if the fiber paths are not parallel or their curvature is too large, the gradient norm and contour curvature radius of the level set functions are considered as constraints in the design optimization. The p-norm method is used to aggregate the constraints to improve the optimization efficiency. The coefficients of CS-RBFs are considered as design variables. Several numerical examples of compliance minimization and eigen-frequency maximization are investigated to verify the proposed method.

P‐4.2: An Image Pre‐distortion Algorithm for Near‐eye Display

With the development of near-eye display technology, optical distortion is often introduced into the design of near-eye optical system to meet people's requirements for high definition and large field of view. However, the larger the field of view, the greater the distortion of the projected image. In order to correct distorted images, this paper proposes an image pre-distortion algorithm based on nonlinear fitting and forward mapping interpolation. The radial basis function fitting algorithm is used to get a more precise mapping relationship between the original image and the pre-distorted image. The compactly supported radial basis function (CSRBF) interpolation algorithm based on forward mapping is used for image gray reconstruction to obtain better correction effect of image distortion. The experimental results show that the fitting error of the radial basis function fitting algorithm can be reduced to less than 1 pixel, reaching sub-pixel level. The pre-distortion algorithm has a good correction effect on the non-rotational symmetry distortion produced by the near-eye optical system, which makes the display image clear and complete.

A partition of unity level set method with moving knot CS-RBFs for optimizing variable stiffness composites

A partition of unity level set method with moving knot Compactly Supported Radial Basis Functions (CS-RBFs) is proposed for optimizing variable stiffness composite structures. The iso-contours of a level set function are utilized to represent the curved fiber paths, and the tangent vector of the iso-contour defines the orientation of fiber. The level set function of the full design domain is constructed according to the Partition of Unity (POU) method by a set of local level set functions defined on an array of overlapping subdomains, and they are constructed by using the CS-RBFs. The positions of knots are iteratively changed during the optimization to improve the performance of composite structures. Several examples of compliance minimization are presented.

Buckling optimization of variable-stiffness composites with multiple cutouts considering manufacturing constraints

The progress of the automatic fiber placement (AFP) technique makes it viable to fabricate variable-stiffness (VS) composites with curvilinear fiber paths and promotes the rise of the VS composite design. However, traditional representation methods of fiber paths are not flexible for composites with cutouts and are also inconvenient to cooperate with the gradient-based solver. In this study, a parameterized angle variable scheme (PAVS) by the compactly supported radial basis functions (CS-RBFs) is proposed to represent the continuous fiber paths so that manufacturing constraints including minimal turning radius and gap/overlap can be conveniently related to the curl operation and divergence of the fiber angle vector field. A level set representation method of fiber paths is also provided for comparison. Two buckling optimization frameworks of the VS composite considering manufacturing constraints are then proposed based on these two representation methods. Numerical examples of two square plates with cutouts are conducted. Different numbers of support points, support radii, manufacturing constraints, and initial designs are investigated. Results indicate that the proposed PAVS needs few support points and a small support radius. It can also set initial design easily and describe the manufacturing constraints directly and quantitively.

A novel CS-RBFs-based parameterization scheme for the optimization design of curvilinear variable-stiffness composites with manufacturing constraints

Recent advancements in additive manufacturing technology have made it convenient to design and fabricate composites with curvilinear fibers and the desired performance. For the optimization design of fiber-reinforced composite, two key issues are the description of spatially varying fiber angles and the mathematical representation of the manufacturing constraints. A novel parameterization scheme based on the compactly supported radial basis functions (CS-RBFs) is established to describe the spatially varying fiber angle vector field. Its main characters, including the nonnegativity, partition of unity, and differentiability, inherently ensure the fiber continuity, the bounds of the design variables and make it compatible with the gradient-based optimization design algorithm. Besides, a mathematical formulation is developed to construct the relationships between two common manufacturing constraints (minimum turning radius and gap/overlap) and the curl and divergence of the vector field. By the proposed parameterization scheme and p-norm aggregation function, these manufacturing constraints can be easily described in all elements and added to the optimization. Several numerical examples are conducted to investigate the influence of the varying support points, the radius of the CS-RBFs, initial design, load condition, norm parameter and the excitation frequency on the final design. The optimized results demonstrate the effectiveness of the proposed method.

Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions

The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.

Two Interpolation Matrix Triangularization Methods for Parametric Level Set-Based Structural Topology Optimization

As an implementation form of basis function, interpolation matrices (IMs) have a crucial impact on parametric level set method (PLSM)-based structural topology optimization (STO). However, there are few studies on compressing IM into triangular matrix (TM) with less storage and computation. Algorithm using LU decomposition and Algorithm using innovative asymmetric basis functions that transform the IMs of compactly supported radial basis functions (CSRBFs) into highly sparse TMs are proposed. Theoretical derivation and numerical experiments show that they effectively improve computational efficiency.

Defect identification in heat transfer problems using boundary data

The identification of internal defects has been widely focused and employed in the field of nondestructive evaluation (NDE), whereas its application in inverse heat transfer is often restricted by connectedness. Thus, in this paper, a novel identification method of multiple defects based on the parametric level set method (PLSM) and the Method of Moving Asymptotes (MMA) is proposed, where the connectedness limitation is released. In PLSM, the compactly supported radial basis functions (CSRBFs) are utilized to decouple the material related level set function, which endows the PLSM to be capable of forming high curvature portions and capturing local details of the defects. After that, the evolution of guessed level set can be tracked by updating the time-dependent expansion coefficients. Then, a least-square problem is defined to find the optimal coefficients that correspond to the actual defects. To obtain the controllable convergence and stabilize the solving of the ill-posed optimization problem, the MMA is employed to minimize the objective function. Furthermore, no requirement for the mesh reconstruction after the usage of ersatz material model, one also renders the sensitivity prone to compute. The efficiency and accuracy of the proposed method are further validated in several numerical examples by discussing the identification of single and multiple defects.

Galerkin-based finite strain analysis with enriched radial basis interpolation

We present an enriched radial basis function (RBF) interpolation, which is combined with a symmetric Galerkin method to solve finite strain problems. Specifically, Wendland compact support radial basis functions (RBF) are enriched with a quadratic polynomial, whose parameters are condensed-out during the shape-function construction phase. A total Lagrangian technique is adopted and for finite strain plasticity the formalism with the Mandel stress is adopted. These recent developments are especially convenient from the implementation perspective, as RBF formulations for finite strain plasticity have been limited by interpolation updating. We also note that, although tetrahedra are adopted for quadrature in the undeformed configuration, mesh deformation is of no consequence for the results. Four benchmark tests are successfully solved, with extensive convergence studies and showing clear advantages with respect to finite-element in terms of robustness and with respect to classical Element-Free Galerkin (EFG) formulations in terms of efficiency.

A parameterized level set method for structural topology optimization based on reaction diffusion equation and fuzzy PID control algorithm

<abstract> <p>We propose a parameterized level set method (PLSM) for structural topology optimization based on reaction diffusion equation (RDE) and fuzzy PID control algorithm. By using the proposed method, the structural compliance minimization problem under volume constraints is studied. In this work, the RDE is used as the evolution equation of level set function, and the topological derivative of the material domain is used as the reaction term of the RDE to drive the evolution of level set function, which has little dependence on the initial design domain, and can generate holes in the material domain; the compactly supported radial basis function (CS-RBF) is used to interpolate the level set function and modify the RDE, which can improve the computational efficiency, and keep the boundary smooth in the optimization process. Meanwhile, the fuzzy PID control algorithm is used to deal with the volume constraints, so that the convergence process of the structure volume is relatively stable. Furthermore, the proposed method is applied to 3D structural topology optimization. Several typical numerical examples are provided to demonstrate the feasibility and effectiveness of this method.</p> </abstract>

Implicit Progressive-Iterative Algorithm of Curves and Surfaces with Compactly Supported Radial Basis Functions

<p indent=0mm>To reconstruct curves and surfaces robustly from scattered data, an implicit progressive-iterative algorithm with compactly supported radial basis functions based on the variational quasi-interpolation method is proposed. Firstly, the non-zero constraint of the implicit function is constructed using normal vectors at the given points, an iterative scheme for calculating coefficients of the implicit function is developed and its convergence is discussed. Secondly, by introducing an acceleration factor, the implicit progressive-iteration algorithm is sped up, and the convergence is analyzed. Finally, the accelerated algorithm is modified to decrease the space and time complexity. Numerical experiments show that the algorithm is effective for curve and surface reconstruction, and it also achieves good results for reconstructing from data with missing samples, non-uniform distribution, and noises. Moreover, it is simple to implement and easy to process in parallel.

Comparison of RBFs Interpolation for Load Transfer on Wing Structure

Load transfer is of importance in simulation of aeroelasticity by coupling with CFD and FEM. Different RBFs (Radial Basis Function) with variant parameters are tested in load transfer and their results are compared in this paper. Result comparison shows that different RBFs with variant parameter will lead to different results. More approximate load on structure can be gained by MQB and TPS functions. The compact support RBFs should be used with wisely chosen support radius to gain approximate results.

A well-conditioned and efficient implementation of dual reciprocity method for Poisson equation

&lt;abstract&gt;&lt;p&gt;One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.&lt;/p&gt;&lt;/abstract&gt;

Radial basis Taylor series method and its applications

Purpose The study aims to present a new meshless method based on the Taylor series expansion. The compact supported radial basis functions (CSRBFs) are very attractive, can be considered as a numerical tool for the engineering problems and used to obtain the trial solution and its derivatives without differentiating the basis functions for a meshless method. A meshless based on the CSRBF and Taylor series method has been developed for the solutions of engineering problems. Design/methodology/approach This paper is devoted to present a truly meshless method which is called a radial basis Taylor series method (RBTSM) based on the CSRBFs and Taylor series expansion (TSE). The basis function and its derivatives are obtained without differentiating CSRBFs. Findings The RBTSM does not involve differentiation of the approximated function. This property allows us to use a wide range of CSRBF and weight functions including the constant one. By using a different number of terms in the TSE, the global convergence properties of the RBTSM can be improved. The global convergence properties are satisfied by the RBTSM. The computed results based on the RBTSM shows excellent agreement with results given in the open literature. The RBTSM can provide satisfactory results even with the problem domains which have curved boundaries and irregularly distributed nodes. Originality/value The CSRBFs have been widely used for the construction of the basic function in the meshless methods. However, the derivative of the basis function is obtained with the differentiation of the CSRBF. In the RBTSM, the derivatives of the basis function are obtained by using the TSE without differentiating the CSRBF.

Non-probabilistic Reliability-based Topology Optimization (NRBTO) Scheme for Continuum Structures Based on the parameterized Level-Set method and Interval Mathematics

In this paper, a study on non-probabilistic reliability-based topology optimization (NRBTO) scheme for continuum structures based on the parameterized Level-Set method (PLSM) is conducted, in which the unknown-but-bounded (UBB) uncertainties of material and external loads are taken into account simultaneously. By interpolating the level set function (LSF) with the compactly supported radial basis functions (CSRBFs), the partial differential equation (PDE) is transformed into an ordinary differential equation (ODE). Based on the interval-set model, the displacement constraint is transformed into the non-probabilistic reliability-based scheme and the reliability is evaluated by the optimization feature distance (OFD). Moreover, the interval parametric vertex approach, the concept of shape derivative and the adjoint vector method are employed to obtain the sensitivity between the optimization model and the pseudo time to obtain the evolution velocity field of LSF. By utilizing the optimization criterion (OC) method, the optimization problem can be solved iteratively. To verify the validity and applicability of the proposed NRBTO method, three examples are presented, and numerical results show that taking the UBB uncertainties effects into account during the topology optimization may have a significant influence on the final structural configurations.

A meshfree formulation for large deformation analysis of flexoelectric structures accounting for the surface effects

In this work, we present a compactly supported radial basis function (CSRBF) based meshfree method to analyse geometrically nonlinear flexoelectric nanostructures considering surface effects. Flexoelectricity is the polarization of dielectric materials due to the gradient of strain, which is different from piezoelectricity in which polarization is dependent linearly on strain. The surface effects gain prominence as the size of the structure tends to nanoscale and so their consideration is inevitable when flexoelectric nanostructures are analysed. First, the proposed meshfree formulation is validated and the influence of nonlinear strain terms on the energy conversion ability of flexoelectric beams made of a non-piezoelectric material like cubic Strontium Titanate is studied. Subsequently, the meshfree formulation for nonlinear flexoelectricity is extended to include nonlinear surface effects. It is determined that the surface effects can have notable influence on the output flexoelectric voltage of nano-sized cantilever structures in the nonlinear regime.

Subpixel image registration regularised by and norms

In this study, the authors propose a subpixel image registration framework that detects and matches feature points. Rigid and nonrigid registration models are employed to solve the problem of subpixel image registration problem. A rigid registration model based on the l 2 norm is proposed to regularise the rotation coefficients using the indicator function to estimate the rigid transformation parameters. The latter estimation simplified is made easy by the reduction in the rigid transformation from two dimensions to one dimension. Furthermore, a non-rigid registration model based on the l 1 and l 2 norms is proposed to estimate the elastic coefficients of the compact support radial basis functions. Due to the linear representation of the transformation function, the rigid and nonrigid subpixel image registration models can be solved efficiently using the fast iterative shrinkage-thresholding algorithm. Experiments on a demosaicing data set, the ocean of remote sensing data set, a brain data set and the fundus image registration data set show that the proposed rigid and non-rigid registration models can accurately perform subpixel image registration.