Scattered Data Interpolation Research Articles

An Improved [formula omitted] rational Bernstein–Bézier triangular patch for scattered data interpolation

Scattered data interpolation is important in sciences, engineering, and medical-based problems. The degree smoothness attained is either C1 or C2. However, based on our simulations, we found that, Hussain and Hussain (2011) scheme is not producing C1 everywhere. Motivated by this, in this study we will propose an improve sufficient condition for the C1 on each adjacent triangle. Our scheme is guaranteed to produce C1 everywhere. To support our claim, we have implemented both schemes and perform some numerical comparison including error measurement. From the results, our proposed scheme is producing C1 interpolating surface while the interpolating surface produced by Hussain and Hussain (2011) is not C1 everywhere. Furthermore, our proposed scheme give smaller error compared with Hussain and Hussain (2011) scheme. All numerical and graphical results are presented by using MATLAB programming.

Cubic q-Bézier Triangular Patch for Scattered Data Interpolation and Its Algorithm

This paper presents an approach to scattered data interpolation using q-Bézier triangular patches via an efficient algorithm. While existing studies have formed q-Bézier triangular patches through convex combination, their application to scattered data interpolation has not been previously explored. Therefore, this study aims to extend the use of q-Bézier triangular patches to scattered data interpolation by achieving C1 continuity throughout the data points. We test the proposed scheme using both established data points and real-life engineering problems. We compared the performance of the proposed interpolation scheme with a well-known existing scheme by varying the q parameter. The comparison was based on visualization and error analysis. Numerical and graphical results were generated using MATLAB. The findings indicate that the proposed scheme outperforms the existing scheme, demonstrating a higher coefficient of determination (R2), smaller root mean square error (RMSE), and faster central processing unit (CPU) time. These results highlight the potential of the proposed q-Bézier triangular patches scheme for more accurate and reliable scattered data interpolation via the proposed algorithm.

A local meshless method for the numerical solution of multi-term time-fractional generalized Oldroyd-B fluid model

This work presents an accurate and efficient method, for solving a two dimensional time-fractional Oldroyd-B fluid model. The proposed method couples the Laplace transform (LT) with a radial basis functions based local meshless method (LRBFM). The suggested numerical scheme first uses the LT which transform the given equation to an elliptic equation in LT space, and then it utilizes the LRBFM to solve transformed equation in LT space, and then the solution is converted back into the time domain via the improved Talbot's scheme. The local meshless methods are widely recognized for scattered data interpolation and for solving PDEs in complex shaped domains. The adaptability, simplicity, and ease of use are features that have led to the popularity of local meshless methods. The local meshless methods are easy and straightforward, they only requires to solve linear system of equations. The main objective of using the LT is to avoid the computation of costly convolution integral in time-fractional derivative and the effect of time stepping on accuracy and stability of numerical solution. The stability and the convergence of the proposed numerical scheme are discussed. Further, the Ulam-Hyers (UH) stability of the proposed model is discussed. The accuracy and efficiency of the suggested numerical approach have been demonstrated using numerical experiments on five different domains with regular nodes distribution.

Generalized multilevel B-spline approximation for scattered data interpolation in image processing

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%).

Scattered data interpolation on the 2-dimensional surface through Shepard-like technique

In the current paper, the problem of interpolation of scattered data on two-dimensional surfaces is considered by proposing an extension to the Shepard method and its modified version to surfaces. Each proposed operator is a linear combination of basis functions whose coefficients are the values of the function or its Taylor of first-order expansions at the interpolation points using both functional and derivative data. Numerical tests are given to show the interpolation performance, where several numerical results show a good approximation accuracy of the proposed operator.

A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation

A Scalable Optimization Approach for the Multilinear System Arising from Scattered Data Interpolation

Cubic Bézier-Like Triangular Patches for Rainfall Scattered Data Interpolation and Visualization

Scattered data interpolation plays an important role in computer graphics and scientific visualization. This method can be used to interpolate any regular or irregular data sets. For instance, rainfall distribution is an example of an irregular data that usually appear in statistics. In this study, cubic Bézier-Like triangular patches defined on triangular domain is used to interpolate rainfall data sets. From graphical results, we obtain a very smooth surface. Therefore, the proposed scheme can be used to interpolate and provide a good visualization to the given irregular data sets.

Determination of compactly supported functions in shift-invariant space by single-angle Radon samples

While traditionally the computerized tomography of a function f∈L2(R2) depends on the samples of its Radon transform at multiple angles, the real-time imaging sometimes requires the reconstruction of f by the samples of its Radon transform Rpf at a single angle θ, where p=(cos⁡θ,sin⁡θ) is the direction vector. This naturally leads to the question of identifying those functions that can be determined by their Radon samples at a single angle θ. The shift-invariant space V(φ,Z2) generated by φ is a type of function space that has been widely considered in many fields including wavelet analysis and signal processing. In this paper we examine the single-angle reconstruction problem for compactly supported functions f∈V(φ,Z2). The central issue for the problem is to identify the eligible p and sampling set Xp⊆R such that f can be determined by its single-angle Radon (w.r.t. p) samples at Xp. For the general generator φ, we address the eligible p for the two cases: (1) φ being nonvanishing (∫R2φ(x)dx≠0) and (2) being vanishing (∫R2φ(x)dx=0). We prove that eligible Xp exists for general φ. In particular, Xp can be explicitly constructed if φ∈C1(R2). Positive definite functions form an important class of functions that have been widely applied in scattered data interpolation. For the case that φ is positive definite, the corresponding single-angle problem in SIS V(φ,Z2) is addressed such that Xp can be constructed easily. Besides using the samples of the single-angle Radon transform, another common feature for our recovery results is that the number of the required samples is minimum.

An Extension of Multiquadric Method Based on Trend Analysis for Surface Construction

The representation of spatial discrete data is crucial for the data analysis of meteorology, agriculture, geological exploration and other fields. Among various methods for scattered data interpolation, the multiquadric (MQ) method is the most favorable in the field of surface construction. However, the classical MQ method has accuracy concern on the boundary and is time consuming for large data set. This study proposes an algorithm by integrating the MQ method with trend surface analysis. In which, the low-order polynomial trend surface equation is firstly used to model the overall trend. Then, the MQ equation is applied to fit the residual surface after removal the trend from the data. Our implementation can eliminate the distortion in data missing areas by the classical MQ method, and the modeling efficiency can be improved significantly since the local MQ method divide the residual surface into a group of subsurfaces. The accuracy and efficiency of the proposed algorithm are validated on a synthetic model. The performance of the developed algorithm is further examined on the elevation data collected in Tibet and the seabed of a strait in Norway. The results show that with an equivalent resolution, the developed algorithm can be much more efficient than the classical MQ method and well-developed Kriging method.

Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations

Radial basis function (RBF) is a basis function suitable for scattered data interpolation and high dimensional function interpolation, wherein the independent shape parameters have a direct impact on the accuracy of calculation results. Research in this field is mostly concerned with the shape parameter selection strategy based on the premise of global distribution or block regional distribution. In this paper, a shape parameter selection strategy is proposed, which is used for the local RBF collocation method (LRBF) for solving partial differential equations. It overcomes many limitations of the traditional methods applied to LRBF. In this strategy, a set of twin matrices similar to the interpolation matrix are constructed to evaluate the error of the model. In addition, the penalty term contained in the twin matrix is used to relax the influence of the far end region on the target node. Since the objective problem is nonlinear, a particle swarm optimization algorithm (PSO) is employed to minimize the training objective and adjust the shape parameters of the basis function at each iteration. Extensive numerical results showed the effectiveness of the error estimation strategy, by providing a good shape parameter and better solution accuracy. At the end of the paper, the generality of the shape parameter optimization framework based on this strategy is discussed through three examples.

Refined error estimates for Green kernel-based interpolation

Positive-definite kernels are probably best known for their application in many problems driven by scattered data interpolation. Fasshauer and Ye introduced constructive theory of reproducing kernels of generalized Sobolev spaces in 2011 to provide insight into the types of functions being well approximated by these kernels on a set of scattered points. In this approach, the reproducing kernel is viewed as the Green kernel of a suitable differential operator with some boundary conditions. Sampling inequalities and the minimum norm property in reproducing kernel Hilbert spaces (RKHSs) bring out the standard error bound; however, this estimate is valid only when the target functions belong to the native spaces of the Green kernels. In this paper we provide Sobolev-type error estimates for cases in which the target functions are smoother than functions in the native space. The results are useful and effective for the error analysis of Green kernel-based interpolation problems.

Polynomial mapped bases: theory and applications

Abstract In this paper, we collect the basic theory and the most important applications of a novel technique that has shown to be suitable for scattered data interpolation, quadrature, bio-imaging reconstruction. The method relies on polynomial mapped bases allowing, for instance, to incorporate data or function discontinuities in a suitable mapping function. The new technique substantially mitigates the Runge’s and Gibbs effects.

On the Numerical Approximation of Three-Dimensional Time Fractional Convection-Diffusion Equations

In this paper, we present an efficient method for the numerical investigation of three-dimensional non-integer-order convection-diffusion equation (CDE) based on radial basis functions (RBFs) in localized form and Laplace transform (LT). In our numerical scheme, first we transform the given problem into Laplace space using Laplace transform. Then, the local radial basis function (LRBF) method is employed to approximate the solution of the transformed problem. Finally, we represent the solution as an integral along a smooth curve in the complex left half plane. The integral is then evaluated to high accuracy by a quadrature rule. The Laplace transform is used to avoid the classical time marching procedure. The radial basis functions are important tools for scattered data interpolation and for solving partial differential equations (PDEs) of integer and non-integer order. The LRBF and global radial basis function (GRBF) are used to produce sparse collocation matrices which resolve the issue of the sensitivity of shape parameter and ill conditioning of system matrices and reduce the computational cost. The application of Laplace transformation often leads to the solution in complex plane which cannot be generally inverted. In this work, improved Talbot’s method is utilized which is an efficient method for the numerical inversion of Laplace transform. The stability and convergence of the method are discussed. Two test problems are considered to validate the numerical scheme. The numerical results highlight the efficiency and accuracy of the proposed method.

Characterisation of freeform, structured surfaces in T-spline spaces and its applications

In advanced manufacturing, surface topographical designs with deterministic freeform and embedded structures have proven to contain effective, additive functionalities. These surfaces need to be geometrically characterised regarding the designed form and structures. However, this is problematic since existing characterisation techniques such as polynomial form removal, Gaussian/spline/wavelet filtration, field-based statistical parameterisation, spectral and fractal analysis do not provide satisfying results. In this paper, we, therefore, propose to characterise the complex surfaces in T-spline spaces, i.e. basis spline spaces along with T-junctions, using an efficient T-spline fitting algorithm. Several case studies show that the proposed method is compatible and has notable potentials for the challenging characterisation tasks, including non-Euclidean freeform removal, edge-reserving filtration with multiscale analysis, scattered data interpolation and smoothing, and smart large-data downsampling or compression.

Scattered data interpolation: Strictly positive definite radial basis/cardinal functions

Radial basis functions are a simple and accurate method for multivariate interpolation but the ill-conditioning situation due to their interpolation matrices, discourages an acceptable approximation for both large number of nodes or flat function interpolation. In current work, a new type of basis named well-conditioned RBFs (WRBFs) was created by adding the strictly positive definite Radial Basis Functions (SPD-RBFs) to cardinal functions, was introduced and applied for interpolation. To light up this manner, two classes of global cardinal functions were used for adding by SPD-RBFs. These cardinal functions are Shepard functions and Quasi-cardinal RBFs. Theoretical and numerical analyses prove that utilizing WRBFs has some advantages such as eliminating the ill-conditioning system which has arisen from the global positive definite RBFs interpolation, improving the convergence of pure cardinal functions interpolation and also working better than pure RBFs for small shape parameters.

A Coupled RBF Method for the Solution of Elastostatic Problems

Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.

Scattered Data Interpolation Using Cubic Trigonometric B閦ier Triangular Patch

This paper discusses scattered data interpolation using cubic trigonometric Bézier triangular patches with continuity everywhere. We derive the condition on each adjacent triangle. On each triangular patch, we employ convex combination method between three local schemes. The final interpolant with the rational corrected scheme is suitable for regular and irregular scattered data sets. We tested the proposed scheme with 36,65, and 100 data points for some well-known test functions. The scheme is also applied to interpolate the data for the electric potential. We compared the performance between our proposed method and existing scattered data interpolation schemes such as Powell–Sabin (PS) and Clough–Tocher (CT) by measuring the maximum error, root mean square error (RMSE) and coefficient of determination (). From the results obtained, our proposed method is competent with cubic Bézier, cubic Ball, PS and CT triangles splitting schemes to interpolate scattered data surface. This is very significant since PS and CT requires that each triangle be splitting into several micro triangles.

A Numerical Investigation on a Hybrid-Parameterless Radial Basis Function Applied with a Meshless Method

A Hybrid RBF has recently been proposed and tested with some scattered data interpolation problems and the results have appeared promising whereas the appearance of the shape parameter remains a difficulty when deploying. This work, therefore, focuses on three objectives; firstly, it is aimed to extend the use of the newly proposed-RBF to application of RBF-collocation method. Secondly, realizing the burden attributed to the lack of information on choosing an optimum shape parameter, the hybrid RBF is then modified where the shape parameter is no longer included. Thirdly, it is to investigate its application/implementation towards solving PDEs particularly those in both linear and non-linear form. It has been found in this work that the new RBF of this HyBrid form with no parameter can well be a good candidate and truly deserves further study with more complex problems.

Fast GPU 3D diffeomorphic image registration

3D image registration is one of the most fundamental and computationally expensive operations in medical image analysis. Here, we present a mixed-precision, Gauss–Newton–Krylov solver for diffeomorphic registration of two images. Our work extends the publicly available CLAIRE library to GPU architectures. Despite the importance of image registration, only a few implementations of large deformation diffeomorphic registration packages support GPUs. Our contributions are new algorithms to significantly reduce the run time of the two main computational kernels in CLAIRE: calculation of derivatives and scattered-data interpolation. We deploy (i) highly-optimized, mixed-precision GPU-kernels for the evaluation of scattered-data interpolation, (ii) replace Fast-Fourier-Transform (FFT)-based first-order derivatives with optimized 8th-order finite differences, and (iii) compare with state-of-the-art CPU and GPU implementations. As a highlight, we demonstrate that we can register 2563 clinical images in less than 6 s on a single NVIDIA Tesla V100. This amounts to over 20× speed-up over the current version of CLAIRE and over 30× speed-up over existing GPU implementations.

A variational flexible multibody formulation for partitioned fluid–structure interaction: Application to bat-inspired drones and unmanned air-vehicles

We present a three-dimensional (3D) partitioned aeroelastic formulation for a flexible multibody system interacting with incompressible turbulent fluid flow. While the incompressible Navier–Stokes system is discretized using a stabilized Petrov–Galerkin procedure, the multibody structural system consists of a generic interaction of multiple components such as rigid body, beams and flexible thin shells along with various types of joints and connections among them. A co-rotational framework is utilized for the category of small strain problems where the displacement of the body is decomposed into a rigid body rotation and a small strain component. This assumption simplifies the structural equations and allows for the incorporation of multiple bodies (rigid as well as flexible) in the system. The displacement and rotation constraints at the joints are imposed by a Lagrange multiplier method. The equilibrium conditions at the fluid–structure interface are satisfied by the transfer of tractions and structural displacements via the radial basis function approach, a scattered data interpolation technique, which is globally conservative. For the coupled stability in low structure-to-fluid mass ratio regimes, a nonlinear iterative force correction scheme is employed in the partitioned staggered predictor–corrector scheme. The convergence and generality of the radial basis function mapping are analyzed by carrying out systematic error analysis of the transfer of fluid traction across the non-matching fluid–structure interface where a third-order of convergence is observed. The proposed aeroelastic framework is then validated by considering a flow across a flexible pitching plate configuration with serration at the trailing edge. Finally, we demonstrate the flow across a flexible flapping wing of a bat modeling the bone fingers as beams and the flexible membrane as thin shells in the multibody system along with the joints.