Wendland Functions Research Articles

Generalised Wendland functions for the sphere

In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from d-dimensional Euclidean space to the (d − 1)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain 3F2 hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where d is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when d is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.

RBF Moment computation and meshless cubature on general polygonal regions

We obtain analytical formulas for the computation of integrals (moments) of all the most common Radial Basis Functions (usually shortened as RBFs) on polygonal regions that may be nonconvex or even multiply connected. With RBFs of finite regularity, such as Thin-Plate Splines, Wendland functions and Radial Powers, our Matlab codes, based on standard linear solvers for the corresponding moment matching systems, provide cubature rules that are reasonably accurate and numerically stable.

Chicken Sertoli cells culture based detection of chemical damage of the environment

We obtain analytical formulas for the computation of integrals (moments) of all the most common Radial Basis Functions (usually shortened as RBFs) on polygonal regions that may be nonconvex or even multiply connected. With RBFs of finite regularity, such as Thin-Plate Splines, Wendland functions and Radial Powers, our Matlab codes, based on standard linear solvers for the corresponding moment matching systems, provide cubature rules that are reasonably accurate and numerically stable.

Estimation and prediction of Gaussian processes using generalized Cauchy covariance model under fixed domain asymptotics

We study estimation and prediction of Gaussian processes with covariance model belonging to the generalized Cauchy (GC) family, under fixed domain asymptotics. Gaussian processes with this kind of covariance function provide separate characterization of fractal dimension and long range dependence, an appealing feature in many physical, biological or geological systems. The results of the paper are classified into three parts. In the first part, we characterize the equivalence of two Gaussian measures with GC covariance functions. Then we provide sufficient conditions for the equivalence of two Gaussian measures with Matern (MT) and GC covariance functions and two Gaussian measures with Generalized Wendland (GW) and GC covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GC covariance model, under fixed domain asymptotics. The last part focuses on optimal prediction with GC model and specifically, we give conditions for asymptotic efficiency prediction and asymptotically correct estimation of mean square error using a misspecified GC, MT or GW model. Our findings are illustrated through a simulation study: the first compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter of the GC model with the given asymptotic distribution. We then compare the finite-sample behavior of the prediction and its associated mean square error when the true model is GC and the prediction is performed using the true model and a misspecified GW model.

Topology analysis of global and local RBF transformations for image registration

For elastic registration, topology preservation is a necessary condition to be satisfied, especially for landmark-based image registration. In this paper, we focus on the topology preservation properties of two different families of radial basis functions (RBFs), known as Gneiting and Matérn functions. Firstly, we consider a small number of landmarks, dealing with the cases of one, two and four landmark matching; in all these situations we analyze topology preservation and compare numerical results with those obtained by Wendland functions. Secondly, we discuss the registration properties of these two families of functions, when we have a larger number of landmarks. Finally, we analyze the behavior of Gneiting and Matérn functions, considering some test examples known in the literature and a real application.

Multiphase simulation of liquid jet breakup using smoothed particle hydrodynamics

This paper deals with numerical modeling of two-phase liquid jet breakup using the smoothed particle hydrodynamics (SPH) method. Simulation of multiphase flows involving fluids with a high-density ratio causes large pressure gradients at the interface and subsequently divergence of numerical solutions. A modified procedure extended by Monaghan and Rafiee is employed to stabilize the sharp interface between the fluids. Various test cases such as Rayleigh–Taylor instability, two-phase still water and air bubble rising in water have been conducted, by which the capability of accurately capturing the physics of multiphase flows is verified. The results of these simulations are in a good agreement with analytical and previous numerical solutions. Finally, the simulation of the breakup process of liquid jet into surrounding air is accomplished. The whole numerical solutions are accomplished for both Wendland and cubic spline kernel functions and Wendland kernel function gave more accurate results. Length of liquid breakup in Rayleigh regime is calculated for various flow conditions such as different Reynolds and Weber numbers. The results of breakup length demonstrate in satisfactory agreement with the experimental correlation. Finally, impinging distance and breakup length of a simple multijet setup are analyzed. The two-jet multijet has a longer breakup length than a three-jet one.

$C^\infty$ Compactly Supported and Positive Definite Radial Kernels

A family of $C^\infty$ compactly supported radial kernels is presented. These positive definite kernels can be generated numerically using convolutions of compactly supported radial functions. An alternative proof that shows the infinitely smooth limit of the well-known Wendland functions is not a suitable $C^\infty$ compactly supported kernel is also presented. Numerical experiments are provided to demonstrate the accuracy of approximations of the proposed kernels, including comparisons with Gaussians and Wendland radial basis functions of finite smoothness.

Local interpolation schemes for landmark-based image registration: A comparison

In this paper we focus, from a mathematical point of view, on properties and performances of some local interpolation schemes for landmark-based image registration. Precisely, we consider modified Shepard's interpolants, Wendland's functions, and Lobachevsky splines. They are quite unlike each other, but all of them are compactly supported and enjoy interesting theoretical and computational properties. In particular, we point out some unusual forms of the considered functions. Finally, detailed numerical comparisons are given, considering also Gaussians and thin plate splines, which are really globally supported but widely used in applications.

Reliable approximation of separatrix manifolds in competition models with safety niches

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey–predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants.

Closed form representations and properties of the generalised Wendland functions

In this paper we investigate the generalisation of Wendland’s compactly supported radial basis functions to the case where the smoothness parameter is not assumed to be a positive integer or half-integer and the parameter ℓ, which is chosen to ensure positive definiteness, need not take on the minimal value. We derive sufficient and necessary conditions for the generalised Wendland functions to be positive definite and deduce the native spaces that they generate. We also provide closed form representations for the generalised Wendland functions in the case when the smoothness parameter is an integer and where the parameter ℓ is any suitable value that ensures positive definiteness, as well as closed form representations for the Fourier transform when the smoothness parameter is a positive integer or half-integer.

Strictly and non-strictly positive definite functions on spheres

Isotropic positive definite functions on spheres play important roles in spatial statistics, where they occur as the correlation functions of homogeneous random fields and star-shaped random particles. In approximation theory, strictly positive definite functions serve as radial basis functions for interpolating scattered data on spherical domains. We review characterizations of positive definite functions on spheres in terms of Gegenbauer expansions and apply them to dimension walks, where monotonicity properties of the Gegenbauer coefficients guarantee positive definiteness in higher dimensions. Subject to a natural support condition, isotropic positive definite functions on the Euclidean space $\mathbb{R}^3$, such as Askey's and Wendland's functions, allow for the direct substitution of the Euclidean distance by the great circle distance on a one-, two- or three-dimensional sphere, as opposed to the traditional approach, where the distances are transformed into each other. Completely monotone functions are positive definite on spheres of any dimension and provide rich parametric classes of such functions, including members of the powered exponential, Mat\'{e}rn, generalized Cauchy and Dagum families. The sine power family permits a continuous parameterization of the roughness of the sample paths of a Gaussian process. A collection of research problems provides challenges for future work in mathematical analysis, probability theory and spatial statistics.

Wendland functions with increasing smoothness converge to a Gaussian

The Wendland functions are a class of compactly supported radial basis functions with a user-specified smoothness parameter. We prove that with an appropriate rescaling of the variables, both the original and the “missing” Wendland functions converge uniformly to a Gaussian as the smoothness parameter approaches infinity. We also explore the convergence numerically with Wendland functions of different smoothness.

L p error estimates for approximation by Sobolev splines and Wendland functions on ℝ d

It is known that a Green's function-type condition may be used to derive rates for approximation by radial basis functions (RBFs). In this paper, we introduce a method for obtaining rates for approximation by functions which can be convolved with a finite Borel measure to form a Green's function. Following a description of the method, rates will be found for two classes of RBFs. Specifically, rates will be found for the Sobolev splines, which are Green's functions, and the perturbation technique will then be employed to determine rates for approximation by Wendland functions.

Closed form representations for a class of compactly supported radial basis functions

In this paper we re-examine Wendland's strategy for the construction of compactly supported positive definite radial basis functions. We acknowledge that this strategy can be modified to capture a much larger range of functions, including the so-called missing Wendland functions which have been the subject of a recent paper by Schaback (Adv Comput Math 34:67---81, 2011). Our approach is to focus on a general integral representation of such functions and we will show how a careful evaluation of this integral leads to new closed form expressions for both Wendland's original functions and the missing ones. The resulting expressions are easy to code and so provide the potential user with a quick way of accessing a desired example for a given application.

Topology preservation evaluation of compact-support radial basis functions for image registration

Many Radial Basis Function (RBF)-based transformations are used to model the deformations in image registration, and they have different topology preservation properties. This paper compares analytically and experimentally the topology preservation performance of compact-support thin-plate Spline (CSTPS), locally constrained cosine (Cos), Wendland, Gaussian, Buhmann and Wu functions in landmark-based image registration. In addition, the topology preservation characteristics of thin-plate Spline (TPS) and elastic body Spline (EBS)-based transformations are compared for global-support deformations. The comparative results show that, for local deformation CSTPS and Buhmann preserve topology better than others. The Cos and Gaussian functions could easily produce topology violations for relatively dense-landmark matching. For global-support transformations, CSTPS, Wendland ψ 3,1, Buhmann, Wu and TPS outperformed others because they preserve topology better. The Cos, Gaussian and EBS functions perform poorly because folds and tears of the deformation surface occur easily. With very large support, CSTPS produces similar results as TPS, and Wendland ψ 3,1 has similar performance with Wu functions. Also, Cos and Gaussian performed similarly in this case. In the experiments, these theoretical results are evaluated extensively using transformations on random point sets, artificial images, and medical images.

The missing Wendland functions

The Wendland radial basis functions (Wendland, Adv Comput Math 4:389–396, 1995) are piecewise polynomial compactly supported reproducing kernels in Hilbert spaces which are norm–equivalent to Sobolev spaces. But they only cover the Sobolev spaces 1 $$\label{eqstartrep} H^{d/2+k+1/2}({\mathbf{R}}^d),\;k\in {\mathbf{N}} $$ and leave out the integer order spaces in even dimensions. We derive the missing Wendland functions working for half-integer k and even dimensions, reproducing integer-order Sobolev spaces in even dimensions, but they turn out to have two additional non-polynomial terms: a logarithm and a square root. To give these functions a solid mathematical foundation, a generalized version of the “dimension walk” is applied. While the classical dimension walk proceeds in steps of two space dimensions taking single derivatives, the new one proceeds in steps of single dimensions and uses “halved” derivatives of fractional calculus.

TU‐C‐ValA‐05: Assessment of a Model‐Based Deformable Image Registration Approach for Radiotherapy Planning

Purpose: To assess the accuracy of a surface‐based deformable image registration strategy as a function of the elasticity model for the integration of multi‐modality imaging, image‐guided radiation therapy, and quantification of geometrical change during and following therapy.Method and Materials: A surface‐model based deformable image registration system has been developed that enables quantitative description of geometrical change in multi‐modal images. Based on the deformation of organ surfaces represented by triangular surface meshes, a volumetric deformation field is derived using different volumetric elasticity models (Thin‐Plate Splines, Wendland functions, Elastic Body Splines) as alternatives to finite‐element modeling. Results: The system was demonstrated on five liver cancer patients, ten prostate cancer patients, thorax in five healthy volunteers, and abdomen in five healthy volunteers. The accuracy of the system was assessed by tracking visible fiducials (bronchial bifurcations in the lung, vessel bifurcations in the liver, implanted gold markers in the prostate). The maximum displacements for lung, liver and prostate were 5.3 cm, 3.2 cm, and 1.8 cm respectively. The largest registration error (direction, mean ± standard deviation) for lung, liver and prostate were (inferior‐superior, −0.21 ± 0.38 cm), (anterior‐posterior, −0.09 ± 0.34 cm), and (left‐right, 0.04 ± 0.38 cm) respectively, which was within the image resolution regardless of the deformation model. The computation time (2.7 GHz Intel Xeon) was on the order of seconds (e.g. 10 seconds for two prostate data sets), and image deformation results could be viewed at interactive speed (less than 1 second for 512×512 voxels). Conclusion: Surface‐based deformable image registration enables the quantification of geometrical change in normal tissue and tumor with acceptable accuracy and speed.

Matrix-valued radial basis functions: stability estimates and applications

Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980’s. Several applications require that certain physical properties are satisfied by the interpolant, for example, being divergence-free in case of incompressible data. In this paper we consider a class of customized (e.g., divergence-free) RBFs that are matrix-valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier–Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution.