Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n +1 "failed to produce acceptable results". In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions' values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).