After being introduced to approximate two-dimensional geographical surfaces in 1971, the multivariate radial basis functions (RBFs) have been receiving a great amount of attention from scientists and engineers. Over decades, RBFs have been applied to a wide variety of problems. Approximation, interpolation, classification, prediction, and neural networks are inevitable in nowadays science, engineering, and medicine. Moreover, numerically solving partial differential equations (PDEs) is also a powerful branch of RBFs under the name of the ‘Meshfree/Meshless’ method. Amongst many, the so-called ‘Generalized Multiquadric (GMQ)’ is known as one of the most used forms of RBFs. It is of (ɛ 2 + r 2) β form, where r = ║x-x Θ║2 for x, x Θ ∈ ℝ n represents the distance function. The key factor playing a very crucial role for MQ, or other forms of RBFs, is the so-called ‘shape parameter ɛ’ where selecting a good one remains an open problem until now. This paper focuses on measuring the numerical effectiveness of various choices of ɛ proposed in literature when used in image reconstruction problems. Condition number of the interpolation matrix, CPU-time and storage, and accuracy are common criteria being utilized. The results of the work shall provide useful information on selecting a ‘suitable and reliable choice of MQ-shape’ for further applications in general.

The purpose of this study was to propose a new interpolation scheme that was designed to remedy the shortcomings encountered in two popular interpolation methods; the triangle-based blending (TBB) method and the inverse distance weighted (IDW) method. At the same time, the proposed method combines their desirable aspects, which are the local nature and non-use of quadratic surface construction, making it comparatively less time-consuming and more independent of the global effect. Because of these properties, the new scheme was named the ‘Modified Local Distance-Weighted (MLD)’ method, and it was tested in detail with different sizes of datasets. The datasets involved in this investigation were of two types; uniformly and non-uniformly distributed. The performances were carefully monitored and assessed via several criteria; accuracy, sensitivity to parameters, CPU-time, storage requirement and ease of implementation. For comparative purposes, three alternative interpolation methods were simultaneously carried out, i.e. TBB, IDW and Radial Basis Function-Based (RBF) method. The investigation clearly revealed promising aspects of the proposed scheme where a good quality of results was anticipated, and CPU-time and storage were seen to have been significantly reduced. The research strongly indicates the benefits of the proposed method for larger sized datasets for real practical scientific and engineering uses in the future.