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. In 1987 the idea was extended into the construction of neural networks corresponding to the beginning of the era of artificial intelligence, forming what is now called ‘Radial Basis Function Neural Networks (RBFNs)’. Ever since, RBFNs have been developed and applied to a wide variety of problems; approximation, interpolation, classification, prediction, in nowadays science, engineering, and medicine. This also includes numerically solving partial differential equations (PDEs), another essential branch of RBFNs under the name of the ‘Meshfree/Meshless’ method. Amongst many, the so-called ‘Multiquadric (MQ)’ is known as one of the mostly-used forms of RBFs and yet only a couple of its versions have been extensively studied. This study aims to extend the idea toward more general forms of MQ. At the same time, the key factor playing a very crucial role for MQ called ‘shape parameter’ (where selecting a reliable one remains an open problem until now) is also under investigation. The scheme was applied to tackle the problem of function recovery as well as an approximation of its derivatives using six forms of MQ with two choices of the variable shape parameter. The numerical results obtained in this study shall provide useful information on selecting both a suitable form of MQ and a reliable choice of MQ-shape for further applications in general.

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.

Under the architecture of a neural network, this work proposes and applies three multiquadric radial basis function (MQ-RBF) interpolation schemes; The Common Local Radial Basis Function Scheme (CLRBF), The Iterative Local Radial Basis Function Scheme (ILRBF), and The Radius Local Radial Basis Function Scheme (RLRBF). The schemes are designed to perform locally to overcome drawbacks normally encountered when using a global one. The famous Franke function in two dimensions is numerically tackled. It is revealed in this work that all three local methods outperform the traditional MQ interpolation in terms of both CPU-time and condition number, while the accuracy is overall acceptable, particularly when the number of nodes increases. This finding indicates their potential for dealing with bigger datasets and more complex problems.

With the growth of artificial intelligence technologies, the research on artificial neural networks (ANNs) has been paid much more attention. Radial basis function neural networks (RBFNs) are a type of ANNs that are referred to as models that replicate the role of biological neural networks. While their applications are growing in a wide range of areas, conventional forms of RBFs contain a highly problem-dependent shape parameter, making it not as convenient as one would expect. This work investigates the numerical effectiveness of RBFs containing no shapes, so they are referred to as ‘shapefree’, under the application of image reconstruction. Nine forms of shapefree RBFs have been gathered and implemented in conjunction with the RBFNs. Two popular images (known as Lena and Plane) are damaged in Salt-and-Pepper manner before being repaired by the networks using these shapefree RBFs. The overall performances are monitored based on error norm, CPU-time and storage, and condition number. This aims to provide useful information regarding choices of RBFs for future uses, to overcome the pain one faces from choosing a suitable value of shape parameter.

This work aims to numerically investigate the performance of the multiquadric (MQ) radial basis function in more general formats for image reconstruction applications. Desired features, i.e., accuracy and shape parameter sensitivity, of each form is numerically compared and explored. The famous Lena image is damaged using two levels of damage: 20% and 40%, in a Salt-and-Pepper manner. It has been discovered in this work that β=3/2 produces reasonably good accuracy and is least affected by the change in shape parameter while keeping both the CPU time and the condition number reasonably acceptable. This finding is promising and useful for further applications of MQ in more complex contexts.