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.
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.
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