On Multiquadric Shape Determining Strategies in Image Reconstruction Applications: A Comparative Study

Abstract

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.

Generalized Multiquadric Neural Networks in Image Reconstruction

“…the randomly chosen pixels are replaced either by 0 or 255, as shown in Figure 2. Other attempts under this context can also be found in [7,[10][11][12]. The root mean square error (RMSE), CPU-time and storage, and the mean condition number ( ( )…”

“…The multivariate property has made RBF attractive in a wide range of applications and one of the most popular RBF choices is the so-called multiquadric (MQ-RBF). The original and nonnormalized formula under this type of RBFs is have been receiving attention where the solutions are known to be highly dependent on H , [4][5][6][7]. This challenge has come to our attention as whether there is a E that is less affected by the choice of H .…”

ON A LOCAL $k$-POINT MULTIQUADRIC NEURAL NETWORK IN FUNCTION RECOVERY APPLICATIONS

“…It can be clearly seen from the figure that different RBFs with different shape parameters produce different surfaces, indicating directly the differences in the final results quality. Recent attempts to provide useful information on how to choose a good shape for a particular problem can be found in [6], [7]. Over decades, RBFs have been receiving a great amount of attention from scientists and engineers.…”