Two kinds of parametric piecewise rational interpolation kernels for image magnification

TL;DR

This work addresses image magnification by constructing finite-support, symmetric $C^1$ rational interpolation kernels on $[-2,2]$ that satisfy the partition of unity. It introduces a new cubic/linear kernel $S_{3/1}(t)$ and five quartic/linear kernels $S_{4/1}(t)$, including subfamilies with zero and nonzero derivatives at $t=1$, all with explicit forms and rigorous continuity and unity properties. Numerical experiments show that certain quartic/linear kernels can outperform the classical cubic kernel in PSNR, SSIM, and FSIM on standard test images, underscoring the benefit of parameterized flexibility. The results offer practical, verifiable tools for image magnification and lay groundwork for higher-order rational kernels, with code and data publicly available.

Abstract

We study the constructions of piecewise rational interpolation kernels that are supported on the interval $[-2,2]$, and present one novel rational cubic/linear and five quartic/linear interpolation kernels. All proposed kernels are symmetric, $C^1$ continuous, and possess certain degrees of approximation order. The proposed quartic/linear interpolation kernels include the cubic and the cubic/linear interpolation kernel as special cases. Our numerical results show that one of the quartic/linear interpolation kernels can outperform the cubic interpolation kernel in terms of PSNR, SSIM, and FSIM.

Figures (3)

• Figure 1: Ten test images.
• Figure 2: Normalized PSNR values for the ten test images (Left $S^4_{4/1}$, Right $S^5_{4/1}$).
• Figure 3: Parameter distributions of $S_{4/1}^4$(First row) and $S_{4/1}^5$ (Second row) corresponding better PSNR values than those of three kinds of interpolation kernels.