Frequency-domain weighted-least-squares design of quadratic interpolators

Abstract

Existing finite-support interpolators are derived from continuities in the time-domain. In this study, the authors optimally design a quadratic interpolator using two second-degree piecewise polynomials in the frequency-domain. The optimal coefficients of the piecewise polynomials are found by minimising the weighted least-squares error between the ideal and actual frequency responses of the quadratic interpolator subject to a few constraints. Adjusting the weighting functions in different frequency bands can yield accurate frequency response in a specified passband and even can ignore `don`t care` bands so that various quadratic interpolators can be designed for interpolating various discrete signals containing different frequency components. One-dimensional and two-dimensional examples have shown that the quadratic interpolator can achieve much higher interpolation accuracy than the existing interpolators for wide-band signals, and various images have been tested to verify that the quadratic interpolator can achieve comparable interpolation accuracy as the Catmull-Rom cubic for narrow-band signals (images), but the computational complexity can be reduced to about 70%. Therefore both narrow-band and wide-band signals can be interpolated with high accuracy.