Shifting Interpolation Kernel Toward Orthogonal Projection

Abstract

Orthogonal projection offers the optimal solution for many sampling-reconstruction problems in terms of the least square error. In the standard interpolation setting where the sampling is assumed to be ideal, however, the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited. In this paper, we propose a notion of shifting kernel toward the orthogonal projection. For a given interpolation kernel, we formulate optimization problems whose solutions lead to shifted interpolations that, while still being interpolatory, are closest to the orthogonal projection in the sense of the minimax regret. The quality of interpolation is evaluated in terms of the average approximation error over input shift. For the standard linear interpolation, we obtain several values of optimal shift, dependent on a priori information on input signals. For evaluation, we apply the new shifted linear interpolations to a Gaussian signal, an ECG signal, a speech signal, a two-dimensional signal, and three natural images. Significant improvements are observed over the standard and the 0.21-shifted linear interpolation proposed early.