H-infinity Optimal Approximation for Causal Spline Interpolation

TL;DR

This paper develops a causal, robust spline interpolation filter using H-infinity optimization, providing a closed-form solution for cubic splines and numerical methods for higher orders, suitable for real-time applications.

Contribution

It introduces a causal H-infinity optimal approximation method for spline interpolation, including a closed-form solution for cubic splines and numerical solutions for higher-order splines.

Findings

Closed-form H-infinity solution for cubic splines

Numerical methods for higher-order spline filters

Design of robust FIR filters via LMI

Abstract

In this paper, we give a causal solution to the problem of spline interpolation using H-infinity optimal approximation. Generally speaking, spline interpolation requires filtering the whole sampled data, the past and the future, to reconstruct the inter-sample values. This leads to non-causality of the filter, and this becomes a critical issue for real-time applications. Our objective here is to derive a causal system which approximates spline interpolation by H-infinity optimization for the filter. The advantage of H-infinity optimization is that it can address uncertainty in the input signals to be interpolated in design, and hence the optimized system has robustness property against signal uncertainty. We give a closed-form solution to the H-infinity optimization in the case of the cubic splines. For higher-order splines, the optimal filter can be effectively solved by a numerical computation. We also show that the optimal FIR (Finite Impulse Response) filter can be designed by an LMI (Linear Matrix Inequality), which can also be effectively solved numerically. A design example is presented to illustrate the result.