Optimization of gridding algorithms for FFT by vector optimization

TL;DR

This paper introduces a vector optimization framework for designing optimal gridding kernels in FFT applications, leading to significant accuracy improvements over traditional methods like PSWF and MIRT-NUFFT.

Contribution

It redefines kernel optimality through vector optimization, providing a rigorous method to construct tailored kernels with superior error performance.

Findings

Proposed kernels outperform PSWF and MIRT-NUFFT in specific regions.

Achieved orders-of-magnitude improvements in mean absolute errors.

Validated the effectiveness of VO-based kernel design through experiments.

Abstract

The Fast Fourier Transform (FFT) is widely used in applications such as MRI, CT, and interferometry; however, because of its dependence on uniformly sampled data, it requires the use of gridding techniques for practical implementation. The performance of these algorithms strongly depends on the choice of the gridding kernel, with the first prolate spheroidal wave function (PSWF) regarded as optimal. This work redefines kernel optimality through the lens of vector optimization (VO), introducing a rigorous framework that characterizes optimal kernels as Pareto-efficient solutions of an error shape operator. We establish the continuity of such operator, study the existence of solutions, and propose a novel methodology to construct kernels tailored to a desired target error function. The approach is implemented numerically via interior-point optimization. Comparative experiments demonstrate that the proposed kernels outperform both the PSWF and the state-of-the-art methods (MIRT-NUFFT) in specific regions of interest, achieving orders-of-magnitude improvements in mean absolute errors. These results confirm the potential of VO-based kernel design to provide customized accuracy profiles aligned with application-specific requirements. Future research will extend this framework to multidimensional cases and relative error minimization, with potential integration of machine learning for adaptive target error selection.