Optimization of gridding algorithms for FFT by vector optimization

TL;DR

The paper reframes gridding kernel design for nonuniform FFTs through vector optimization, introducing the error-shape operator Λ that maps a kernel to an error profile. It proves Λ is continuous and shows how to obtain approximate Pareto-optimal kernels by restricting to a Slepian/PSWF basis, while enabling tailored error shapes via a target function η and scalarizing penalties. A calibration-evaluation pipeline using interior-point optimization yields kernels that outperform PSWF and state-of-the-art MIRT-NUFFT in specified regions of interest, sometimes by orders of magnitude in mean absolute error. The methodology promises ROI-focused accuracy customization and lays groundwork for multidimensional extensions and machine-learning-assisted η selection.

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.

Figures (10)

• Figure 1: The target profile error functions $\eta$ used for the tests.
• Figure 2: An example of the Fourier transform of a signal for Test 3. Notice that the meaningful frequencies concentrate on the areas in which the target error function $\eta$ is smaller.
• Figure 3: MAE errors without oversampling. In the first row are shown the MAEs of Test 1, in the second row those of Test 2, and in the third row those of Test 3. In all the Tests and for all $W = 1, 2, 3$, our methods clearly outperforms both the MIRT-NUFFT and the PSWF-based gridding algorithm.
• Figure 4: The MAE for Test 3, with $W = 4$ and $\gamma = 1$. Our methods performs slightly better than the PSWF-based near the frequencies $x_0 = -0.3$, $x_1 = 0.25$, and $x_2 = 0.4$, but the improvement is significant only for the latter.
• Figure 5: The MAE errors for the case with oversampling $\gamma = 2$. In the first row are shown the MAEs of Test 1, in the second row those of Test 2, and in the third row those of Test 3. In several cases our method outperforms both the PSWF-based one and the MIRT-NUFFT. However, in some cases MIRT-NUFFT seems to perform slightly better.

Theorems & Definitions (34)

• Remark 2.1.1: Computational cost
• Lemma 2.2.1
• proof
• Theorem 2.2.2
• Example 3.1.1
• Example 3.1.2: Example 1.51, jahn_vector_2011
• Definition 3.1.3
• Definition 3.1.4
• Example 3.1.5
• Proposition 3.1.6: Lemma 5.14, jahn_vector_2011