发表机构
School of Mathematical Sciences, Fudan University; Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University（复旦大学数学科学学院; 复旦大学当代应用数学上海市重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出将二维II型非均匀离散傅里叶变换矩阵分解为核矩阵与DFT矩阵的乘积，利用分层半可分离近似实现直接求逆，离线复杂度O(M+N^{3/2}log^3 N)，在线复杂度O(M+N log^3 N)，并可作为迭代法预条件子。
AI 中文摘要
本文提出了一种二维II型非均匀离散傅里叶变换（NUDFT）的直接求逆方法。NUDFT矩阵$A$被分解为$A = G F$，其中$G$可表示为核矩阵，$F$是二维DFT矩阵。我们证明$G$可以用分层半可分离（HSS）矩阵近似，并给出了HSS秩的估计。然后，利用HSS矩阵的最小二乘求解器和二维逆快速傅里叶变换，可以高效求解逆NUDFT问题。我们的算法离线复杂度为$O\bigl(M+ N^{3 / 2} \log^{3} N\bigr)$，其中$M$和$N$分别是NUDFT矩阵的行和列大小。一旦直接求解器构建完成，它可以以在线复杂度$O\bigl(M+ N \log^{3} N\bigr)$应用于向量。所提方法可作为迭代法的预条件子，特别是当采样点分布在网格上导致$A$病态时。数值结果展示了求逆方法的扩展性能，并证明了其作为预条件子的效率和鲁棒性。
英文摘要
Reconstructing images on Cartesian grids from nonuniform Fourier samples is a basic computational task in Fourier reconstruction methods for computed tomography. This paper proposes a direct solver for the 2D type-II nonuniform discrete Fourier transform~(NUDFT) based on the face-splitting product of hierarchically semiseparable~(HSS) matrices. We define the face-splitting product of two HSS trees and prove that the face-splitting product of two HSS matrices admits an HSS representation on the resulting tree. The proof is constructive, providing explicit formulas for the HSS generators and bounds on the off-diagonal ranks at each level. For the NUDFT, a Fourier change of basis expresses the transformed 2D matrix as a face-splitting product of two transformed 1D NUDFT matrices. This identity yields an algebraic construction of a 2D HSS approximation from two 1D HSS approximations, for which we establish a Frobenius-norm approximation error bound. Combining this construction with recompression, URV factorization, and a two-dimensional inverse fast Fourier transform gives a direct solver for the associated least-squares problem. For fixed accuracy, square frequency grids, and balanced HSS trees with fixed leaf column sizes, the solver has an offline complexity of $O(M \log^{2} N + N^{3 / 2} \log^{3} N)$ and an online complexity of $O(M + N \log^{3} N)$ per right-hand side, where $M$ and $N$ are the numbers of sample points and unknown coefficients, respectively. Numerical experiments on random and polar grids illustrate the computational scaling and reconstruction accuracy of the method, and demonstrate a substantial reduction in lsqr iteration counts when the direct solver is used as a preconditioner.
