Non-uniform FFT (NUFFT): Principles & Applications

• Non-uniform FFT (NUFFT) is a generalization of the classical FFT designed to compute Fourier transforms efficiently for nonuniformly sampled data.
• It employs convolutional gridding, advanced kernel functions, and oversampling to achieve near-uniform FFT performance despite arbitrary sample locations.
• NUFFT is widely applied in medical imaging, computational physics, and time-series analysis to enable scalable, accurate transform-based solutions.

The non-uniform fast Fourier transform (NUFFT), also called the nonequispaced FFT or NFFT in the literature, generalizes the classical FFT to efficiently and accurately compute discrete Fourier transforms when either input data or output frequencies are located on nonuniform grids. The NUFFT achieves nearly the same computational complexity as the uniform-grid FFT while supporting arbitrary sample locations. Through developments in convolutional gridding, advanced window functions, and error analysis, the NUFFT has enabled scalable Fourier-based algorithms for a wide array of applications including inverse problems, computational imaging, PDE solvers, medical imaging, and time-series analysis.

1. Mathematical Definition and Problem Statement

Given a collection of $M$ nonuniform points $x_j \in [a,b]^d$ with associated complex weights $c_j$, and a set of $N$ frequency modes $\{k\}$ (which may lie on a uniform or nonuniform spectrum), the central NUFFT problem is to compute sums of the form

$$F_k = \sum_{j=1}^M c_j \, e^{i k \cdot x_j}, \quad k \in \Omega$$

where $\Omega$ is the set of frequencies of interest. This encompasses several types:

• Type 1: nonuniform spatial samples $\rightarrow$ uniform spectrum.
• Type 2: uniform spatial samples $\rightarrow$ nonuniform spectrum.
• Type 3: both data and spectrum nonuniform.

Direct evaluation is $O(MN)$, prohibitive for large-scale problems. The NUFFT achieves $x_j \in [a,b]^d$0 (or $x_j \in [a,b]^d$1, depending on context), up to multiplicative constants depending on dimension and desired accuracy (Barnett et al., 2018). The key to acceleration is to use localized, rapidly decaying interpolation kernels, controlled oversampling in the spectral domain, and efficient deconvolution.

2. Algorithmic Principles and Kernel Design

Modern NUFFT implementations follow a gridding-based pipeline:

1. Spreading/Gridding: Each nonuniform sample $x_j \in [a,b]^d$2 is convolved with a compactly supported kernel $x_j \in [a,b]^d$3 and spread onto a nearby subset of an oversampled uniform grid. The kernel is typically chosen for rapid decay and convenient Fourier-analytic properties, with commonly used examples including the Kaiser–Bessel (KB) window, the exponential of semicircle (ES) kernel (Barnett et al., 2018, Barnett, 2020), and the min–max window (Lin, 2017). The oversampling factor $x_j \in [a,b]^d$4 is typically in the range 1.25–2.
2. Fast Fourier Transform: An FFT is used on the oversampled fine grid (size $x_j \in [a,b]^d$5), reducing complexity from $x_j \in [a,b]^d$6 to $x_j \in [a,b]^d$7 in 1D.
3. Deconvolution (Deapodization): The effect of the spreading kernel is removed in Fourier space by dividing by the (possibly numerically tabulated) Fourier transform $x_j \in [a,b]^d$8.

This core procedure is adapted across spatial dimensions and for both forward/inverse and hybrid transform types. Vectorized, cache-optimized, and parallelized implementations are essential for large-scale performance (Barnett et al., 2018). For nonuniform targets (Type 2/Type 3), the process is essentially reversed, interpolating from the uniform Fourier domain to nonuniform outputs using the same kernel (Barnett et al., 2018, Lin, 2017).

Extensive error analysis has shown that the ES, KB, and prolate spheroidal wavefunction (PSWF) kernels achieve optimal exponential convergence in kernel width $x_j \in [a,b]^d$9 for aliasing error, e.g., $c_j$0 for a suitable constant $c_j$1 (Barnett, 2020). Parameter selection for the kernel width, oversampling, and window choice is critical for controlling accuracy and computational cost.

3. Applications in Inverse Problems and Computational Imaging

The NUFFT's ability to decouple sampling constraints from the Fourier transform underpins its indispensability in many inverse problems:

• Magnetic Resonance Imaging (MRI): Non-Cartesian (e.g., radial, spiral) k-space trajectories require Type 1/2 NUFFT for both image reconstruction and forward modeling. Efficient gridding kernels, density compensation, and multi-coil extensions are implemented in packages such as PyNUFFT (Lin, 2017), while optimization and learning frameworks exploit efficient derivative/Jacobian computation via the NUFFT (Wang et al., 2021).
• Computed Tomography (CT): Forward and backprojection steps in cone-beam CT leverage multidimensional NUFFT operators for efficient Radon-space resampling and projection geometry flexibility. This reduces volumetric complexity from $c_j$2 to $c_j$3, making iterative and 3D methods numerically feasible (Tang, 2016).
• Photoacoustic Tomography: Exact Fourier inversion formulas derived for planar detectors in arbitrary sensor geometries are efficiently realized using multidimensional NUFFTs (NED-NER), yielding higher-resolution reconstructions and supporting flexible sensor layouts (Schmid et al., 2015, Schmid et al., 2015).
• Nonlocal PDEs and Micromagnetics: Convolutions with singular or slowly decaying kernels (e.g., Newton, Coulomb, dipole) in micromagnetic potential evaluation (Exl et al., 2013), nonlinear Schrödinger, and Davey-Stewartson solvers (Bao et al., 2014, Mauser et al., 2014, Jiang et al., 2013, Bao et al., 2015) are transformed to smooth integrals via coordinate change (polar/spherical). NUFFT then evaluates the resulting sums at $c_j$4 cost, eliminating “locking” errors seen in standard FFTs (Jiang et al., 2013, Bao et al., 2014).
• Optics and Coherent Diffraction: The angular spectrum and Fresnel diffraction integrals, which require Fourier transforms between nonuniform grids in source/target planes, are accelerated and generalized by NUFFT (Shimobaba et al., 2013, Barnett, 2020, Sultan et al., 9 Jan 2025). Applications range from coherent holography to non-line-of-sight imaging.
• Time-Series and Spectral Estimation: For unevenly sampled data (e.g., astrophysics, finance), the NUFFT is critical in fast periodogram computation (e.g., Lomb–Scargle periodogram (Garrison et al., 2024)), spectral covariance estimation (e.g., Malliavin–Mancino estimators (Chang et al., 2020)), and FT spectrometer pipelines (Wen et al., 2022).

4. Implementation, Kernel Optimization, and Acceleration

State-of-the-art codes such as FINUFFT (Barnett et al., 2018) and PyNUFFT (Lin, 2017) implement all major transform types (1/2/3) in up to three dimensions, supporting CPU and GPU parallelism, cache-aware blocking, minimal RAM overhead (no plan-stage), and vectorized kernel evaluation using piecewise-polynomial approximations.

Key design aspects include:

• Kernel Parameter Auto-tuning: Kernel width $c_j$5 and shape parameter $c_j$6 (for ES, KB) set based on the target accuracy $c_j$7, balancing aliasing error with computational load.
• Parallelism and SIMD: Modern codes achieve near-ideal scaling up to the memory bandwidth limit and large $c_j$8 by bin-sorting and parallel thread blocking.
• Sparse Gridding Matrices: In finite element, surface integral, or unstructured domain settings (e.g., P$c_j$9-FEM in micromagnetics (Exl et al., 2013)), efficient sparse assembly and tensor-train compression further mitigate memory bottlenecks.
• FMM-NUFFT Hybridization: For specialized kernels (e.g., in 1D on the unit circle), fast multipole–based NUFFTs can reach near-linear complexity for machine-precision targets (Gumerov et al., 2016).

Empirical performance benchmarks consistently favor high-precision, advanced NUFFT implementations over both legacy NFFT/CMCL codes and dense summation, with acceleration factors of $N$0–$N$1 typical for large-scale problems (Barnett et al., 2018, Garrison et al., 2024).

5. Error Analysis, Limitations, and Rigorous Guarantees

Theoretical error bounds in modern NUFFT are rigorous, with aliasing errors decaying as $N$2, and explicit formulas for kernel parameters ensuring the user-specified tolerance is achieved (Barnett, 2020, Barnett et al., 2018). For multidimensional transforms or mixed spatial/frequency nonuniformity, the main limitations remain:

• Kernel-Induced Local Averaging: Wider kernels reduce aliasing but can introduce local smoothing if too wide (relevant in spectral estimators (Chang et al., 2020)).
• Memory Overhead: The necessity to maintain oversampled grids and precomputed kernels imposes $N$3 memory requirements.
• Adaptive and Anisotropic Scenarios: Highly clustered or variable-density input points can degrade parallel scaling; adaptive schemes and advanced tree-based hybridizations are being developed (Gumerov et al., 2016).
• Bandlimiting and Boundary Effects: For propagation or PDE problems where truncation and periodicity are artificial, error control requires careful treatment of windowing and effective domain extension (Sultan et al., 9 Jan 2025, Exl et al., 2013).
• Extreme Undersampling: Severe undersampling can raise the noise floor, though the nonperiodic nature of irreular sampling in NUFFT-based pipelines usually suppresses aliasing compared to interpolation-based FFTs (Wen et al., 2022).

6. Domain-Specific Extensions and Open Research Problems

Active research and extensions include:

• Trajectory Optimization in MRI: Efficient Jacobian computation with respect to k-space locations enables gradient-based optimization in model-based and deep-learning pipelines (Wang et al., 2021).
• Advanced Reconstruction in NLOS Imaging: NUFFT generalizes convolution-based propagation to nonuniform sampling in non-line-of-sight and holographic pipelines, supporting arbitrary relay-surface geometries and scalable volume reconstructions (Sultan et al., 9 Jan 2025).
• Sparse Spatiotemporal Arrays in Imaging: Flexible sensor placement (e.g., equi-angular, equi-steradian) and corresponding NUFFT-accelerated inversion improve limited-view and sparse-acquisition performance in ultrasound and photoacoustic imaging (Schmid et al., 2015, Schmid et al., 2015).
• Higher-order and Fractional Transforms: Scaled FFTs (fractional, chirp-Z) and their integration with NUFFT expand the class of efficiently computable transforms, supporting arbitrary rescaling in spectral estimation and diffraction (Sultan et al., 9 Jan 2025).
• Kernel Selection and Data-Driven Gridding: Exploration of application-specific optimal kernels, adaptive windowing, and learned density compensation remains a topic of ongoing study, particularly in highly nonuniform or noise-prone settings (Wen et al., 2022, Chang et al., 2020).

7. Summary Table of Representative Algorithms and Their Complexities

Application	NUFFT Variant / Key Features	Complexity	Cited Reference
Type 1/2/3 general NUFFT	ES/KB/PSWF kernel, σ=2	$N$4	(Barnett et al., 2018)
FMM-NUFFT (1D, cot/tan kernel)	Multilevel FMM, series expansions	$N$5	(Gumerov et al., 2016)
Finite element micromagnetics	Kernel splitting, FFT+adjoint NUFFT	$N$6	(Exl et al., 2013)
Photoacoustic inversion	NED-NER multidimensional NUFFT	$N$7	(Schmid et al., 2015)
PDE convolution (Coulomb/dipole)	Spherical/polar split, quadrature + NUFFT	$N$8	(Jiang et al., 2013)
MRI, non-Cartesian k-space	Gridding, density comp., multi-coil	$N$9	(Lin, 2017)
Fresnel/ASM/NLOS diffraction	Type 1/2/3 NUFFT, scaled FFT	$\{k\}$0	(Barnett, 2020, Sultan et al., 9 Jan 2025)
Spectral estimation (Lomb–Scargle)	Type 1/adjoint, GPU, batch	$\{k\}$1	(Garrison et al., 2024)
Fourier time-series estimators	Kernel-averaged NUFFT	$\{k\}$2	(Chang et al., 2020)

In conclusion, the NUFFT, built on localized kernel gridding, oversampling, and rigorous deconvolution, is the foundation for a vast family of algorithms generalizing FFT-based techniques to arbitrary sampling and evaluation geometries. By ensuring $\{k\}$3 or better complexity and controlling approximation errors to target tolerances, NUFFT-based methods are now ubiquitous across computational science domains, enabling both theoretical advances and practical large-scale simulation, reconstruction, and inference (Barnett et al., 2018, Lin, 2017, Exl et al., 2013, Jiang et al., 2013, Barnett, 2020, Schmid et al., 2015, Garrison et al., 2024, Wen et al., 2022, Sultan et al., 9 Jan 2025, Mauser et al., 2014, Wang et al., 2021).