NUFFT: Fast Transforms for Irregular Data

• Non-Uniform Fast Fourier Transforms (NUFFT) are algorithms that efficiently compute Fourier transforms on irregularly sampled data using spreading, FFT application, and deconvolution.
• NUFFT leverages convolution with smooth kernels, such as Kaiser–Bessel and exponential-of-semicircle, to map nonuniform samples onto a regular grid, ensuring high numerical accuracy and stability.
• Applications of NUFFT span multiple fields including MRI image reconstruction, computational physics, and financial modeling, where efficient DFT computation is crucial.

Non-Uniform Fast Fourier Transforms (NUFFT) are a class of algorithms enabling the efficient computation of discrete Fourier transforms (DFT) when data are sampled at arbitrary, nonuniform locations in space or time. While the standard FFT provides O(N log N) complexity for uniformly-sampled datasets, the NUFFT extends this efficiency and numerical stability to a range of scenarios where the classical FFT is not directly applicable, including irregular geometry, compressed acquisition schemes, or adaptive sampling. The NUFFT is extensively used in computational physics, signal processing, medical imaging, inverse problems, computational finance, and more, and supports a wide range of practical and theoretical research.

1. Mathematical Foundations and Algorithmic Principles

The fundamental problem addressed by the NUFFT is to compute sums of the form

$$f_j = \sum_{k=0}^{N-1} c_k\, e^{-2\pi i\, x_j\, k},$$

where $\{x_j\}$ are (potentially nonuniform) locations and $\{c_k\}$ are input coefficients. More generally, the NUFFT supports transforms where both source and frequency locations are arbitrary (“type 3 NUFFT”).

The dominant strategy in most modern NUFFT algorithms is to reinterpret the nonuniform DFT as a convolution problem. Nonuniform input data is “spread” onto a regular grid via convolution with a smooth, rapidly-decaying kernel (e.g., Gaussian, Kaiser–Bessel, or exponential-of-semicircle), facilitating a subsequent FFT. Deconvolution is then applied in Fourier space to correct for the blurring induced by the spreading kernel. This approach is characterized by three principal steps:

1. Spreading/Gridding: Each sample is mapped onto a uniform grid using a localized convolution kernel; this may involve evaluating the kernel at multiple grid points for each nonuniform datum (Barnett et al., 2018, Ruiz-Antolin et al., 2017).
2. FFT Application: The FFT is applied to the regularized grid, inheriting the O(N log N) complexity.
3. Deconvolution/Correction: After the FFT, the effects of the kernel are inverted via division in the frequency domain. The choice and properties of the convolution kernel critically determine the numerical accuracy and stability (Barnett, 2020).

An alternative class of algorithms is based on low-rank approximations and matrix factorizations (Ruiz-Antolin et al., 2017), or multipole expansions (Gumerov et al., 2016). These may decouple the nonuniform transform into sums of standard FFTs with modulating phase factors, allowing for both theoretical analysis and practical parallelization.

The three canonical types of NUFFT are:

• Type 1: Nonuniform-to-uniform (synthesis) — compute Fourier coefficients from nonuniform data.
• Type 2: Uniform-to-nonuniform (interpolation) — evaluate a Fourier series at nonuniform locations.
• Type 3: Nonuniform-to-nonuniform — general case, where both input and output locations are arbitrary.

2. Kernel Design and Error Analysis

Crucial to the implementation of efficient NUFFTs is the selection of the spreading kernel. The kernel must be compactly supported, smooth, and have a Fourier transform that decays rapidly in the out-of-band region to minimize aliasing error and facilitate accurate deconvolution. The exponential-of-semicircle (ES) kernel,

$$\phi_{\text{ES}}(z) = \exp(\beta \sqrt{1-z^2} - \beta), \qquad |z|\leq 1,$$

has emerged as a highly effective choice, offering nearly optimal exponential decay of aliasing error and facilitating efficient, on-the-fly kernel evaluation (Barnett et al., 2018, Barnett, 2020). The classic Kaiser–Bessel kernel (Barnett et al., 2018) is also widely used for its theoretical performance guarantees and closed-form Fourier transform.

Detailed error analysis reveals that the aliasing error in the NUFFT decays exponentially with the kernel width. The error constant and rate are controlled by kernel design, typically parametrized by kernel width $w$, oversampling factor $\sigma$, and a safety parameter $\gamma$: $\epsilon_\infty \sim \sqrt{w}\, e^{-c w},$ where $c$ can be made arbitrarily close to the optimal value achieved by prolate spheroidal wavefunctions (PSWF), the theoretical “gold standard” for time-frequency localization (Barnett, 2020).

Theoretical analyses have rigorously established these bounds via asymptotic expansions, contour integration, and connections to the PSWF, confirming that the ES, KB, and PSWF kernels offer nearly identical convergence rates for practical purposes.

3. Algorithmic Variants and Efficient Implementations

NUFFT algorithms have evolved along several lines to address different computational constraints and data structures:

• Standard gridding-based methods implement the three-step pipeline using efficient data structures, load balancing (for parallelization), and on-the-fly kernel evaluation to minimize RAM usage (Barnett et al., 2018).
• Low-rank approximations reinterpret the NUDFT as a Hadamard (entrywise) product between a correction matrix and the standard DFT matrix. The correction matrix is shown to be low-rank when the nonuniform nodes are near-uniform; diagonal scaling and multiple FFTs combine to give the desired transform, with the overall cost $$O(N \log N\, \log(1/\epsilon)/\log\log(1/\epsilon))$$ (Ruiz-Antolin et al., 2017).
• Fast multipole-based algorithms operate on a hierarchical decomposition of space (e.g., the unit circle in 1D) to accelerate nonuniform sums using near–field/far–field expansions and precise error control, with complexity $O(N \log N + N \log \epsilon^{-1})$ and even $O(N)$ for machine–precision (Gumerov et al., 2016).
• High-order quadrature and coordinate transformation are effective for applications such as nonlocal PDEs where the singularity in the Fourier symbol can be canceled by variable changes; a hybrid FFT/NUFFT approach then attains optimal complexity (Jiang et al., 2013, Bao et al., 2014).

Public libraries such as FINUFFT (Barnett et al., 2018), NFFT (Potts et al.), PyNUFFT (Lin, 2017), and others provide multi-dimensional, multi-threaded support, optimizations for cache-aware load balancing, and robust numerical parameterization for high-dimensional applications.

4. Applications Across Scientific and Engineering Disciplines

The broad utility of the NUFFT is established in several major scientific domains:

• Imaging and Tomography: In magnetic resonance imaging (MRI), non-Cartesian k-space sampling (spiral, radial, etc.) necessitates efficient and accurate forward and adjoint NUFFT operators for reconstruction, trajectory optimization, and model-based inversion (Lin, 2017, Wang et al., 2021). In X-ray CT and other tomographic modalities, the Fourier Slice Theorem links projections to slices of the object’s spectrum, requiring NUFFTs for efficient projection/backprojection (Tang, 2016).
• Computational Physics and Chemistry: Nonlocal Coulomb and dipole-dipole interactions are computed via convolution in the Fourier domain; NUFFT-based algorithms effectively handle singularities via variable transformations and high-order quadrature (Jiang et al., 2013, Bao et al., 2014, Bao et al., 2014).
• Photoacoustic Tomography and Ultra-fast Optics: Analytic backprojections in non-Cartesian geometries, flexible detector arrangements, and reduced sensor counts are enabled by NUFFT-based reconstructions (Schmid et al., 2015, Schmid et al., 2015).
• Time Series Analysis and Spectral Estimation: Power spectrum estimation in nonuniform sampling scenarios is addressed via multitapered NUFFT approaches, achieving nearly optimal bias and variance at dramatically reduced computational cost (Cui et al., 2 Jul 2024, Chang et al., 2020).
• Computational Finance: The COS method for European option pricing, which relies on Fourier–cosine expansions of densities, can be reformulated as a type-2 NUFFT to price large batches of options with different strikes extremely efficiently (LeFloc'h, 17 Jul 2025).
• Non-Line-of-Sight Imaging: Inverse problems involving light transport propagation from irregularly sampled surfaces (SPAD arrays, relay walls, etc.) exploit both NUFFT and scaled FFT (SFFT) for efficient volumetric reconstruction under nonuniform acquisition, preserving the O(N log N) complexity (Sultan et al., 9 Jan 2025).
• Fourier Transform Spectrometers: NUFFT methods enable direct processing of interferometric data acquired at nonuniform optical path differences, outperforming traditional interpolation schemes in amplitude fidelity and aliasing suppression (Wen et al., 2022).

5. Accuracy–Efficiency Trade-offs and Practical Considerations

The primary advantage of the NUFFT is the preservation of FFT-level (O(N log N)) efficiency when handling irregular geometries, missing data, or variable resolution requirements. Key practical considerations include:

• Overhead versus Batch Size: For small output sets (e.g., few strikes in COS pricing), classic direct methods may be marginally faster due to NUFFT overhead (LeFloc'h, 17 Jul 2025).
• Parameterization: Grid size, oversampling ratio, kernel width, and error tolerance affect both accuracy and performance; proper tuning is crucial for balancing throughput and approximation error (Barnett et al., 2018, Barnett, 2020).
• Edge Cases: For data near the boundary of the spatial or frequency domain, aliasing artifacts can arise; these are controlled by kernel width and spacing, or alternative weighted NUFFT formulations (Wen et al., 2022).
• Parallelization: Most modern NUFFT algorithms are highly parallelizable due to spatial locality in spreading/interpolation and the inherent parallel structure of the FFT (Barnett et al., 2018, Lin, 2017).
• Differentiability: When integrating NUFFT in optimization loops (e.g., trajectory learning in MRI), care must be taken with frequency-domain interpolation, which is not necessarily differentiable almost everywhere. Analytic Jacobians and approximate gradient techniques have been proposed to overcome the limitations of computational auto-differentiation (Wang et al., 2021).

6. Recent Developments and Research Directions

The ongoing development of NUFFT algorithms is characterized by several trends:

• Kernel Theory and Optimality: Analytical connections between practical kernels (KB, ES) and PSWF have prompted new designs aiming for both optimal error rates and minimal compute overhead (Barnett, 2020).
• Low-Rank and Operator-Based Approaches: Decomposing the nonuniform DFT into sums/products of FFTs with diagonal and permutation matrices enables more flexible, adaptive, and parallel implementations (Ruiz-Antolin et al., 2017).
• Domain-Specific Applications: Extensions to Fresnel and Rayleigh–Sommerfeld diffraction in computational optics (Shimobaba et al., 2013, Barnett, 2020), application-specific compressed sensing, and robust solutions in inverse problems (e.g., smoothing, regularization, and Jacobian-vector products in learned MRI sampling (Wang et al., 2021)).
• Spectral Estimation in Nonuniform Time Series: Multitapered NUFFT methods confidently generalize classical multitaper spectral analysis from uniform to irregular time grids, facilitating large-scale analysis of biomedical and geophysical signals (Cui et al., 2 Jul 2024).
• Public Library Development and Scalability: FINUFFT (Barnett et al., 2018) and PyNUFFT (Lin, 2017) exemplify robust, high-performance, multi-platform libraries supporting, respectively, both research and industrial applications. Efficient parallel algorithms, GPU support, and convenient multi-language bindings broaden practical accessibility.

7. Summary Table: Core Features of NUFFT in Modern Applications

Feature/Aspect	Description	Example Use Case
Kernel Type	ES, KB, Gaussian, PSWF	Signal gridding, deconvolution, error control
Supported Transform Types	Type 1, 2, 3	Tomography, MRI, finance, imaging
Complexity	$O(N \log N)$ (with log factors, see above)	All large-scale nonuniform DFT problems
Parallelism and Libraries	Multi-threaded, GPU, C++, Python/Julia	FINUFFT, PyNUFFT, NFFT
Typical Accuracy	6–14 digits with moderate kernel widths	MRI, tomography, spectral estimation
Memory Usage	Minimal if on-the-fly kernels, low precompute	Large-scale 3D reconstructions
Differentiability	Analytic Jacobians or approximate adjoints	Learned MRI sampling, inverse problems

The Non-Uniform Fast Fourier Transform constitutes a central algorithmic tool in computational mathematics and the physical sciences, enabling the transfer of Fourier-based methods to settings with irregular data, adaptive sampling, and non-Cartesian problem domains, with applications ranging from simulation and spectral analysis to image reconstruction, machine learning, and quantitative finance. Ongoing research continues to refine its efficiency, accuracy, and domain-specific integrations.