---
title: 'Type 3 NUFFT: Arbitrary Grid Fourier Transform'
url: https://www.emergentmind.com/topics/type-3-nufft
type: topic
---

# Type 3 NUFFT: Arbitrary Grid Fourier Transform

The Type 3 Nonuniform Fast Fourier Transform (NUFFT)—also referred to as the "nonuniform-to-nonuniform" transform—computes the discrete Fourier transform (DFT) between a set of arbitrary nonuniform spatial (or temporal) input positions and another set of nonuniform frequency points. Distinguished from Type 1 (nonuniform to uniform frequency grid) and Type 2 (uniform to nonuniform frequency grid) NUFFT, Type 3 is the most general and versatile, enabling the transformation of data sampled on arbitrary grids to arbitrary target frequency locations. It is central to a wide range of applications requiring flexible sampling and spectral analysis and has motivated substantial algorithmic research due to the computational complexity of the direct (quadratic-time) approach.

## 1. Formal Definition and Mathematical Structure

Let $x_j \in \mathbb{R}^d$ for $j=1,\ldots,M$ be the nonuniform input points, $f(x_j)\in\mathbb{C}$ the data at those points, and $\omega_k \in \mathbb{R}^d$ for $k=1,\ldots,K$ the nonuniform target frequencies. The Type 3 NUFFT computes

\[
F(\omega_k) = \sum_{j=1}^M f(x_j) e^{-2\pi i \langle x_j,\,\omega_k\rangle},\qquad k=1,\ldots,K.
\]

In one-dimensional scenarios, this reduces to $f_j = \sum_{k=1}^{N} u_k\,e^{2\pi i x_j \omega_k}$ for given nonuniform $\{ x_j \}$ and $\{ \omega_k \}$, encapsulating the core algebraic structure [2512.03733, 2501.05244].

The corresponding transformation matrix $A$ with entries $A_{j,k} = e^{2\pi i x_j \omega_k}$ is typically full-rank with no exploitable Toeplitz or circulant structure, making naive direct computation an $O(MK)$ operation. The inverse problem, often overdetermined (i.e., $M\ge K$), is commonly solved in a least-squares sense.

## 2. Fast Algorithms and Kernel-Based Approximations

To achieve computational tractability, fast Type 3 NUFFT algorithms combine clever interpolation (gridding), oversampling, and windowing techniques with uniform FFTs:

1. **Gridding/Spreading:** Input data is spread onto a uniform oversampled grid using a localized kernel $\varphi(t)$ (e.g., Kaiser–Bessel or exponential-of-semicircle functions).
2. **Uniform FFT:** A standard FFT of the oversampled grid is performed; this leverages highly optimized FFT implementations for uniform data.
3. **Degridding/Regridding:** The transform is evaluated at nonuniform target frequencies using interpolation with the same kernel, together with an explicit deconvolution by the kernel's Fourier transform [2501.05244, 1808.06736].

Parameter selection—particularly the oversampling ratio $\sigma$ and kernel width $\tau$—controls accuracy and performance. For $d$-dimensional data, the total computational cost is $O(N^d \log N)$ for balanced regimes ($M+K\lesssim N^d$), with the error controlled by $O(e^{-c(\sigma-1)\tau})$ for an appropriate choice of kernel $\varphi$.

Libraries such as FINUFFT implement the spreading and degridding steps using the exponential-of-semicircle (ES) kernel. ES offers compact support and exponential alias suppression, facilitating minimal RAM usage and high throughput for large-scale 1D, 2D, and 3D problems. The underlying error analysis is precise, with kernel support $w\sim O(\log(1/\epsilon))$ for target accuracy $\epsilon$ [1808.06736].

## 3. Low-Rank and Structured Matrix Decompositions

Another approach exploits numerical low-rank structure in the Type 3 NUDFT matrix. Key steps:

- **Matrix Factorization:** Each entry can be expressed as a product of a smooth low-rank "quotient" and a uniform DFT entry.
- **Low-Rank Approximation:** The smooth factor is approximated via a bivariate Chebyshev expansion, yielding a rank-$K=O(\log(1/\epsilon)/\log\log(1/\epsilon))$ separation [1701.04492].
- **Implementation:** The transform is evaluated as a sum of $K$ diagonally-scaled FFTs, with an overall cost of 
  \[
  O\bigl(N \log N\, \tfrac{\log(1/\epsilon)}{\log\log(1/\epsilon)}\bigr).
  \]
  The approximation error is bounded entrywise by $\|\widetilde{F}_3-\widetilde{F}_{3,K}\|_{\max}\le\epsilon$, with total $L_2$-error $\leq N\epsilon\|c\|_2$.

This approach is robust in higher dimensions and can be extended to 2D by dimension-by-dimension low-rank approximations, leading to a sum of $K_1K_2$ 2D FFTs for required accuracy.

## 4. Direct and Inverse Type 3 NUDFT Solvers

Superfast direct solvers for the Type 3 inverse NUDFT (INUDFT) have been proposed based on hierarchical matrix techniques:

- **Matrix Factorization:** The NUDFT-III matrix $A$ is decomposed approximately as $A \approx B H$, where $B$ is a Type II NUDFT matrix and $H$ an explicit matrix embodying the residual structure. Both $B$ and $H$ are amenable to hierarchical semiseparable (HSS) approximation [2512.03733].
- **Forward and Inverse Application:** Forward application leverages fast Type II NUDFT algorithms and HSS matrix multiplication. The inversion proceeds efficiently due to quasi-linear complexity of HSS-based direct solvers, with the step $u= H_HSS^\dagger (B_{fast}^\dagger f)$ requiring only one type-II direct solve and one HSS solve.

Given $M\ge N+2R$ and for i.i.d. uniform samples, Frobenius-norm error bounds for the approximation $A\approx BQ$ and $A\approx B H$ are sharp, and in practice the truncation parameter $R=0$ suffices for small errors.

Numerical performance exhibits $O(N \log^2 N)$ construction and factorization, and $O(N\log N)$ per-solve runtime. As a preconditioner for conjugate gradients on the normal equations, the method dramatically accelerates convergence—reducing iteration counts from $\gg 100$ (unpreconditioned) to $2$–$10$ (preconditioned) [2512.03733].

## 5. Error Estimates, Parameter Choices, and Parallel Performance

Rigorous error analysis underpins kernel-based and low-rank approaches:

- **Kernel Interpolation Error:** For a kernel of half-width $\tau$ and oversampling $\sigma$, the error decays exponentially as $C \exp(-c(\sigma-1)\tau)$, with $C$ and $c$ depending on the kernel [2501.05244, 1808.06736].
- **Low-Rank Approximation Error:** The Chebyshev expansion rank $K$ is chosen so $\|A - A_K\|_{\max}\le\epsilon$, where $K$ is a slowly growing function of $1/\epsilon$ [1701.04492].
- **Hierarchical Compression Error:** HSS approximation achieves quasi-linear complexity with the rank $k = O(\log(1/\rho)\log N + R)$ for target accuracy $\rho$ [2512.03733].

Optimized libraries exploit cache and memory access patterns, multithreading, and on-the-fly kernel evaluation. For example, FINUFFT achieves 80–90% parallel efficiency up to one thread per physical core, with no plan/precompute stage and minimal RAM escalation [1808.06736].

## 6. Applications and Practical Implementations

Type 3 NUFFT is pivotal for scientific computing tasks where both data and reconstruction loci are aperiodic or non-Cartesian:

- **Non-Line-of-Sight (NLOS) Imaging:** Type 3 NUFFT enables flexible reconstruction and sampling strategies in phasor-field-based NLOS imaging, accommodating arbitrary, irregular sensor and reconstruction grids. This flexibility is crucial for compressed measurements and scalable 3D reconstruction [2501.05244].
- **Signal Processing and Medical Imaging:** Arbitrary k-space trajectories in MRI and advanced image reconstruction techniques necessitate Type 3 NUFFT algorithms for efficient and accurate resampling.
- **Numerical PDE Solvers:** The requirement to map between spatially and spectrally nonuniform discretizations arises in high-fidelity scientific simulations.

Contemporary open-source implementations such as FINUFFT [1808.06736] provide highly efficient, high-accuracy routines for Type 3 transforms in dimensions 1–3, with demonstrated throughput and robustness for both scattered and clustered data.

## 7. Summary Table: Algorithmic Approaches for Type 3 NUFFT

| Approach             | Main Reference                     | Complexity           |
|----------------------|------------------------------------|----------------------|
| Kernel-based (gridding)| [2501.05244], [1808.06736] | $O(N^d \log N)$      |
| Low-rank/Chebyshev   | [1701.04492]                       | $O(N \log N \tfrac{\log(1/\epsilon)}{\log\log(1/\epsilon)})$ |
| HSS direct solver    | [2512.03733]                       | Setup: $O(N \log^2 N)$,<br> Solve: $O(N \log N)$ |

Each approach supports high-precision transforms for arbitrary nonuniform grids, with different balances between setup time, memory, and runtime, depending on application context and underlying data structure. 

---
References:  
- [2512.03733] Li & Liu, "A Superfast Direct Solver for Type-III Inverse Nonuniform Discrete Fourier Transform"  
- [2501.05244] Sultan et al., "Optimized Sampling for Non-Line-of-Sight Imaging Using Modified Fast Fourier Transforms"  
- [1808.06736] Barnett et al., "A parallel non-uniform fast Fourier transform library based on an 'exponential of semicircle' kernel"  
- [1701.04492] Ruiz-Antolín & Townsend, "A nonuniform fast Fourier transform based on low rank approximation"

Source: https://www.emergentmind.com/topics/type-3-nufft