# A Frame Theoretic Approach to the Nonuniform Fast Fourier Transform

@article{Gelb2014AFT, title={A Frame Theoretic Approach to the Nonuniform Fast Fourier Transform}, author={Anne Gelb and Guohui Song}, journal={SIAM J. Numer. Anal.}, year={2014}, volume={52}, pages={1222-1242} }

Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the nonuniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to…

## 20 Citations

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

- MathematicsJ. Sci. Comput.
- 2018

This paper demonstrates that the inverse polynomial reconstruction method is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted $$L_2$$L2 space of orthogonal polynomials.

Using frame theoretic convolutional gridding for robust synthetic aperture sonar imaging

- Computer ScienceOCEANS 2017 – Anchorage
- 2017

This work proposes using the frame theoretic convolution gridding (FTCG) algorithm to handle the non-uniform Fourier data, and outlines how the FTCG can be used to enhance current SAS processing.

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

- Computer ScienceJ. Sci. Comput.
- 2014

The convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges and carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients.

Detecting Edges from Non-uniform Fourier Data Using Fourier Frames

- MathematicsJ. Sci. Comput.
- 2017

This investigation further develops an existing approach to discontinuity detection, and involves the use of concentration factors, and produces concentration factors which can more precisely identify jump locations than those previously developed in both one and two dimensions.

On stable reconstructions from univariate nonuniform Fourier measurements

- Mathematics, Computer Science
- 2013

It is proved that a linear scaling of the dimension of the space with the sampling bandwidth is both necessary and sufficient for stable and accurate recovery, and wavelets are up to constant factors optimal spaces for reconstruction.

Edge-adaptive l2 regularization image reconstruction from non-uniform Fourier data

- Computer Science, Mathematics
- 2018

A non-iterative weighted regularization method that uses a pre-processing edge detection to find exactly where the sparsity should be in the edge domain has the potential to outperform reweighted TV regularization methods.

Direct inversion of the nonequispaced fast Fourier transform

- Computer Science, MathematicsLinear Algebra and its Applications
- 2019

Detecting Edges from Non-uniform Fourier Data via Sparse Bayesian Learning

- Computer ScienceJ. Sci. Comput.
- 2019

This paper uses the Bayesian framework to design an improved algorithm for detecting edges from non-uniform Fourier data that employs what is known as type-II Bayesian estimation, specifically a method called sparse Bayesian learning.

Thresholded Non-Uniform Fourier Frame-Based Reconstruction for Stripmap SAR

- Mathematics, Computer ScienceArXiv
- 2019

This paper applies a frame reconstruction algorithm, extending the non-uniform fast Fourier transform, to stripmap SAR data and presents an improved thresholded frame reconstruction algorithms for robust performance and improved computational efficiency.

A Practical Guide to the Recovery of Wavelet Coefficients from Fourier Measurements

- Computer ScienceSIAM J. Sci. Comput.
- 2016

It is shown that generalized sampling has a computational complexity of $\mathcal{O}\left(M(N)\log N\right)$ when recovering the first $N$ boundary-corrected wavelet coefficients of an unknown compactly supported function from M(N)$ Fourier samples.

## References

SHOWING 1-10 OF 45 REFERENCES

Edge Detection from Non-Uniform Fourier Data Using the Convolutional Gridding Algorithm

- Computer ScienceJ. Sci. Comput.
- 2014

The convolutional gridding edge detection algorithm developed in this paper provides an efficient and robust way to calculate edges and carefully chosen parameters ensure that the algorithm retains accuracy in the high frequency coefficients.

A Fast Sinc Function Gridding Algorithm for Fourier Inversion in Computer Tomography

- GeologyIEEE Transactions on Medical Imaging
- 1985

This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels.

Nonuniform fast Fourier transforms using min-max interpolation

- Engineering, Computer ScienceIEEE Trans. Signal Process.
- 2003

This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm and indicates that the proposed method easily generalizes to multidimensional signals.

Non-Equispaced Fast Fourier
Transforms with Applications
to Tomography

- Mathematics
- 2003

AbstractIn this article we describe a non-equispaced fast Fourier transform. It is similar
to the algorithms of Dutt and Rokhlin and Beylkin but is based on an exact Fourier series
representation.…

Sparsity Enforcing Edge Detection Method for Blurred and Noisy Fourier data

- Computer Science, MathematicsJ. Sci. Comput.
- 2012

A new method for estimating the edges in a piecewise smooth function from blurred and noisy Fourier data is presented that only needs the solution of a single l1 minimization to remove artifacts.

Selection of a convolution function for Fourier inversion using gridding [computerised tomography application].

- MathematicsIEEE transactions on medical imaging
- 1991

The authors compare the artifact introduced into the image for various convolving functions of different sizes, including the Kaiser-Bessel window and the zero-order prolate spheroidal wave function (PSWF).

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

- MathematicsSIAM Rev.
- 2001

A unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces is provided by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling.

Density compensation functions for spiral MRI

- MathematicsMagnetic resonance in medicine
- 1997

An analytic density compensation function (DCF) for spiral MRI, based on the Jacobian determinant for the transformation between Cartesian coordinates and the spiral sampling parameters of time and interleaf rotation angle, is derived and the reconstruction accuracy achieved using this function is compared with that obtained using several previously published expressions.

Recovering Exponential Accuracy from Non-harmonic Fourier Data Through Spectral Reprojection

- MathematicsJ. Sci. Comput.
- 2012

It is proved that spectral reprojection can reduce the Gibbs phenomenon in non-harmonic reconstruction as well as remove reconstruction artifacts due to erratic sampling in the case where the Fourier samples form a frame.