Nonequispaced Fast Fourier Transform (NFFT)

NFFT algorithm

The nonequispaced fast Fourier transform (NFFT or NUFFT), see [Keiner, Kunis, Potts, 2006], overcomes one of the main shortcomings of the FFT - the need for an equispaced sampling grid. Considering the evaluation of the $d$-dimensional trigonometric polynomial

\[ f \colon \mathbb{T}^d \to \mathbb{C}, \; \pmb{x} \mapsto \sum_{\pmb{k} \in I_{\pmb{N}}^d} \hat{f}_{\pmb{k}} \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}}\]

with multibandlimit $\pmb{N} \in 2 \mathbb{N}^d$ and index set

\[ I_{\pmb{N}}^d \coloneqq \left\{ \pmb{k} \in \mathbb{Z}^d: - \frac{N_i}{2} \leq k_i \leq \frac{N_i}{2} - 1, \, i = 1,2,\ldots,d \right\}.\]

The first approximation is a linear combination of a shifted periodized window function $\tilde{\varphi}$

\[ s_1(\pmb{x}) = \sum_{\pmb{\ell} \in I_{\pmb{n}}^d} g_{\pmb{\ell}} \, \tilde{\varphi}\left( \pmb{x} - \frac{1}{\pmb{n}} \odot \pmb{\ell} \right)\]

where $\frac{1}{\pmb{n}}$ is the elementwise inversion of the vector $\pmb{n}$. We choose an oversampling vector $\pmb{\sigma} > 1$ componentwise and obtain the index set by

\[ \pmb{n} \coloneqq \pmb{\sigma} \odot \pmb{N}.\]

Here, $\odot$ denotes the componentwise product. Note that one could skip the choice of $\pmb{\sigma}$ entirely and directly choose $n_i > N_i$ for $i= 1, 2, \ldots,d$. The standard choice in the C library is at the moment

\[ n_i = 2^{\lceil \log_2 N_i \rceil + 1}, i= 1, 2, \ldots, d. \]

If the window function $\varphi: \R^d \to \R$ has a one-periodization

\[ \tilde{\varphi}(\pmb{x}) = \sum_{\pmb{r} \in \mathbb{Z}^d} \varphi(\pmb{x}+\pmb{r}) \]

with a uniformly convergent Fourier series, we use the Poisson summation formula and obtain the Fourier coefficients

\[ c_{\pmb{k}}(\tilde{\varphi}) = \hat{\varphi}(\pmb{k}).\]

Here, $\hat{\varphi}$ is the Fourier transform of $\varphi$ defined by

\[ \hat{\varphi}(\pmb{k}) = \int_{\mathbb{T}^d} \varphi(\pmb{x}) \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}} \mathrm{d} \pmb{x}.\]

Replacing $\tilde{\varphi}$ by its Fourier series and splitting the sum in $s_1(x)$ yields

\[\begin{aligned} s_1(\pmb{x}) & = \sum_{\pmb{\ell} \in I_{\pmb{n}}^d} g_{\pmb{\ell}} \sum_{\pmb{k} \in \mathbb{Z}^d} c_{\pmb{k}}(\tilde{\varphi}) \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \left( \pmb{x} - \frac{1}{\pmb{n}} \odot \pmb{\ell}\right) } \\ & = \sum_{\pmb{k} \in \mathbb{Z}^d} c_{\pmb{k}} (\tilde{\varphi}) \underbrace{ \left(\sum_{\pmb{\ell} \in I_{\pmb{n}}^d } g_{\pmb{\ell}} \, \mathrm{e}^{2 \pi \mathrm{i} \, \frac{1}{\pmb{n}} \odot (\pmb{k} \cdot \pmb{\ell})} \right)}_{ \eqqcolon \hat{g}_{\pmb{k}}} \mathrm{e}^{-2 \pi \mathrm{i} \ \pmb{k} \cdot \pmb{x}}\\ & = \sum_{\pmb{k} \in I_{\pmb{n}}^d } c_{\pmb{k}}(\tilde{\varphi}) \hat{g}_{\pmb{k}} \, \mathrm{e}^{-2 \pi \mathrm{i} \ \pmb{k} \cdot \pmb{x} } + \sum_{\pmb{r} \in \mathbb{Z}^d \setminus \{ \pmb{0} \}} \sum_{\pmb{k} \in I_{\pmb{n}}^d } c_{\pmb{k}}(\tilde{\varphi}) \hat{g}_{\pmb{k}} \, \mathrm{e}^{-2 \pi \mathrm{i} \, (\pmb{k} + \pmb{n} \odot \pmb{r})\cdot \pmb{x} }. \end{aligned}\]

Furthermore, a comparison of the equation above and $f$ suggests the choice

\[ \hat{g}_{\pmb{k}} = \begin{cases} \frac{\hat{f}_{\pmb{k}}}{c_{\pmb{k}}(\tilde{\varphi})} \quad &\text{for } \pmb{k} \in I_{\pmb{N}}^d \\ 0 &\text{for } \pmb{k} \in I_{\pmb{n}}^d \setminus I_{\pmb{N}}^d \end{cases}.\]

Now, we are able to compute $g_{\pmb{\ell}}$ by an FFT of size $n_1 \times n_2 \times \cdots \times n_d$. The error of the first approximation $f \approx s_1$ is called aliasing error. For further analysis, we refer to Chapter 7 in [Plonka, Potts, Steidl, Tasche, 2018].
We obtain an approximation $\psi$ for $\varphi$ if the window function is well localized in the spatial domain by the truncation

\[ \psi(\pmb{x}) \coloneqq \varphi(\pmb{x}) \, \mathbb{1}_{\times_{i = 1}^d [-m/n_i,m/n_i]}(\pmb{x})\]

with a chosen window size $m \ll \min_{i\in\{1,2,\dots,d\}}\{n_i\}, m \in \mathbb{N}$. Following the same scheme as above, we can use the periodization

\[ \tilde{\psi}(\pmb{x}) = \sum_{\pmb{r} \in \mathbb{Z}^d} \psi(\pmb{x}+\pmb{r}) \]

and look at the linear combination

\[ s(\pmb{x}_j) \coloneqq \sum_{\pmb{\ell} \in I_{\pmb{n}}^d } g_{\pmb{\ell}} \, \tilde{\psi} \left( \pmb{x}_j - \frac{1}{\pmb{n}} \odot \pmb{\ell} \right).\]

The following calculations show that $s \approx s_1$

\[\begin{aligned} s(\pmb{x}_j) &= \sum_{\pmb{\ell} \in I_{\pmb{n}}^d } g_{\pmb{\ell}} \sum_{\pmb{r} \in \mathbb{Z}^d} \psi \left( \pmb{x}_j - \frac{1}{\pmb{n}} \odot \pmb{\ell} + \pmb{r} \right) \\ &= \sum_{\pmb{\ell} \in I_{\pmb{n}}^d } g_{\pmb{\ell}} \sum_{\pmb{r} \in \mathbb{Z}^d} \varphi \left(\pmb{x}_j - \frac{1}{\pmb{n}} \odot \pmb{\ell} + \pmb{r} \right) \mathbb{1}_{\times_{i = 1}^d [-m/n_i,m/n_i]}(\pmb{x}_j) \\ &= \sum_{\pmb{\ell} \in I_{\pmb{n}}^d} g_{\pmb{\ell}} \, \mathbb{1}_{\times_{i = 1}^d [-m/n_i,m/n_i]}(\pmb{x}_j) \, \tilde{\varphi} \left(\pmb{x}_j - \frac{1}{\pmb{n}} \odot \pmb{\ell} \right) \\ &= \sum_{\pmb{\ell} \in I_{\pmb{n},m}^d (\pmb{x}_j)} g_{\pmb{\ell}} \, \tilde{\varphi} \left(\pmb{x}_j - \frac{1}{\pmb{n}} \odot \pmb{\ell} \right) \end{aligned}\]

with the index set

\[ I_{\pmb{n},m}(\pmb{x}_j)^d = \left\{ \pmb{\ell} \in I_{\pmb{n}}^d \colon \pmb{n} \odot \pmb{x}_j - m \pmb{1} \leq \pmb{\ell} \leq \pmb{n} \odot \pmb{x}_j +m \pmb{1} \right\}\]

for a fixed node $\pmb{x}_j$. This is motivated by

\[ -\frac{m}{n_i} \leq \left( \pmb{x}_j \right)_i \leq \frac{m}{n_i} \]

in order to ensure that $\pmb{x}_j$ is within the support. This second approximation error is called the truncation error. Summarizing, we have $f \approx s_1 \approx s$.

Pseudocode

Input: $M \in \mathbb{N}, \pmb{N} \in (2\mathbb{N})^d, \pmb{n} \in (2\mathbb{N})^d, m \in \mathbb{N}, \pmb{x}_j \in [-0.5,0.5)^d \text{ for } j =1,\ldots,M, \hat{f}_{\pmb{k}} \in \mathbb{C} \text{ for } \pmb{k} \in I^{d}_{\pmb{N}}.$

Precomputation: Compute the nonzero Fourier coefficients $c_{\pmb{k}}(\tilde{\varphi}) \text{ for all } \pmb{k} \in I^{d}_{\pmb{N}}.$ Compute the function values $\tilde{\psi}_{j,\pmb{\ell}} \coloneqq \tilde{\psi} (\pmb{x}_j -\frac{1}{\pmb{n}}\odot\pmb{\ell}) \text{ for } j=1,\ldots,M \text{ and } \pmb{\ell} \in I_{\pmb{n}, m}^d(\pmb{x}_j).$

Set $\hat{g}_{\pmb{k}} = |I^{d}_{\pmb{n}}|^{-1} \, \hat{f}_{\pmb{k}} / c_{\pmb{k}}(\tilde{\varphi}) \text{ for } \pmb{k} \in I^{d}_{\pmb{N}}.$
Compute $g_{\pmb{\ell}} = \sum_{k \in I^{d}_{\pmb{N}}} \hat{g}_{\pmb{k}} \, \mathrm{e}^{2 \pi \mathrm{i} \, \pmb{k} \cdot \left( \frac{1}{\pmb{n}} \odot \pmb{\ell}\right)}$ for $\pmb{\ell} \in I^{d}_{\pmb{n}}$ using a d-variate FFT.
Compute $s(\pmb{x}_j) = \sum_{\pmb{\ell} \in I_{\pmb{n}, m}^d(\pmb{x}_j)} g_{\pmb{\ell}} \, \tilde{\psi}_{j,\pmb{\ell}}$ for $j =1,\ldots,M$.

Output: $s(\pmb{x}_j), \, j =1,\ldots,M$, approximate values of $f(\pmb{x}_j)$.

Computational cost: $\mathcal{O}(N_1 \cdot N_2 \cdots N_d \cdot \log{N} + m^d \ M)$

Adjoint algorithm

Using the transposed index set

\[ I_{\pmb{n},m}^\top(\pmb{\ell}) = \{ j= 1,2, \ldots, M : \pmb{\ell} - m\pmb{1} \leq \pmb{n} \odot \pmb{x}_j \leq \pmb{\ell} + m \pmb{1} \},\]

we obtain the adjoint NFFT algorithm for the fast evaluation the of

\[ \hat{h}_{\pmb{k}} = \sum_{j = 1}^{M} f_j \, \mathrm{e}^{2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}_j}, \pmb{k} \in I_{\pmb{N}}^d,\]

for given coefficients $f_j \in \mathbb{C}, j =1,\ldots,M$.

Plan Structure

NFFT3.NFFT — Type

NFFT{D}

A NFFT (nonequispaced fast Fourier transform) plan, where D is the dimension.

Considering a D-dimensional trigonometric polynomial

\[f \colon \mathbb{T}^D \to \mathbb{C}, \; f(\pmb{x}) \colon = \sum_{\pmb{k} \in I_{\pmb{N}}^D} \hat{f}_{\pmb{k}} \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}}\]

with an index set $I_{\pmb{N}}^D \coloneqq \left\{ \pmb{k} \in \mathbb{Z}^D: - \frac{N_i}{2} \leq k_i \leq \frac{N_i}{2} - 1, \, i = 1,2,\ldots,D \right\}$ where $\pmb{N} \in (2\mathbb{N})^{D}$ is the multibandlimit. The NDFT (nonequispaced discrete Fourier transform) is its evaluation at $M \in \mathbb{N}$ arbitrary points $\pmb{x}_j \in [-0.5,0.5)^D$ for $j = 1, \ldots, M$,

\[f(\pmb{x}_j) \colon = \sum_{\pmb{k} \in I^D_{\pmb{N}}} \hat{f}_{\pmb{k}} \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}_j}\]

with given coefficients $\hat{f}_{\pmb{k}} \in \mathbb{C}$. The NFFT is an algorithm for the fast evaluation of the NDFT and the adjoint problem, the fast evaluation of the adjoint NDFT

\[\hat{h}_{\pmb{k}} \coloneqq \sum^{M}_{j = 1} f_j \, \mathrm{e}^{-2 \pi \mathrm{i} \, \pmb{k} \cdot \pmb{x}_j}, \, \pmb{k} \in I_{\pmb{N}}^D,\]

for given coefficients $f_j \in \mathbb{C}, j =1,2,\ldots,M$. Note that in general, the adjoint NDFT is not the inverse transform of the NDFT.

Fields

N - the multibandlimit $(N_1, N_2, \ldots, N_D)$ of the trigonometric polynomial $f$.
M - the number of nodes.
n - the oversampling $(n_1, n_2, \ldots, n_D)$ per dimension.
m - the window size. A larger m results in more accuracy but also a higher computational cost.
f1 - the NFFT flags.
f2 - the FFTW flags.
init_done - indicates if the plan is initialized.
finalized - indicates if the plan is finalized.
x - the nodes $\pmb{x}_j \in [-0.5,0.5)^D, j = 1, \ldots, M$.
f - the values $f(\pmb{x}_j)$ for the NFFT or the coefficients $f_j \in \mathbb{C}, j = 1, \ldots, M,$ for the adjoint NFFT.
fhat - the Fourier coefficients $\hat{f}_{\pmb{k}} \in \mathbb{C}$ for the NFFT or the values $\hat{h}_{\pmb{k}}, \pmb{k} \in I_{\pmb{N}}^D,$ for the adjoint NFFT.
plan - plan (C pointer).

Constructor

NFFT{D}( N::NTuple{D,Int32}, M::Int32, n::NTuple{D,Int32}, m::Int32, f1::UInt32, f2::UInt32 ) where D

Additional Constructor

NFFT( N::NTuple{D,Int32}, M::Int32, n::NTuple{D,Int32}, m::Int32, f1::UInt32, f2::UInt32 ) where {D}
NFFT( N::NTuple{D,Int32}, M::Int32 ) where {D}

source

Functions

NFFT3.nfft_trafo — Function

nfft_trafo(P)

computes the NDFT via the fast NFFT algorithm for provided nodes $\pmb{x}_j, j =1,2,\dots,M,$ in P.X and coefficients $\hat{f}_{\pmb{k}} \in \mathbb{C}, \pmb{k} \in I_{\pmb{N}}^D,$ in P.fhat.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_trafo_direct

source

NFFT3.nfft_trafo_direct — Function

nfft_trafo_direct(P)

computes the NDFT via naive matrix-vector multiplication for provided nodes $\pmb{x}_j, j =1,2,\dots,M,$ in P.X and coefficients $\hat{f}_{\pmb{k}} \in \mathbb{C}, \pmb{k} \in I_{\pmb{N}}^D,$ in P.fhat.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_trafo

source

NFFT3.nfft_adjoint — Function

nfft_adjoint(P)

computes the adjoint NDFT via the fast adjoint NFFT algorithm for provided nodes $\pmb{x}_j, j =1,2,\dots,M,$ in P.X and coefficients $f_j \in \mathbb{C}, j =1,2,\dots,M,$ in P.f.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_adjoint_direct

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NFFT3.nfft_adjoint_direct — Function

nfft_adjoint_direct(P)

computes the adjoint NDFT via naive matrix-vector multiplication for provided nodes $\pmb{x}_j, j =1,2,\dots,M,$ in P.X and coefficients $f_j \in \mathbb{C}, j =1,2,\dots,M,$ in P.f.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_adjoint

source

NFFT3.nfft_finalize_plan — Function

nfft_finalize_plan(P)

destroys a NFFT plan structure.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_init

source

NFFT3.nfft_init — Function

nfft_init(P)

intialises the NFFT plan in C. This function does not have to be called by the user.

Input

P - a NFFT plan structure.

See also

NFFT{D}, nfft_finalize_plan

source

Literature

[Keiner, Kunis, Potts, 2006] J. Keiner, S. Kunis, and D. Potts. Fast summation of radial functions on the sphere. Computing, 78:1–15, 2006.

[Plonka, Potts, Steidl, Tasche, 2018] G. Plonka, D. Potts, G. Steidl and M. Tasche. Numerical Fourier Analysis: Theory and Applications. Springer Nature Switzerland AG, 2018. doi: 10.1007/978-3-030-04306-3.