TorchKbNufft Documentation¶

Documentation | GitHub | Notebook Examples

About¶

torchkbnufft implements a non-uniform Fast Fourier Transform [1, 2] with Kaiser-Bessel gridding in PyTorch. The implementation is completely in Python, facilitating flexible deployment in readable code with no compilation. NUFFT functions are each wrapped as a torch.autograd.Function, allowing backpropagation through NUFFT operators for training neural networks.

This package was inspired in large part by the NUFFT implementation in the Michigan Image Reconstruction Toolbox (Matlab).

Installation¶

Simple installation can be done via PyPI:

pip install torchkbnufft

torchkbnufft only requires numpy, scipy, and torch as dependencies.

Operation Modes and Stages¶

The package has three major classes of NUFFT operation mode: table-based NUFFT interpolation, sparse matrix-based NUFFT interpolation, and forward/backward operators with Toeplitz-embedded FFTs [3]. Table interpolation is the standard operation mode, whereas the Toeplitz method is always the fastest for forward/backward NUFFTs. For some problems, sparse matrices may be fast. It is generally best to start with Table interpolation and then experiment with the other modes for your problem.

Sensitivity maps can be incorporated by passing them into a KbNufft or KbNufftAdjoint object. Auxiliary functions for calculating sparse interpolation matrices, density compensation functions, and Toeplitz filter kernels are also included.

For examples, see Basic Usage.

References¶

Fessler, J. A., & Sutton, B. P. (2003). Nonuniform fast Fourier transforms using min-max interpolation. IEEE Transactions on Signal Processing, 51(2), 560-574.
Beatty, P. J., Nishimura, D. G., & Pauly, J. M. (2005). Rapid gridding reconstruction with a minimal oversampling ratio. IEEE Transactions on Medical Imaging, 24(6), 799-808.
Feichtinger, H. G., Gr, K., & Strohmer, T. (1995). Efficient numerical methods in non-uniform sampling theory. Numerische Mathematik, 69(4), 423-440.

Basic Usage¶

torchkbnufft works primarily via PyTorch modules. You create a module with the properties of your imaging setup. The module will calculate a Kaiser-Bessel kernel and some interpolation parameters based on your inputs. Then, you apply the module to your data stored as PyTorch tensors. NUFFT operations are wrapped in torch.autograd.Function classes for backpropagation and training neural networks.

The following code loads a Shepp-Logan phantom and computes a single radial spoke of k-space data:

import torch
import torchkbnufft as tkbn
import numpy as np
from skimage.data import shepp_logan_phantom

x = shepp_logan_phantom().astype(np.complex)
im_size = x.shape
# convert to tensor, unsqueeze batch and coil dimension
# output size: (1, 1, ny, nx)
x = torch.tensor(x).unsqueeze(0).unsqueeze(0).to(torch.complex64)

klength = 64
ktraj = np.stack(
   (np.zeros(64), np.linspace(-np.pi, np.pi, klength))
)
# convert to tensor, unsqueeze batch dimension
# output size: (2, klength)
ktraj = torch.tensor(ktraj).to(torch.float)

nufft_ob = tkbn.KbNufft(im_size=im_size)
# outputs a (1, 1, klength) vector of k-space data
kdata = nufft_ob(x, ktraj)

The package also includes utilities for working with SENSE-NUFFT operators. The above code can be modified to include sensitivity maps.

smaps = torch.rand(1, 8, 400, 400) + 1j * torch.rand(1, 8, 400, 400)
sense_data = nufft_ob(x, ktraj, smaps=smaps.to(x))

This code first multiplies by the sensitivity coils in smaps, then computes a 64-length radial spoke for each coil. All operations are broadcast across coils, which minimizes interaction with the Python interpreter, helping computation speed.

Sparse matrices are an alternative to table interpolation. Their speed can vary, but they are a bit more accurate than standard table mode. The following code calculates sparse interpolation matrices and uses them to compute a single radial spoke of k-space data:

adjnufft_ob = tkbn.KbNufftAdjoint(im_size=im_size)

# precompute the sparse interpolation matrices
interp_mats = tkbn.calc_tensor_spmatrix(
   ktraj,
   im_size=im_size
)

# use sparse matrices in adjoint
image = adjnufft_ob(kdata, ktraj, interp_mats)

Sparse matrix multiplication is only implemented for real numbers in PyTorch, which can limit their speed.

The package includes routines for calculating embedded Toeplitz kernels and using them as FFT filters for the forward/backward NUFFT operations. This is very useful for gradient descent algorithms that must use the forward/backward ops in calculating the gradient. The following code shows an example:

toep_ob = tkbn.ToepNufft()

# precompute the embedded Toeplitz FFT kernel
kernel = tkbn.calc_toeplitz_kernel(ktraj, im_size)

# use FFT kernel from embedded Toeplitz matrix
image = toep_ob(image, kernel)

All of the examples included in this repository can be run on the GPU by sending the NUFFT object and data to the GPU prior to the function call, e.g.,

adjnufft_ob = adjnufft_ob.to(torch.device('cuda'))
kdata = kdata.to(torch.device('cuda'))
ktraj = ktraj.to(torch.device('cuda'))

image = adjnufft_ob(kdata, ktraj)

Similar to programming low-level code, PyTorch will throw errors if the underlying dtype and device of all objects are not matching. Be sure to make sure your data and NUFFT objects are on the right device and in the right format to avoid these errors.

For more details, please examine the API in torchkbnufft or the notebooks in the GitHub repository.

torchkbnufft¶

NUFFT Modules¶

These are the primary workhorse modules for applying NUFFT operations.

`KbInterp`	Non-uniform Kaiser-Bessel interpolation layer.
`KbInterpAdjoint`	Non-uniform Kaiser-Bessel interpolation adjoint layer.
`KbNufft`	Non-uniform FFT layer.
`KbNufftAdjoint`	Non-uniform FFT adjoint layer.
`ToepNufft`	Forward/backward NUFFT with Toeplitz embedding.

Utility Functions¶

Functions for calculating density compensation and Toeplitz kernels.

`calc_density_compensation_function`	Numerical density compensation estimation.
`calc_tensor_spmatrix`	Builds a sparse matrix for interpolation.
`calc_toeplitz_kernel`	Calculates an FFT kernel for Toeplitz embedding.

Math Functions¶

Complex mathematical operations (gradually being removed as of PyTorch 1.7).

`absolute`	Complex absolute value.
`complex_mult`	Complex multiplication.
`complex_sign`	Complex sign function value.
`conj_complex_mult`	Complex multiplication, conjugating second input.
`imag_exp`	Imaginary exponential.
`inner_product`	Complex inner product.

torchkbnufft.functional¶

fft_filter¶

fft_filter(image, kernel, norm='ortho')[source]¶

FFT-based filtering on an oversampled grid.

This is a wrapper for the operation

\[\text{output} = iFFT(\text{kernel}*FFT(\text{image})) \]

where \(iFFT\) and \(FFT\) are both implemented as oversampled FFTs.

Parameters

image (Tensor) – The image to be filtered.
kernel (Tensor) – FFT-domain filter.
norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

Return type

Tensor

Returns

Filtered version of image.

Interpolation Functions¶

kb_spmat_interp¶

kb_spmat_interp(image, interp_mats)[source]¶

Kaiser-Bessel sparse matrix interpolation.

See KbInterp for an overall description of interpolation.

To calculate the sparse matrix tuple, see calc_tensor_spmatrix().

Parameters

image (Tensor) – Gridded data to be interpolated to scattered data.
interp_mats (Tuple[Tensor, Tensor]) – 2-tuple of real, imaginary sparse matrices to use for sparse matrix KB interpolation.

Return type

Tensor

Returns

image calculated at scattered locations.

kb_spmat_interp_adjoint¶

kb_spmat_interp_adjoint(data, interp_mats, grid_size)[source]¶

Kaiser-Bessel sparse matrix interpolation adjoint.

See KbInterpAdjoint for an overall description of adjoint interpolation.

To calculate the sparse matrix tuple, see calc_tensor_spmatrix().

Parameters

data (Tensor) – Scattered data to be interpolated to gridded data.
interp_mats (Tuple[Tensor, Tensor]) – 2-tuple of real, imaginary sparse matrices to use for sparse matrix KB interpolation.

Return type

Tensor

Returns

data calculated at gridded locations.

kb_table_interp¶

kb_table_interp(image, omega, tables, n_shift, numpoints, table_oversamp, offsets)[source]¶

Kaiser-Bessel table interpolation.

See KbInterp for an overall description of interpolation and how to construct the function arguments.

Parameters

image (Tensor) – Gridded data to be interpolated to scattered data.
omega (Tensor) – k-space trajectory (in radians/voxel).
tables (List[Tensor]) – Interpolation tables (one table for each dimension).
n_shift (Tensor) – Size for fftshift, usually im_size // 2.
numpoints (Tensor) – Number of neighbors to use for interpolation.
table_oversamp (Tensor) – Table oversampling factor.
offsets (Tensor) – A list of offsets, looping over all possible combinations of numpoints.

Return type

Tensor

Returns

image calculated at scattered locations.

kb_table_interp_adjoint¶

kb_table_interp_adjoint(data, omega, tables, n_shift, numpoints, table_oversamp, offsets, grid_size)[source]¶

Kaiser-Bessel table interpolation adjoint.

See KbInterpAdjoint for an overall description of adjoint interpolation.

Parameters

data (Tensor) – Scattered data to be interpolated to gridded data.
omega (Tensor) – k-space trajectory (in radians/voxel).
tables (List[Tensor]) – Interpolation tables (one table for each dimension).
n_shift (Tensor) – Size for fftshift, usually im_size // 2.
numpoints (Tensor) – Number of neighbors to use for interpolation.
table_oversamp (Tensor) – Table oversampling factor.
offsets (Tensor) – A list of offsets, looping over all possible combinations of numpoints.
grid_size (Tensor) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size.

Return type

Tensor

Returns

data calculated at gridded locations.

NUFFT Functions¶

kb_spmat_nufft¶

kb_spmat_nufft(image, scaling_coef, im_size, grid_size, interp_mats, norm=None)[source]¶

Kaiser-Bessel NUFFT with sparse matrix interpolation.

See KbNufft for an overall description of the forward NUFFT.

To calculate the sparse matrix tuple, see calc_tensor_spmatrix().

Parameters

image (Tensor) – Image to be NUFFT’d to scattered data.
scaling_coef (Tensor) – Image-domain coefficients to pre-compensate for interpolation errors.
im_size (Tensor) – Size of image with length being the number of dimensions.
grid_size (Tensor) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size.
interp_mats (Tuple[Tensor, Tensor]) – 2-tuple of real, imaginary sparse matrices to use for sparse matrix KB interpolation.
norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

Return type

Tensor

Returns

image calculated at scattered Fourier locations.

kb_spmat_nufft_adjoint¶

kb_spmat_nufft_adjoint(data, scaling_coef, im_size, grid_size, interp_mats, norm=None)[source]¶

Kaiser-Bessel adjoint NUFFT with sparse matrix interpolation.

See KbNufftAdjoint for an overall description of the adjoint NUFFT.

To calculate the sparse matrix tuple, see calc_tensor_spmatrix().

Parameters

data (Tensor) – Scattered data to be iNUFFT’d to an image.
scaling_coef (Tensor) – Image-domain coefficients to compensate for interpolation errors.
im_size (Tensor) – Size of image with length being the number of dimensions.
grid_size (Tensor) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size.
interp_mats (Tuple[Tensor, Tensor]) – 2-tuple of real, imaginary sparse matrices to use for sparse matrix KB interpolation.
norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

Return type

Tensor

Returns

data transformed to an image.

kb_table_nufft¶

kb_table_nufft(image, scaling_coef, im_size, grid_size, omega, tables, n_shift, numpoints, table_oversamp, offsets, norm=None)[source]¶

Kaiser-Bessel NUFFT with table interpolation.

See KbNufft for an overall description of the forward NUFFT.

Parameters

image (Tensor) – Image to be NUFFT’d to scattered data.
scaling_coef (Tensor) – Image-domain coefficients to pre-compensate for interpolation errors.
im_size (Tensor) – Size of image with length being the number of dimensions.
grid_size (Tensor) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size.
omega (Tensor) – k-space trajectory (in radians/voxel).
tables (List[Tensor]) – Interpolation tables (one table for each dimension).
n_shift (Tensor) – Size for fftshift, usually im_size // 2.
numpoints (Tensor) – Number of neighbors to use for interpolation.
table_oversamp (Tensor) – Table oversampling factor.
offsets (Tensor) – A list of offsets, looping over all possible combinations of numpoints.
norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

Return type

Tensor

Returns

image calculated at scattered Fourier locations.

kb_table_nufft_adjoint¶

kb_table_nufft_adjoint(data, scaling_coef, im_size, grid_size, omega, tables, n_shift, numpoints, table_oversamp, offsets, norm=None)[source]¶

Kaiser-Bessel NUFFT adjoint with table interpolation.

See KbNufftAdjoint for an overall description of the adjoint NUFFT.

Parameters

data (Tensor) – Scattered data to be iNUFFT’d to an image.
scaling_coef (Tensor) – Image-domain coefficients to compensate for interpolation errors.
im_size (Tensor) – Size of image with length being the number of dimensions.
grid_size (Tensor) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size.
omega (Tensor) – k-space trajectory (in radians/voxel).
tables (List[Tensor]) – Interpolation tables (one table for each dimension).
n_shift (Tensor) – Size for fftshift, usually im_size // 2.
numpoints (Tensor) – Number of neighbors to use for interpolation.
table_oversamp (Tensor) – Table oversampling factor.
offsets (Tensor) – A list of offsets, looping over all possible combinations of numpoints.
norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

Return type

Tensor

Returns

data transformed to an image.

TorchKbNufft Documentation¶

About¶

Installation¶

Operation Modes and Stages¶

References¶

Basic Usage¶

torchkbnufft¶

NUFFT Modules¶

Utility Functions¶

Math Functions¶

torchkbnufft.functional¶

fft_filter¶

Interpolation Functions¶

kb_spmat_interp¶

kb_spmat_interp_adjoint¶

kb_table_interp¶

kb_table_interp_adjoint¶

NUFFT Functions¶

kb_spmat_nufft¶

kb_spmat_nufft_adjoint¶

kb_table_nufft¶

kb_table_nufft_adjoint¶

Indices and tables¶