calc_toeplitz_kernel(omega, im_size, weights=None, norm=None, grid_size=None, numpoints=6, table_oversamp=1024, kbwidth=2.34, order=0.0)[source]

Calculates an FFT kernel for Toeplitz embedding.

The kernel is calculated using a adjoint NUFFT object. If the adjoint applies \(A'\), then this script calculates \(D\) where \(F'DF \approx A'WA\), where \(F\) is a DFT matrix and \(W\) is a set of non-Cartesian k-space weights. \(D\) can then be used to approximate \(A'WA\) without any interpolation operations.

For details on Toeplitz embedding, see Efficient numerical methods in non-uniform sampling theory (Feichtinger et al.).

This function has optional parameters for initializing a NUFFT object. See KbNufftAdjoint for details.


This function is intended to be used in conjunction with ToepNufft for forward operations.

  • omega should be of size (len(im_size), klength) or (N, len(im_size), klength), where klength is the length of the k-space trajectory.

  • omega (Tensor) – k-space trajectory (in radians/voxel).

  • im_size (Sequence[int]) – Size of image with length being the number of dimensions.

  • weights (Optional[Tensor]) – Non-Cartesian k-space weights (e.g., density compensation). Default: torch.ones(omega.shape[1])

  • norm (Optional[str]) – Whether to apply normalization with the FFT operation. Options are "ortho" or None.

  • grid_size (Optional[Sequence[int]]) – Size of grid to use for interpolation, typically 1.25 to 2 times im_size. Default: 2 * im_size

  • numpoints (Union[int, Sequence[int]]) – Number of neighbors to use for interpolation in each dimension.

  • n_shift – Size for fftshift. Default: im_size // 2.

  • table_oversamp (Union[int, Sequence[int]]) – Table oversampling factor.

  • kbwidth (float) – Size of Kaiser-Bessel kernel.

  • order (Union[float, Sequence[float]]) – Order of Kaiser-Bessel kernel.

Return type



The FFT kernel for approximating the forward/adjoint operation.


>>> image = torch.randn(1, 1, 8, 8) + 1j * torch.randn(1, 1, 8, 8)
>>> omega = torch.rand(2, 12) * 2 * np.pi - np.pi
>>> toep_ob = tkbn.ToepNufft()
>>> kernel = tkbn.calc_toeplitz_kernel(omega, im_size=(8, 8))
>>> image = toep_ob(image, kernel)