KbNufft

class KbNufft(im_size, grid_size=None, numpoints=6, n_shift=None, table_oversamp=1024, kbwidth=2.34, order=0.0, dtype=None, device=None)[source]

Non-uniform FFT layer.

This object applies the FFT and interpolates a grid of Fourier data to off-grid locations using a Kaiser-Bessel kernel. Mathematically, in one dimension it estimates \(Y_m, m \in [0, ..., M-1]\) at frequency locations \(\omega_m\) from \(X_k, k \in [0, ..., K-1]\), the oversampled DFT of \(x_n, n \in [0, ..., N-1]\). To perform the estimate, this layer applies

\[X_k = \sum_{n=0}^{N-1} s_n x_n e^{-i \gamma k n} \]
\[Y_m = \sum_{j=1}^J X_{\{k_m+j\}_K} u^*_j(\omega_m) \]

In the first step, an image-domain signal \(x_n\) is converted to a gridded, oversampled frequency-domain signal, \(X_k\). The scaling coefficeints \(s_n\) are multiplied to precompensate for NUFFT interpolation errors. The oversampling coefficient is \(\gamma = 2\pi / K, K >= N\).

In the second step, \(u\), the Kaiser-Bessel kernel, is used to estimate \(X_k\) at off-grid frequency locations \(\omega_m\). \(k_m\) is the index of the root offset of nearest samples of \(X\) to frequency location \(\omega_m\), and \(J\) is the number of nearest neighbors to use from \(X_k\). Multiple dimensions are handled separably. For a detailed description see Nonuniform fast Fourier transforms using min-max interpolation (JA Fessler and BP Sutton).

When called, the parameters of this class define properties of the kernel and how the interpolation is applied.

  • im_size is the size of the base image, analagous to \(N\).

  • grid_size is the size of the grid after forward FFT, analogous to \(K\). To reduce errors, NUFFT operations are done on an oversampled grid to reduce interpolation distances. This will typically be 1.25 to 2 times im_size.

  • numpoints is the number of nearest neighbors to use for interpolation, i.e., \(J\).

  • n_shift is the FFT shift distance, typically im_size // 2.

Parameters:

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

  • 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 (Optional[Sequence[int]]) – 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.

  • dtype (Optional[dtype]) – Data type for tensor buffers. Default: torch.get_default_dtype()

  • device (Optional[device]) – Which device to create tensors on. Default: torch.device('cpu')

Examples

>>> image = torch.randn(1, 1, 8, 8) + 1j * torch.randn(1, 1, 8, 8)
>>> omega = torch.rand(2, 12) * 2 * np.pi - np.pi
>>> kb_ob = tkbn.KbNufft(im_size=(8, 8))
>>> data = kb_ob(image, omega)
forward(image, omega, interp_mats=None, smaps=None, norm=None)[source]

Apply FFT and interpolate from gridded data to scattered data.

Input tensors should be of shape (N, C) + im_size, where N is the batch size and C is the number of sensitivity coils. omega, the k-space trajectory, should be of size (len(grid_size), klength) or (N, len(grid_size), klength), where klength is the length of the k-space trajectory.

Note

If the batch dimension is included in omega, the interpolator will parallelize over the batch dimension. This is efficient for many small trajectories that might occur in dynamic imaging settings.

If your tensors are real, ensure that 2 is the size of the last dimension.

Parameters:

  • image (Tensor) – Object to calculate off-grid Fourier samples from.

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

  • interp_mats (Optional[Tuple[Tensor, Tensor]]) – 2-tuple of real, imaginary sparse matrices to use for sparse matrix NUFFT interpolation (overrides default table interpolation).

  • smaps (Optional[Tensor]) – Sensitivity maps. If input, these will be multiplied before the forward NUFFT.

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

Return type:

Tensor

Returns:

image calculated at Fourier frequencies specified by omega.