Model Extensions

Model Extensions#

The basic GROG gridding model can be extended to account for physical effects that are present in real MRI acquisitions. PyGROG implements three main extensions via gadgets that wrap a base SparseFFT operator.

Parallel Imaging (Multi-Coil)#

The multi-coil acquisition model is:

\[ y_{i,\ell} = \int S_\ell(\mathbf{r})\, x(\mathbf{r})\, e^{-2\pi i \mathbf{r} \cdot \mathbf{k}_i}\, d\mathbf{r} + n_{i,\ell}, \quad \ell = 1, \ldots, L, \]

or in operator form:

\[\begin{split} \tilde{\mathbf{y}} = \begin{bmatrix} \mathcal{F}_\Omega S_1 \\ \vdots \\ \mathcal{F}_\Omega S_L \end{bmatrix} x + \mathbf{n} = \tilde{\mathcal{F}}_\Omega x + \mathbf{n}. \end{split}\]

SparseFFT supports coil sensitivity maps via the smaps argument. When smaps is provided, the forward direction (image → k-space, i.e. the adjoint NUFFT direction) expands the image with each coil map before applying the FFT:

\[ (\tilde{\mathcal{F}}_\Omega^* x)_\ell = \mathcal{F}_\Omega^* (S_\ell^* x), \]

and the adjoint direction (k-space → image, i.e. the forward NUFFT direction) performs coil-combination:

\[ \tilde{\mathcal{F}}_\Omega \tilde{y} = \sum_{\ell=1}^L S_\ell^* \mathcal{F}_\Omega \tilde{y}_\ell. \]

Coil sensitivity maps can be estimated from the ACR using nlinv_calib() (NLINV algorithm).

Low-Rank Temporal Subspace#

In dynamic MRI (cardiac, quantitative mapping, etc.) the image evolves over time. Instead of reconstructing each frame independently, the temporal signal is constrained to a low-rank subspace spanned by \(K \ll T\) basis vectors \(\{\phi_k\}_{k=1}^K\):

\[ x(\mathbf{r}, t) \approx \sum_{k=1}^K \alpha_k(\mathbf{r})\, \phi_k(t). \]

The \(K\) spatial coefficient maps \(\{\alpha_k\}\) are the unknowns; the basis \(\Phi \in \mathbb{C}^{K \times T}\) is computed once from simulated or measured signal dictionaries via truncated SVD.

The extended encoding operator maps coefficients to multi-frame k-space:

\[ y_t = \mathcal{F}_\Omega \left( \sum_{k=1}^K \phi_k(t)\, \alpha_k \right) = \sum_{k=1}^K \phi_k(t)\, \mathcal{F}_\Omega \alpha_k. \]

PyGROG implements this with SubspaceSparseFFT, which fuses the basis projection directly into the sparse FFT. The standalone SubspaceProjection class handles the projection/expansion alone (without the FFT), which is useful for post-processing or preconditioning.

Off-Resonance Correction#

B0 field inhomogeneities cause a spatially and temporally varying phase during readout:

\[ y(t) = \int S(\mathbf{r})\, x(\mathbf{r})\, e^{i 2\pi \Delta f(\mathbf{r})\, t}\, e^{-2\pi i \mathbf{r} \cdot \mathbf{k}(t)}\, d\mathbf{r}, \]

where \(\Delta f(\mathbf{r})\) is the B0 field map in Hz and \(t\) is the readout time of sample \(\mathbf{k}(t)\).

The off-resonance exponential is approximated by a low-rank factorisation (Sutton et al., 2003):

\[ e^{i 2\pi \Delta f(\mathbf{r})\, t} \approx \sum_{\ell=1}^{L_\text{orc}} B_\ell(t)\, C_\ell(\mathbf{r}), \]

where \(B_\ell \in \mathbb{C}^{n_\text{samples}}\) and \(C_\ell \in \mathbb{C}^{n_x \times n_y}\) are the temporal and spatial basis functions obtained by SVD of the phase modulation matrix.

The extended operator is:

\[ y(t) \approx \sum_{\ell=1}^{L_\text{orc}} B_\ell(t)\, \mathcal{F}_\Omega [C_\ell(\mathbf{r})\, S(\mathbf{r})\, x(\mathbf{r})], \]

which requires only \(L_\text{orc}\) standard FFT evaluations. PyGROG implements this in OffResonanceCorrection, reusing the factorisation from mri-nufft.

Tip

All three extensions can be combined: use a SubspaceSparseFFT as the base operator for OffResonanceCorrection to obtain a joint subspace + off-resonance corrected operator.

References#

  • Griswold MA, et al. GRAPPA. Magn Reason Med. 2002;47(6):1202-10.

  • Seiberlich N, et al. GROG. Magn Reason Med. 2007;58(6):1257-65.

  • Uecker M, et al. NLINV. Magn Reason Med. 2008;60(3):674-82.

  • Sutton BP, et al. Fast, iterative image reconstruction for MRI in the presence of field inhomogeneities. IEEE Trans Med Imaging. 2003;22(2):178-88.

  • Liang ZP. Spatiotemporal imaging with partially separable functions. IEEE ISBI. 2007:988-91.