Magnetic Resonance Imaging and Compressed Sensing

In MRI, the measured signal at time $t$ is

$$s(t) = \int_{\Omega} m(\mathbf{r})\, e^{-j2\pi \mathbf{k}(t) \cdot \mathbf{r}}\,d\mathbf{r},$$

where $m(\mathbf{r})$ is the magnetization (the "image"), $\mathbf{k}(t) = \frac{\gamma}{2\pi}\int_0^t \mathbf{G}(\tau)\,d\tau$ is the k-space trajectory determined by the gradient waveforms $\mathbf{G}(\tau)$, and $\gamma$ is the gyromagnetic ratio.

Key observation: Each measurement is a sample of the Fourier transform $\tilde{m}(\mathbf{k})$ at position $\mathbf{k}(t)$. The MRI forward operator is thus an undersampled Fourier transform.

Discretize the image on an $N \times N$ grid: $\mathbf{m} \in \mathbb{C}^{N^2}$. The undersampled MRI forward model is

$$\mathbf{y} = \mathbf{F}_{\Omega}\mathbf{m} + \mathbf{w},$$

where $\mathbf{F}_{\Omega} \in \mathbb{C}^{M \times N^2}$ is the Fourier encoding matrix restricted to the $M$ acquired k-space locations ($M < N^2$ for accelerated acquisition), and $\mathbf{w} \sim \mathcal{N}(0, \sigma^2\mathbf{I})$ is additive noise.

Acceleration factor: $R = N^2 / M$. Clinical MRI commonly uses $R = 4$--$8$; research settings push to $R = 16$ or beyond.

Modern MRI uses $C$ receive coils, each with a spatially varying sensitivity map $\mathbf{S}_c(\mathbf{r})$. The signal from coil $c$ at k-space location $\mathbf{k}$ is

$$y_c(\mathbf{k}) = \int \mathbf{S}_c(\mathbf{r})\,m(\mathbf{r})\, e^{-j2\pi\mathbf{k}\cdot\mathbf{r}}\,d\mathbf{r}.$$

In matrix form: $\mathbf{y}_c = \mathbf{F}_{\Omega} \text{diag}(\mathbf{s}_c)\,\mathbf{m}$. Stacking all coils:

$$\mathbf{y} = \underbrace{\begin{bmatrix} \mathbf{F}_{\Omega}\text{diag}(\mathbf{s}_1) \\ \vdots \\ \mathbf{F}_{\Omega}\text{diag}(\mathbf{s}_C) \end{bmatrix}}_{\mathcal{A}_{\mathrm{MRI}}} \mathbf{m} + \mathbf{w}.$$

SENSE inverts this system via the pseudoinverse, exploiting the redundancy from multiple coils to recover undersampled data. The effective acceleration is limited by the number of coils: $R_{\max} \leq C$.

GRAPPA estimates missing k-space data from acquired neighbors using linear interpolation kernels learned from a fully sampled calibration region (autocalibration signal, ACS). For each missing k-space point $k_0$, GRAPPA computes

$$\hat{y}_c(k_0) = \sum_{c'=1}^{C} \sum_{b \in \mathcal{B}} w_{c,c',b}\,y_{c'}(k_0 + b),$$

where $\mathcal{B}$ is the interpolation kernel support and the weights $\{w_{c,c',b}\}$ are fit from the ACS lines.

Parallel to RF imaging: GRAPPA is a k-space interpolation method, analogous to gridding and NUFFT interpolation used in diffraction tomography (Ch 15). Both exploit local k-space correlations to fill missing data.

The E2E VarNet (Sriram et al., 2020) is a learned MRI reconstruction architecture that unrolls the variational formulation

$$\hat{\mathbf{m}} = \arg\min_{\mathbf{m}} \|\mathcal{A}_{\mathrm{MRI}}\mathbf{m} - \mathbf{y}\|_2^2 + \lambda\,R(\mathbf{m})$$

into $T$ gradient-descent layers with learned refinement networks. Each layer $t$ performs:

1. Data consistency: $\mathbf{r}^{(t)} = \mathbf{m}^{(t)} - \eta_t \mathcal{A}_{\mathrm{MRI}}^H(\mathcal{A}_{\mathrm{MRI}}\mathbf{m}^{(t)} - \mathbf{y})$
2. Learned refinement: $\mathbf{m}^{(t+1)} = \mathbf{r}^{(t)} - \mathcal{G}_{\theta_t}(\mathbf{r}^{(t)})$

where $\mathcal{G}_{\theta_t}$ is a U-Net with layer-specific parameters. This is structurally identical to the unrolled OAMP with ProxNet from Ch 18 — the only difference is the forward operator ($\mathcal{A}_{\mathrm{MRI}}$ vs $\mathbf{A}$).