Exercises

$\mathcal{R}(\alpha f + \beta g)(\theta, s) = \int_{-\infty}^{\infty} [\alpha f + \beta g](s\cos\theta - t\sin\theta, s\sin\theta + t\cos\theta)\,dt$

$= \alpha \int f(\cdots)\,dt + \beta \int g(\cdots)\,dt = \alpha\,\mathcal{R}f(\theta,s) + \beta\,\mathcal{R}g(\theta,s). \quad \blacksquare$

$\mathcal{R}f(\theta, s) = \int \delta(s\cos\theta - t\sin\theta - x_0)\,\delta(s\sin\theta + t\cos\theta - y_0)\,dt.$

The sifting property of $\delta$ forces $s = x_0\cos\theta + y_0\sin\theta$, producing a sinusoidal track in the $(\theta, s)$ plane. $\blacksquare$

$\frac{\partial(k_x, k_y)}{\partial(\nu, \theta)} = \begin{pmatrix} \cos\theta & -\nu\sin\theta \\ \sin\theta & \nu\cos\theta \end{pmatrix}.$

The determinant is $\nu\cos^2\theta + \nu\sin^2\theta = \nu$.

$dk_x\,dk_y = |\nu|\,d\nu\,d\theta$.

To cover the full plane with $\theta \in [0, \pi)$ and $\nu \in (-\infty, \infty)$, we need the absolute value. This is the ramp filter. $\blacksquare$

With $s = 10$ and $C \approx 5$ (typical for random Fourier sampling): $M \approx 5 \times 10 \times \log^4(65536) \approx 5 \times 10 \times 16^4 \approx 5 \times 10 \times 65536$. But this is the theoretical worst-case.

In practice (ignoring polylog factors): $M \approx 4s \text{ to } 6s = 40$ to $60$ samples suffice for a 10-sparse signal.

With $N^2 = 65536$ pixels and $M = 50$ samples: $M/N^2 = 50/65536 \approx 0.08\%$.

In practice, medical images are not truly 10-sparse but approximately sparse with $s \approx 1000$--$5000$ significant coefficients, requiring $M \approx 16000$--$32000$ samples ($R = 2$--$4$). This matches the empirical $R = 4$--$8$ used clinically.

Skipping every other k-space line creates a 2-fold aliased image. At each voxel $\mathbf{r}$, the aliased image from coil $c$ is

$I_c^{\text{aliased}}(\mathbf{r}) = S_c(\mathbf{r})\,m(\mathbf{r}) + S_c(\mathbf{r} + \Delta\mathbf{r})\,m(\mathbf{r} + \Delta\mathbf{r}),$

where $\Delta\mathbf{r}$ is the aliasing shift (half the field of view).

Stacking two coils:

$$\begin{pmatrix} I_1^{\text{aliased}} \\ I_2^{\text{aliased}} \end{pmatrix} = \begin{pmatrix} S_1(\mathbf{r}) & S_1(\mathbf{r}+\Delta\mathbf{r}) \\ S_2(\mathbf{r}) & S_2(\mathbf{r}+\Delta\mathbf{r}) \end{pmatrix} \begin{pmatrix} m(\mathbf{r}) \\ m(\mathbf{r}+\Delta\mathbf{r}) \end{pmatrix}.$$

Invert the $2 \times 2$ sensitivity matrix at each voxel pair. $\blacksquare$

$\lambda = 3 \times 10^8 / 77 \times 10^9 = 3.896$ mm. $D = 64 \times 3.896/2 = 124.7$ mm. $F = 5000 / 124.7 = 40.1$. $\delta_{\mathrm{lat}} = 3.896 \times 40.1 = 156$ mm $\approx 15.6$ cm.

Require $\delta_{\mathrm{lat}} = 10$ mm. $F = 10 / 3.896 = 2.57$. $D = z / F = 5000 / 2.57 = 1948$ mm $\approx 2$ m. $N_e = D / (\lambda/2) = 1948 / 1.948 = 1000$ elements.

A 1000-element array at half-wavelength spacing spans 2 m — feasible for a wall-mounted system but impractical for a mobile platform.

In the Fourier domain, $\mathcal{R}^*\mathcal{R}$ acts as multiplication by $1/|\nu|$ (the inverse of the ramp filter). This means $\widetilde{\mathcal{R}^*\mathcal{R}f}(k_x, k_y) = \tilde{f}(k_x, k_y) / \sqrt{k_x^2 + k_y^2}$.

Back-projection without filtering produces a $1/r$ blurred image — each point scatterer is reconstructed as a radially symmetric blur falling off as $1/r$. The ramp filter $|\nu|$ in FBP compensates for this, restoring the correct Fourier weighting. $\blacksquare$

CT: $M = 36 \times 256 = 9216$ measurements for $N = 4096$ unknowns (overdetermined, $M/N = 2.25$). RF: $M = 6 \times 6 \times 10 = 360$ measurements for $N = 4096$ unknowns (severely underdetermined, $M/N = 0.088$).

CT: 36 projections cover $[0, \pi)$ at 5-degree intervals, filling 36 radial lines in the full Fourier plane. RF: 360 measurements sample 360 points on a limited set of Ewald arcs, covering a much smaller region of the Fourier plane with large gaps.

$\mathbf{A}_{\mathrm{CT}}$ is overdetermined with good Fourier coverage: its condition number is moderate ($\kappa \sim 10$--$100$). FBP gives acceptable results.

$\mathbf{A}$ is severely underdetermined with sparse, non-uniform Fourier coverage: its condition number is large or effectively infinite (many singular values near zero). Regularization is mandatory; the reconstruction is fundamentally non-unique without priors.

Randomly split the $M$ Tx-Rx-frequency measurement indices into $\Omega_1$ (training, ~70% of measurements) and $\Omega_2$ (validation, ~30%). Let $\mathbf{A}_{\Omega_k}$ denote the rows of $\mathbf{A}$ indexed by $\Omega_k$.

$\mathcal{L}(\theta) = \|\mathbf{A}_{\Omega_2} f_\theta(\mathbf{A}_{\Omega_1}^{H} \mathbf{y}_{\Omega_1}) - \mathbf{y}_{\Omega_2}\|_2^2,$$where$f_\theta$ is the unrolled OAMP network taking as input the backpropagation image from the training measurements.

(a) The RF forward operator has Kronecker structure, so the sub-operators $\mathbf{A}_{\Omega_k}$ lose this structure — the Kronecker speedup (Ch 07) is partially lost. (b) RF measurements are far fewer than MRI k-space samples, so the partition reduces an already small dataset. (c) The signal-to-noise ratio per measurement is typically lower in RF, making the validation loss noisier.

Replace FBP with the backpropagation (matched-filter) image: $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^{H} \mathbf{D}^{-1} \mathbf{y}$, where $\mathbf{D}$ is a diagonal normalisation matrix.

$\hat{\mathbf{c}}_{\mathrm{CNN}} = \mathcal{G}_\theta(\hat{\mathbf{c}}^{\text{BP}})$. The U-Net $\mathcal{G}_\theta$ maps the artifact-corrupted backpropagation image to the clean reflectivity.

In CT, FBP with full angular coverage gives PSNR $\sim$35--40 dB. The U-Net only removes minor noise and truncation artifacts. In RF, the backpropagation image has PSNR $\sim$10--20 dB with severe sidelobe artifacts from the incomplete PSF. The U-Net must remove much more severe artifacts, requiring larger receptive fields and more training data. This is why model-based architectures (E2E VarNet, unrolled OAMP) that embed the forward model outperform pure post-processing for RF imaging.

1. Both are linear operators mapping an image/scene to measurements.
2. Both sample in a Fourier-like domain (k-space for MRI, Ewald arcs for RF).
3. Both suffer from incomplete coverage that motivates CS/regularized reconstruction.

1. $\mathbf{F}_{\Omega}$ is a subsampled DFT (fast $O(N\log N)$ products via FFT); $\mathbf{A}$ has Kronecker structure ($O(N^{3/2})$ products).
2. MRI k-space is Cartesian or along smooth trajectories; RF k-space samples lie on curved Ewald arcs.
3. $\mathbf{F}_{\Omega}$ is well-conditioned (restricted isometry); $\mathbf{A}$ is typically ill-conditioned with rapidly decaying singular values.

Ultrasound: $\lambda_{\mathrm{US}} = 1540 / (5 \times 10^6) = 0.308$ mm. RF: $\lambda_{\mathrm{RF}} = 3 \times 10^8 / (77 \times 10^9) = 3.896$ mm.

US: $D = 50 \times 0.308 = 15.4$ mm (a small lesion). RF: $D = 50 \times 3.896 = 194.8$ mm (a medium-sized object).

When $D_{\mathrm{obj}} / \lambda \gg 1$, the ray-optics (CT-like) approximation is reasonable. When $D_{\mathrm{obj}} / \lambda \sim 1$--$50$, diffraction is significant and the Born/Rytov model (Ch 05-06) is needed. Both US and RF typically operate in the $D/\lambda \sim 10$--$100$ regime, where diffraction matters but is manageable under the first-order Born approximation.

Each DAS image $I_n = \mathbf{A}_n^H \mathbf{A}_n \boldsymbol{\sigma}$ has PSF $\mathrm{PSF}_n = \mathbf{A}_n^H \mathbf{A}_n$. Coherent compounding sums the images linearly:

$I_{\mathrm{comp}} = \sum_n \mathbf{A}_n^H \mathbf{A}_n \boldsymbol{\sigma}$,

so $\mathrm{PSF}_{\mathrm{comp}} = \sum_n \mathbf{A}_n^H \mathbf{A}_n$.

Each single plane wave illuminates from angle $\alpha_n$, covering a narrow angular cone in k-space. The sum of PSFs covers a wider angular range, narrowing the mainlobe in the lateral direction. With $N_{\mathrm{pw}}$ angles spanning $\pm\alpha_{\max}$, the effective aperture increases, and lateral resolution improves by a factor $\sim \sin(\alpha_{\max})$. This is the ultrasound analogue of multi-view coherent RF imaging.

Each projection is a line integral (one row of $\mathbf{A}_{\mathrm{CT}}$) and provides $N$ samples. With $N_v$ random projections: $M = N_v \times N$. The restricted isometry of the Radon matrix requires $N_v \gtrsim s\,\text{polylog}(N) / N$, giving $M \gtrsim s\,\text{polylog}(N)$.

Each k-space line gives $N$ Fourier coefficients. With $N_l$ random lines: $M = N_l \times N$. Random Fourier sampling requires $M \gtrsim s \log^4(N^2)$ — similar to CT.

Each Tx-Rx pair at one frequency gives a single complex measurement. With $M_{\mathrm{RF}}$ such measurements: $M_{\mathrm{RF}} \gtrsim s \cdot \mu^2 \cdot \text{polylog}(N^2)$, where $\mu$ is the coherence between the sensing matrix columns and the sparsifying basis. For the RF sensing matrix, $\mu$ can be significantly larger than for Fourier-based modalities, requiring more measurements per unit of sparsity.

CT and MRI benefit from structured measurements (each projection or k-space line provides $N$ correlated samples). RF measurements are less structured, leading to higher sampling requirements for the same sparsity level. This partly explains why RF imaging needs more aggressive regularization and benefits more from learned priors.

The U-Net/CNN blocks that learn image-domain priors (edge detection, denoising, artifact removal). These capture generic image processing operations that partially transfer across domains.

(a) Forward/adjoint operators: $\mathcal{A}_{\mathrm{MRI}}$ vs $\mathbf{A}$. (b) Data-consistency layers: must use the correct forward model. (c) Normalization layers: image statistics differ.

Phase 1: Pre-train a MoDL/VarNet on abundant MRI data with $\mathcal{A}_{\mathrm{MRI}}$. Phase 2: Replace the forward operator with $\mathbf{A}$, freeze the U-Net weights, and train only the data-consistency layers on simulated RF data. Phase 3: Unfreeze the U-Net and fine-tune the full network on the (small) RF dataset with a reduced learning rate.

(a) Negative transfer: if MRI priors (smooth tissue) bias the network toward over-smoothing discrete RF scatterers. (b) Domain gap in noise statistics: MRI noise is approximately Gaussian; RF noise may have non-Gaussian components from clutter. (c) Catastrophic forgetting during fine-tuning if the RF dataset is too small.