Inverse Problems, Parallel Imaging Acceleration, Compressed Sensing & Bedside Classifiers
Solving for underlying anatomy $\mathbf{x}$ from noisy measurements $\mathbf{y} = \mathbf{A}\mathbf{x} + \mathbf{n}$.
$\mathcal{R}(\mathbf{x}) = \|\mathbf{x}\|_2^2 \implies \hat{\mathbf{x}} = (\mathbf{A}^H \mathbf{A} + \lambda \mathbf{I})^{-1} \mathbf{A}^H \mathbf{y}$.
Stabilizes ill-conditioned singular values, but causes edge blurring in reconstructed image edges.
$\mathcal{R}(\mathbf{x}) = \|\mathbf{\Psi} \mathbf{x}\|_1$ (Wavelet / Total Variation).
Permits exact image recovery from undersampled k-space when aliasing artifacts are incoherent (pseudo-random trajectory).
Accelerating scans by factor $R$ via multi-channel spatial coil profiles.
Aliased pixel values across $C$ receive coils fold $R$ spatial locations:
$$\mathbf{y}_{\text{alias}} = \mathbf{S} \mathbf{x}_{\text{true}} \implies \hat{\mathbf{x}} = (\mathbf{S}^H \mathbf{\Psi}_n^{-1} \mathbf{S})^{-1} \mathbf{S}^H \mathbf{\Psi}_n^{-1} \mathbf{y}_{\text{alias}}$$Noise penalty during algebraic unfolding:
$$\text{SNR}_{\text{SENSE}} = \frac{\text{SNR}_{\text{full}}}{g \sqrt{R}} \quad (g \ge 1)$$High $g$-factor occurs when coil sensitivity fields overlap collinear spatial directions.
Mechanism: Real-time rolling Pearson correlation between arterial blood pressure (MAP) and cerebral tissue oxygen saturation ($\text{rSO}_2$).
Clinical Value: Identifies loss of myogenic tone ($\text{COx} > 0.3$) during therapeutic hypothermia, forecasting germinal matrix hemorrhage (IVH).
Mechanism: 1D convolutional neural network segments isoelectric suppressions (< 5 µV) in continuous bedside neonatal aEEG.
Clinical Value: Tracks severity of hypoxic injury during cooling and flags subclinical seizures hours before motor signs appear.
Next: Artifact Mitigation, Motion Correction & SNR Optimization