Phase 3 ยท Computational Inversion & Machine Learning
Image Reconstruction, Inverse Problems & Biomarker Extraction
Mathematical formulations of ill-posed inverse problems: SENSE/GRAPPA parallel imaging unfolding, compressed sensing, diffuse optical tomography inversion, and deep learning for bedside biomarker classification in neonatal encephalopathy.
Course: IDNE 701 (Fall 2026)
Date: Oct 23, 2026 (Week 9)
Duration: 80 Minutes
Instructor: Mark Bolding, PhD
01. Formulating the Forward & Inverse Problem
In any neuroimaging system, our measurement vector $\mathbf{y} \in \mathbb{C}^M$ is related to the discretized underlying neural source image $\mathbf{x} \in \mathbb{C}^N$ through a linear system operator $\mathbf{A}$ contaminated by additive noise $\mathbf{n}$:
$$\mathbf{y} = \mathbf{A}\mathbf{x} + \mathbf{n}$$
Equation 1: Generalized discrete forward imaging problem
Because $M < N$ (undersampled parallel imaging) or because $\mathbf{A}$ has rapidly decaying singular values (diffuse optical tomography, EEG source localization), direct matrix inversion is unstable or ill-posed. We formulate the reconstruction as a penalized optimization problem:
$$\hat{\mathbf{x}} = \arg\min_{\mathbf{x}} \left\{ \|\mathbf{A}\mathbf{x} - \mathbf{y}\|_2^2 + \lambda \mathcal{R}(\mathbf{x}) \right\}$$
Equation 2: Regularized Tikhonov / Sparse Inverse Formulation
where $\mathcal{R}(\mathbf{x}) = \|\mathbf{x}\|_2^2$ represents classical Tikhonov smoothing, and $\mathcal{R}(\mathbf{x}) = \|\mathbf{\Psi}\mathbf{x}\|_1$ enforces sparsity in a wavelet or finite-difference gradient domain (Compressed Sensing).
02. Parallel Imaging Acceleration: SENSE & GRAPPA
By skipping k-space phase-encoding lines by an acceleration factor $R$, scan duration is reduced by $1/R$. However, spatial undersampling causes $R$ equidistant aliased voxels to fold onto one another in image space:
$$\mathbf{I}_{\text{aliased},\gamma}(y) = \sum_{r=0}^{R-1} S_\gamma(y + r \cdot \Delta y) \cdot \rho(y + r \cdot \Delta y)$$
Equation 3: SENSE coil sensitivity unfolding equation for channel ฮณ
SENSE inverts this coil sensitivity matrix $\mathbf{S}$ at every pixel. The local noise amplification during unfolding is quantified by the Geometry Factor ($g$-factor):
$$\text{SNR}_{\text{SENSE}} = \frac{\text{SNR}_{\text{full}}}{g \sqrt{R}}$$
where $g \ge 1$ depends on the spatial orthogonality of the receive array coil profiles.
03. Physics-Informed Neural Networks & Biomarker Models
Rather than treating neural networks as black-box image generators, modern neuroengineering employs unrolled physics-informed networks (e.g., MoDL, VarNet) where each network layer alternates between learned regularization and explicit data consistency projection with the forward operator $\mathbf{A}$.
๐ Unrolled Deep Reconstruction
Replaces gradient descent steps with learned convolutional blocks. Enforces strict consistency with acquired k-space or optical fluence data, preventing hallucination of non-existent anatomy.
๐ง Latent Space Biomarker Classifiers
High-dimensional reconstructed volumes or time series are projected into low-dimensional latent embeddings (Variational Autoencoders, 1D/3D CNNs) to classify pathological states: ischemic penumbra, burst-suppression ratio, and vascular autoregulation collapse.
04. Grand Challenge 06: Neonatal Biomarker Extraction in the NICU
๐ถ Automated Bedside Biomarker Scoring for HIE & IVH
In neonatal intensive care, clinicians need continuous quantitative decision support rather than raw images. Computational pipelines must automatically extract and track physiological biomarkers:
๐ Cerebral Autoregulation Index (COx)
Computes the continuous moving Pearson correlation coefficient between slow fluctuations of Mean Arterial Blood Pressure (MAP) and optical $\text{rSO}_2$:
$$\text{COx} = \frac{\text{Cov}(\text{MAP}, \text{rSO}_2)}{\sigma_{\text{MAP}} \sigma_{\text{rSO}_2}}$$
A positive $\text{COx} > 0.3$ flags loss of autoregulatory vasoconstriction (pressure-passive perfusion), directly predicting secondary intraventricular hemorrhage (IVH).
โก Burst-Suppression Ratio (BSR)
In bedside neonatal aEEG, deep learning temporal segmentation models track the percentage of isoelectric silence ($< 5\,\mu\text{V}$) per epoch:
$$\text{BSR} = \frac{T_{\text{suppression}}}{T_{\text{total}}} \times 100\%$$
Rapid automated BSR tracking during 72-hour therapeutic hypothermia identifies evolving hypoxic-ischemic brain damage hours before clinical seizure onset.