A Sheaf-Theoretic View of Dynamic Inverse Problems

Dynamic X-ray CT, as presented by Aryeh Keating at the Virginia Tech Applied Math seminar, frames a question that sits right on the fault line between applied reconstruction and pure structure:

How can we recover a coherent, time-varying image from incomplete, noisy measurements—without assuming known priors or hand-tuned hyperparameters?

His method blends optical flow, expectation–maximization, and Kalman filtering into a sequential algorithm. Each frame is reconstructed locally and then stabilized through dynamic coupling.
But the deeper question remains: what is the global object being reconstructed?

1. The Structural Reformulation

In my reading, dynamic inverse problems are not merely optimization tasks. They are gluing problems.

Each time-slice $(U_t)$ is a local observation domain.
Each reconstruction $(x_t)$ is a section over that domain.
Consistency between neighboring slices—enforced by motion models or flow fields—is the restriction map $\rho_{ts}$.

The collection of all these spaces forms a reconstruction sheaf $\mathcal{F}$.
A global, stable reconstruction corresponds to a global section:
$$
\Gamma(\mathcal{F}) = {U_t ,x_t,\text{consistent on overlaps},}.
$$
Under this view, the optical flow is a connection, EM iterations are derived limits, and Kalman updates are derived pushforwards.
The entire dynamic inverse problem becomes a question about acyclicity:
do higher cohomology groups $(H^{>0}(\mathcal{F}))$ vanish, or do they signal irreducible inconsistency between local reconstructions?

2. Why This Generalization Matters

Keating’s framework works beautifully in practice but still inherits four limitations common to sequential inverse problems:

1. Hyperparameter dependence.
   Regularization strengths, noise covariances, and transition weights must be chosen externally.
2. Model fragility.
   Small errors in motion models can propagate and destabilize later frames.
3. Local–global disconnect.
   Each EM or Kalman iteration only sees local consistency, not the full cohomological landscape.
4. Lack of structural guarantees.
   Convergence and uniqueness rely on empirical heuristics rather than mathematical topology.

The sheaf-theoretic generalization addresses these directly:

In short: we replace trial-and-error regularization with intrinsic structural coherence.

3. Next Steps

1. Formalization.
   Write down the Reconstruction Sheaf $\mathcal{F}$ explicitly for the dynamic CT model
   $(y_t = A_t x_t + \epsilon_t, \quad x_t = F_t x_{t-1} + w_t.)$
2. Derived interpretation.
   Show that EM corresponds to minimizing obstruction energy (approximating $\varprojlim^1 \mathcal{F})$,
   and that the Kalman filter is a derived pushforward ($R^1 A_{t*}$).
3. Numerical experiment.
   Simulate a small dynamic phantom and measure cohomological “instability energy” as the new regularizer.
4. Collaboration.
   Reach out to Keating with a short note:
   “I’m exploring a categorical generalization of your dynamic CT framework using sheaf cohomology to detect ill-posedness. May I reference your slides or preprint?”
5. Paper direction.
   From Sequential EM to Derived Gluing: A Sheaf-Theoretic Framework for Dynamic Inverse Problems.

4. Why It Belongs in the JTPMath Pipeline

This problem lives exactly at the intersection JTPMath was built to explore:
the frontier where computation, topology, and physics collapse into structure.
Keating’s dynamic CT reconstruction is a concrete instantiation of a more universal phenomenon—the failure of naive locality in time-varying systems.
By translating his algorithm into sheaf language, we expose that failure as cohomology and make it measurable, controllable, and eventually removable.

The moral:

Numerical instability isn’t noise. It’s the visible trace of hidden topology.

Keywords: dynamic CT, sheaf theory, inverse problems, EM, Kalman filtering, cohomology, reconstruction geometry.