Per-Bundle Statistical Limits and Learned-Prior Inversion in Multiplexed X-ray Imaging, with Application to Temporal CT

Objective. Temporal computed tomography (TCT) fires N_S=3 X-ray sources simultaneously onto a shared detector, creating a pre-reconstruction inverse problem: each bundle of M=5 Poisson intensities sums unlabeled photon contributions from K=3 line integrals. Because the measurement sums exponentials rather than forming one Beer–Lambert product, log and Σ do not commute and the inversion is nonlinear. We quantify the dose cost this multiplexing imposes and how closely estimators approach that limit. Approach. We treat the 5×3 bundle as a model problem for multiplexed photon aggregation, with TCT the motivating instance. Closed-form Cramér–Rao bounds (CRBs) are expressed as dose-inflation factors against an equal-dose single-source floor, and two estimators—a converged constrained maximum-likelihood estimator and a physics-motivated residual network—are benchmarked on synthetic independent and identically distributed (i.i.d.), analytical-phantom and single-patient bundles. Main results. Aggregation imposes a structural loss: at equal attenuation only 43% of single-source Fisher information survives for the endpoint paths and 23% for the middle path, fixing constant CRB inflation ratios √(7/3)≈1.53 and √(13/3)≈2.08. The constrained maximum-likelihood estimate attains the endpoint CRBs within a few percent but stays up to 1.35× above the middle-path bound. A learned prior narrows this gap where the middle path is photon-starved: on i.i.d. data the network cuts the constrained estimate's error by 27–49% for line-integral values of 6–9, nearly matching the Bayes posterior mean's overall error; on one patient, across separate body regions, it gains comparably for values of 5–8, loses elsewhere, and never falls below the equal-dose floor. Significance. The structure (a Poisson sum of exponentials) and methodology (Fisher-information efficiencies, dose-inflation accounting) are not specific to TCT: they characterize any measurement that aggregates attenuated photons before counting, transferring with only the incidence matrix respecified. Whether a learned prior yields a generalizable dose reduction is the open question a companion paper addresses through a multi-patient corpus.