A Conjugate Bayesian Framework for Fast 3D Positronium Lifetime Estimation with a Partial System Matrix

TL;DR

This paper introduces a scalable Bayesian framework for fast 3D positronium lifetime estimation that leverages a partial system matrix and conjugate updates, significantly reducing computational demands.

Contribution

The authors develop a novel Bayesian method using a time-of-flight-aware partial system matrix and conjugate Gamma--Exponential updates for efficient 3D lifetime imaging.

Findings

Reduced memory and computational requirements while maintaining data model accuracy.

Achieved rapid voxel-wise effective-rate estimation in simulated and real datasets.

Provided a stable, fast estimator suitable for large-scale 3D positronium lifetime imaging.

Abstract

Background: Positronium lifetime imaging extends conventional positron emission tomography by using the time interval between positron emission and annihilation as an additional contrast mechanism. Voxel-wise lifetime estimation in fully three-dimensional settings is computationally difficult because the number of feasible detector-time channels grows rapidly, whereas only a small subset is observed in practice. We developed a scalable statistical framework for three-dimensional positronium lifetime estimation based on a time-of-flight-aware partial system matrix restricted to observed detector-time channels, combined with posterior event-to-voxel weighting and a conjugate Gamma--Exponential update for closed-form voxel-wise effective-rate estimation. Results: Restricting the forward model to observed detector-time channels reduced memory and computational requirements while preserving the Poisson data model for retained detected triple coincidences. In simulated data with 4056 voxels, the analytic Bayesian estimator required 2.76 s versus 74.46 s for 10 L-BFGS-B iterations on the same CPU while accurately recovering the effective-rate map. In a triple-coincidence dataset acquired with a J-PET prototype scanner and a NEMA image-quality phantom, a 234 375-voxel effective-rate map was estimated in approximately 3 s from about $3.64\times10^5$ retained events. Conclusions: Restricting the system matrix to observed detector-time channels makes fully three-dimensional positronium lifetime estimation computationally practical for sparse triple-coincidence data. The proposed posterior-weighted conjugate update provides a fast and stable single-component surrogate estimator of voxel-wise effective lifetime for large-scale three-dimensional positronium lifetime imaging.