Score-based diffusion models for diffuse optical tomography with uncertainty quantification

Score-based diffusion models are a recently developed framework for posterior sampling in Bayesian inverse problems with a state-of-the-art performance for severely ill-posed problems by leveraging a powerful prior distribution learned from empirical data. Despite generating significant interest especially in the machine-learning community, a thorough study of realistic inverse problems in the presence of modelling error and utilization of physical measurement data is still outstanding. In this work, the framework of unconditional representation for the conditional score function (UCoS) is evaluated for linearized difference imaging in diffuse optical tomography (DOT). DOT uses boundary measurements of near-infrared light to estimate the spatial distribution of absorption and scattering parameters in biological tissues. The problem is highly ill-posed and thus sensitive to noise and modelling errors. We introduce a novel regularization approach that prevents overfitting of the score function by constructing a mixed score composed of a learned and a model-based component. Validation of this approach is done using both simulated and experimental measurement data. The experiments demonstrate that a data-driven prior distribution results in posterior samples with low variance, compared to classical model-based estimation, and centred around the ground truth, even in the context of a highly ill-posed problem and in the presence of modelling errors.

💡 Research Summary

This paper investigates the application of score‑based diffusion (SBD) models to the highly ill‑posed inverse problem of diffuse optical tomography (DOT) using the unconditional representation for the conditional score function (UCoS) framework. DOT aims to recover spatial maps of absorption (μ_a) and reduced scattering (μ_s′) from boundary measurements of near‑infrared light. Because the forward model is nonlinear, a first‑order Taylor (Born or Rytov) approximation is employed, yielding a linearized difference imaging formulation y = AΔx + ε, where A is the forward matrix and ε is Gaussian noise. Traditional Bayesian approaches rely on analytically tractable Gaussian priors (e.g., Ornstein‑Uhlenbeck processes) and Tikhonov regularization, which struggle with complex tissue structures, modeling errors, and limited‑view geometries.

The authors adopt a data‑driven prior by training a diffusion model on simulated DOT images. In the diffusion framework, a stochastic differential equation (SDE) progressively adds Gaussian noise to data, and the reverse‑time SDE generates posterior samples when the conditional score s(x,t;μ_y) is known. UCoS circumvents repeated forward evaluations during sampling by expressing the conditional score through a low‑dimensional function r(·) that depends only on the current noisy state and time. Specifically, the score can be written as s(x,t;μ_y)=λ(t)