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

Fabian Schneider, Meghdoot Mozumder, Konstantin Tamarov, Leila Taghizadeh, Tanja Tarvainen, Tapio Helin, Duc-Lam Duong

TL;DR

This work investigates score-based diffusion models for Bayesian inverse problems in diffuse optical tomography (DOT) with uncertainty quantification. It introduces a regularized UCoS framework that convexly combines a learned score with a model-based Gaussian score, enabling robust posterior sampling in this highly ill-posed linearized setting. The authors prove a local approximation: as diffusion time $t$ becomes small, the score of the convex mixture approximates the score of a geometric mixture $p_1^{\alpha} p_2^{1-\alpha}$ of PDFs, and they train a Fourier-Neural Operator to realize the learned score. Validation on simulated DOT with full-view and limited-view geometries and on experimental measurements demonstrates reduced posterior variance and improved localization around inclusions, outperforming DPS and classical model-based priors, and showing robustness to modelling errors. The approach offers practical, scalable uncertainty quantification for DOT and potentially other large-scale inverse problems.

Abstract

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.

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

TL;DR

This work investigates score-based diffusion models for Bayesian inverse problems in diffuse optical tomography (DOT) with uncertainty quantification. It introduces a regularized UCoS framework that convexly combines a learned score with a model-based Gaussian score, enabling robust posterior sampling in this highly ill-posed linearized setting. The authors prove a local approximation: as diffusion time becomes small, the score of the convex mixture approximates the score of a geometric mixture of PDFs, and they train a Fourier-Neural Operator to realize the learned score. Validation on simulated DOT with full-view and limited-view geometries and on experimental measurements demonstrates reduced posterior variance and improved localization around inclusions, outperforming DPS and classical model-based priors, and showing robustness to modelling errors. The approach offers practical, scalable uncertainty quantification for DOT and potentially other large-scale inverse problems.

Abstract

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.
Paper Structure (21 sections, 5 theorems, 48 equations, 7 figures, 2 algorithms)

This paper contains 21 sections, 5 theorems, 48 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Theory
  4. Bayesian 'model-based' estimation with a Gaussian prior
  5. Posterior sampling using conditional score based diffusion models
  6. Conditional score--based diffusion
  7. UCoS: an efficient diffusion posterior sampling
  8. Training in UCoS
  9. Regularization based on Gaussian approximation
  10. Unconditional score approximation
  11. Diffuse optical tomography
  12. Simulations
  13. Data generation
  14. Training and evaluation
  15. Training
  16. ...and 6 more sections

Key Result

Lemma 2.2

Assume that $H = {\mathbb R}^d$ and the posterior $\mu^y$ admits a density $q^y$. Then the score function equals where the convolution kernel $k$ satisfies

Figures (7)

  • Figure 1: 2D simulation domains. Locations of the sources and detectors are shown as red and blue dots ($\cdot$) on the domain boundaries. Axes units are in millimetres.
  • Figure 2: Simulated DOT with one target inclusion with (a) the full view and (b) the limited view setup. Columns from left to right: simulated ground truth (first column), five posterior samples (columns 2-6), an average (column 7), bias (column 8) and standard deviation (column 9), which are computed over 100 generated samples. The methods used from top to bottom: UCoS, regularized UCoS, DPS, Gaussian OU process prior.
  • Figure 3: Simulated DOT for two target inclusions with (a) the full view and (b) the limited view setup. Columns from left to right: simulated ground truth (first column), five posterior samples (columns 2-6), an average (column 7), bias (column 8) and standard deviation (column 9), which are computed over 100 generated samples. The methods used from top to bottom: UCoS, regularized UCoS, DPS, Gaussian OU process prior.
  • Figure 4: Simulated DOT with two out-of-distribution inclusions with (a) the full view and (b) the limited view setup. Columns from left to right: simulated ground truth (first column), five posterior samples (columns 2-6), an average (column 7), bias (column 8) and standard deviation (column 9), which are computed over 100 generated samples. The methods used from top to bottom: UCoS, regularized UCoS, DPS, Gaussian OU process prior.
  • Figure 5: (a) Photograph of the cylindrical phantom used in DOT experiments. (b) Top view of the phantom with inclusions.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • proof : Proof of Theorem \ref{['thm: mixture scores']}