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.

翻译：分数扩散模型是最近发展起来的一种贝叶斯逆问题后验采样框架，它通过利用从经验数据中学习到的强大先验分布，在处理严重不适定问题上达到了最先进的性能。尽管该框架在机器学习领域引起了广泛关注，但在存在建模误差且使用实际测量数据的情况下，对现实逆问题的深入研究仍有待开展。本工作针对漫射光学层析成像中的线性化差分成像问题，评估了条件分数函数的无条件表示框架。漫射光学层析成像利用近红外光的边界测量来估计生物组织中吸收和散射参数的空间分布。该问题具有高度不适定性，因此对噪声和建模误差非常敏感。我们提出了一种新颖的正则化方法，通过构建一个由学习部分和基于模型部分组成的混合分数，来防止分数函数的过拟合。该方法的验证同时使用了模拟和实验测量数据。实验表明，与经典的基于模型的估计方法相比，数据驱动的先验分布能够产生方差较低的后验样本，并且这些样本围绕真实值分布，即使在高度不适定问题和存在建模误差的情况下也是如此。