光声层析成像的三维不确定性量化

中文摘要 AI 辅助

光声层析成像（PAT）是一种很有前景的高分辨率生物医学成像模态，因此需要对重建图像进行可靠的不确定性量化（UQ）。贝叶斯方法为UQ提供了严格框架，但对实际三维PAT来说计算上仍具挑战性，且对波动方程中的数值近似敏感。我们为PAT开发了一个有限元贝叶斯UQ框架，它能适应复杂计算域和探测器几何形状，实现大规模三维推断。提出的方法将随机化然后优化（RTO）采样策略重新表述为无矩阵算法，仅用正反波传播生成独立后验样本。特别关注构建与离散正向算子形成精确转置对且与连续PAT伴随一致的伴随离散化，以便在采样过程中实现高效最小二乘求解器。我们研究了时间离散化、人工边界条件和伴随一致性对后验不确定性的影响，并确定避免数值伪影的离散化策略。该框架通过精确后验统计、现有贝叶斯PAT方法以及使用无回转采样器（NUTS）的哈密顿蒙特卡罗方法进行了验证，并在一般有限元域上的一个约有2×10^5个未知数的三维问题上得到了证明。据我们所知，这是首次在一般三维有限元几何上进行的大规模贝叶斯PAT研究，该方法自然地扩展到了一类广泛的线性偏微分方程约束的反问题。

英文摘要

Photoacoustic tomography (PAT) is a promising modality for high-resolution biomedical imaging, motivating the need for reliable uncertainty quantification (UQ) of reconstructed images. Bayesian approaches provide a rigorous framework for UQ but remain computationally challenging for realistic three-dimensional PAT and are sensitive to numerical approximations in the governing wave equation. We develop a finite-element Bayesian UQ framework for PAT that accommodates complex computational domains and detector geometries while enabling large-scale three-dimensional inference. The proposed methodology reformulates the randomize-then-optimize (RTO) sampling strategy as a matrix-free algorithm that generates independent posterior samples using only forward and adjoint wave propagations. Particular attention is given to constructing an adjoint discretization that forms an exact transpose pair with the discrete forward operator while remaining consistent with the continuous PAT adjoint, enabling efficient least-squares solvers within the sampling procedure. We investigate the influence of temporal discretization, artificial boundary conditions, and adjoint consistency on posterior uncertainty and identify discretization strategies that avoid numerical artifacts. The framework is validated against exact posterior statistics, existing Bayesian PAT methods, and Hamiltonian Monte Carlo using the No-U-Turn Sampler (NUTS), and is demonstrated on a three-dimensional problem with approximately $2\times 10^5$ unknowns on a general finite-element domain. To the best of our knowledge, this is the first large-scale Bayesian PAT study on general three-dimensional finite-element geometries, and the methodology extends naturally to a broad class of linear PDE-constrained inverse problems.

发表机构

• University of Oulu（奥卢大学）
• King Fahd University of Petroleum and Minerals（法赫德国王石油与矿产大学）
• Technical University of Denmark（丹麦技术大学）
• University of Eastern Finland（芬兰东部大学）

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