Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating {\em a priori} information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $\ell^1$ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities. In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions.

翻译：扩散光学层析成像（DOT）是一个严重病态的非线性逆问题，旨在通过边界测量值估计光学参数。在贝叶斯框架下，可通过先验分布引入光学参数的先验信息来缓解病态性。当目标具有稀疏性或锐利边缘时，常用的先验模型是非可微的全变分和ℓ¹先验。另一种方法是通过对高斯先验的方差进行层次扩展，获得可微的稀疏促进先验，将方差视为未知量，从而使估计能够定位不连续性。本研究针对非线性DOT逆问题，采用指数型、标准伽马型和逆伽马型超先验，构建了层次先验模型。基于不同的超先验及其超参数，层次模型可促进不同程度的稀疏性与平滑性。为计算最大后验估计，将先前提出的交替算法改进以适配非线性模型。随后提出基于超先验累积分布函数的超参数选择方法。通过数值仿真评估各超先验性能，结果表明层次模型可提升重建结果的定位精度、对比度与边缘锐度。