Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime
Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime
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DOI：
10.3390/computation10020015
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发表时间：
2022-01
期刊：
影响因子：
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通讯作者：
Qin Li;Kit Newton;Li Wang
中科院分区：
文献类型：
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作者：
Qin Li;Kit Newton;Li Wang
对于物理模型中的逆问题，人们测量解并使用来自收集的数据的信息来推断模型参数。通常情况下，这些数据是不充分的，使反问题不适定。我们研究的上下文中的光学成像，这是一种医学成像技术，使用光探测（生物）组织结构的不适定性。根据光的强度，正问题可以用不同类型的方程来描述。高能光散射非常小，人们使用辐射传输方程（RTE）作为模型;低能光散射频繁，因此扩散方程（DE）足以作为一个很好的近似。多尺度近似将双曲线型RTE与抛物线型DE联系起来。这两个方程的反问题也有一个多尺度通道，所以人们可以预期，随着光子能量的减少，反问题从适定性变为不适定性。我们使用贝叶斯推理研究这种稳定性恶化。特别是，我们使用基于RTE的先验分布和后验分布之间的Kullback-Leibler散度来证明测量的信息增益随着光子能量的减小而消失，因此逆问题在扩散状态下是不适定的。在线性化的设置，我们还表明，后验分布的均方误差增加，因为我们接近扩散制度。
For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well- to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback–Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.