We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite parameter field representing the objects. To construct the prior distribution we use a topological sensitivity analysis. We demonstrate the approach on the Bayesian solution of 2D inverse problems in light and acoustic holography with synthetic data. Statistical information on objects such as their center location, diameter size, orientation, as well as material properties, are extracted by sampling the posterior distribution. Assuming the number of objects known, comparison of the results obtained by Markov Chain Monte Carlo sampling and by sampling a Gaussian distribution found by linearization about the maximum a posteriori estimate show reasonable agreement. The latter procedure has low computational cost, which makes it an interesting tool for uncertainty studies in 3D. However, MCMC sampling provides a more complete picture of the posterior distribution and yields multi-modal posterior distributions for problems with larger measurement noise. When the number of objects is unknown, we devise a stochastic model selection framework.

中文翻译：

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具有拓扑先验的逆散射贝叶斯方法

我们提出了一个贝叶斯推理框架来估计逆散射问题中的不确定性。鉴于观察到的数据、前向模型及其不确定性，我们找到了代表对象的有限参数域上的后验分布。为了构建先验分布，我们使用拓扑敏感性分析。我们使用合成数据演示了光和声全息中二维逆问题的贝叶斯解决方案。通过对后验分布进行采样，可以提取对象的统计信息，例如其中心位置、直径大小、方向以及材料特性。假设已知物体的数量，通过马尔可夫链蒙特卡罗采样和通过采样通过线性化发现的关于最大后验估计的高斯分布获得的结果的比较显示出合理的一致性。后一个过程具有较低的计算成本，这使其成为 3D 不确定性研究的有趣工具。然而，MCMC 采样提供了更完整的后验分布图，并为具有较大测量噪声的问题生成多模态后验分布。当对象数量未知时，我们设计了一个随机模型选择框架。MCMC 采样提供了更完整的后验分布图，并为具有较大测量噪声的问题生成多模态后验分布。当对象数量未知时，我们设计了一个随机模型选择框架。MCMC 采样提供了更完整的后验分布图，并为具有较大测量噪声的问题生成多模态后验分布。当对象数量未知时，我们设计了一个随机模型选择框架。