一种用于反散射问题的计算几何方法
A computational geometry method for the inverse scattering problem
AI总结：
本文提出基于计算几何的反散射问题求解方法，通过点云近似散射体支撑集，并利用贝叶斯方法联合建模点云非凸包和散射体折射率，结合样条控制点计算体积势，提出概率转移核以高效采样后验分布，验证了方法在同时恢复支撑集和折射率方面的可靠性。
摘要：
本文展示了一种用于求解二维星形、光滑、穿透性障碍物反散射问题的计算方法。我们的方法基于计算几何的经典思想。首先，我们用点云近似散射体的支撑集。其次，我们利用贝叶斯范式建模给定近场数据下点云非凸包和散射体常数折射率的联合条件概率分布。值得注意的是，我们使用点云的非凸包作为样条控制点，在更细的网格上计算出现在直接问题积分方程公式中的体积势。最后，为了采样出现的后验分布，我们提出一个与空间仿射变换兼容的概率转移核。我们的发现表明，我们的方法能够可靠地同时恢复散射体的支撑集和常数折射率。确实，我们的采样方法在估计诸如散射体面积等感兴趣量时表现出鲁棒性。我们最后指出了一系列方法的推广。
英文摘要：
In this paper we demonstrate a computational method to solve the inverse scattering problem for a star-shaped, smooth, penetrable obstacle in 2D. Our method is based on classical ideas from computational geometry. First, we approximate the support of a scatterer by a point cloud. Secondly, we use the Bayesian paradigm to model the joint conditional probability distribution of the non-convex hull of the point cloud and the constant refractive index of the scatterer given near field data. Of note, we use the non-convex hull of the point cloud as spline control points to evaluate, on a finer mesh, the volume potential arising in the integral equation formulation of the direct problem. Finally, in order to sample the arising posterior distribution, we propose a probability transition kernel that commutes with affine transformations of space. Our findings indicate that our method is reliable to retrieve the support and constant refractive index of the scatterer simultaneously. Indeed, our sampling method is robust to estimate a quantity of interest such as the area of the scatterer. We conclude pointing out a series of generalizations of our method.
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