AI 中文总结
本文研究三维空间中从衍射系数等数据恢复严格凸锥形障碍物的逆问题，通过阿达马参数式等技术建立了对应几何衍射理论的严格逆理论。
AI 中文摘要
考虑在三维欧氏空间中，从衍射系数以及入射方向（透镜数据）或衍射波的到达时间中恢复严格凸锥形障碍物的逆问题。入射波是从一点发出的球形脉冲，衍射波的测量在放置于反射阴影内的任意大小接收器上进行。具体而言，透镜数据或到达时间确定锥顶的位置，而衍射系数则重构锥的形状。由于衍射系数由二维单位球面锥底补集上的半波描述，我们将三维空间中锥的逆衍射问题简化为识别二维单位球面上的反射波前，并利用球面上的反射射线恢复障碍物。前者通过构造波前附近半波的阿达马参数式完成，后者依赖于球面上断裂测地线的拓扑性质。本文提出的框架利用了几何衍射理论所表征的衍射波场的解析与几何结构，首次建立了对应于几何衍射理论的严格逆理论。
英文摘要
Consider the inverse problem of recovering a strictly convex conical obstacle in $\mathbb{R}^3$ from the diffraction coefficients along with arrival directions (lens data) or arrival times of diffracted waves. The incident wave is a spherical pulse emanating from a point, and the measurements of diffracted waves are taken at an arbitrarily sized receiver placed within the reflection shadow. Specifically, the lens data or arrival times determine the location of the tip, whereas the diffraction coefficients reconstruct the shape of the cone. Since diffraction coefficients are described by half waves over the complement of the cone base in $\mathbb{S}^2$, we reduce inverse diffraction by a cone in $\mathbb{R}^3$ to identifying the reflected wavefront in $\mathbb{S}^2$ and recovering the obstacle using reflected rays on the sphere. The former is accomplished by constructing the Hadamard parametrix for half waves near the wavefront, whereas the latter relies on the topological properties of broken geodesics on $\mathbb{S}^2$. The framework developed in this paper exploits the analytic and geometric structures of diffracted wave fields characterized in the Geometrical Theory of Diffraction, and establishes, for the first time, a rigorous inverse theory corresponding to GTD.
