三维声软轴对称不可穿透目标的逆散射重建
Inverse scattering reconstruction of a three dimensional sound-soft axis-symmetric impenetrable object
摘要：
在这项工作中，我们考虑的问题重建的三维不可穿透的声音软轴对称障碍物的形状从测量的散射场在多个频率。这个问题在定位和识别一般具有轴对称性的障碍物（如地雷）方面有重要的应用。我们提出了一个由两部分组成的框架，用于恢复障碍物的形状。在第一部分中，我们介绍了一个算法，找到对称轴的障碍，利用远场图案。在第二部分中，我们恢复的形状的障碍物应用递归线性化算法（RLA）与多频测量的散射场。在RLA中，一系列的逆散射问题，使用增加单频测量解决。这些问题都是不适定的和非线性的。不适定性是通过使用带限表示的障碍物的形状，而非线性是通过施加阻尼高斯-牛顿法处理。当使用RLA时，必须解决大量的前向散射问题。因此，它是至关重要的，有一个高效和准确的正向问题求解器。对于前向问题，我们在方位角坐标中应用分离变量，并对所产生的问题进行傅立叶分解，从而使我们得到一系列解耦的更简单的前向散射问题来解决。反问题的数值例子显示我们的两部分框架在不同的情况下，特别是对于非光滑边界的对象的可行性。
英文摘要：
In this work, we consider the problem of reconstructing the shape of a three dimensional impenetrable sound-soft axis-symmetric obstacle from measurements of the scattered field at multiple frequencies. This problem has important applications in locating and identifying obstacles with axial symmetry in general, such as, land mines. We present a two-part framework for recovering the shape of the obstacle. In part 1, we introduce an algorithm to find the axis of symmetry of the obstacle by making use of the far field pattern. In part 2, we recover the shape of the obstacle by applying the recursive linearization algorithm (RLA) with multifrequency measurements of the scattered field. In the RLA, a sequence of inverse scattering problems using increasing single frequency measurements are solved. Each of those problems is ill-posed and nonlinear. The ill-posedness is treated by using a band-limited representation for the shape of the obstacle, while the nonlinearity is dealt with by applying the damped Gauss-Newton method. When using the RLA, a large number of forward scattering problems must be solved. Hence, it is paramount to have an efficient and accurate forward problem solver. For the forward problem, we apply separation of variables in the azimuthal coordinate and Fourier decompose the resulting problem, leaving us with a sequence of decoupled simpler forward scattering problems to solve. Numerical examples for the inverse problem are presented to show the feasibility of our two-part framework in different scenarios, particularly for objects with non-smooth boundaries.
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