We are concerned with the inverse scattering problem of recovering an inhomogeneous medium by the associated acoustic wave measurement. We prove that under certain assumptions, a single far-field pattern determines the values of a perturbation to the refractive index on the corners of its support. These assumptions are satisfied for example in the low acoustic frequency regime. As a consequence if the perturbation is piecewise constant with either a polyhedral nest geometry or a known polyhedral cell geometry, such as a pixel or voxel array, we establish the injectivity of the perturbation to far-field map given a fixed incident wave. This is the first unique determinancy result of its type in the literature, and all of the existing results essentially make use of infinitely many measurements.

中文翻译：

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通过单个远场模式恢复分段恒定折射率

我们关注通过相关声波测量恢复非均匀介质的逆散射问题。我们证明，在某些假设下，单个远场模式决定了其支撑角上折射率的扰动值。例如在低声频范围内满足这些假设。因此，如果扰动是多面体嵌套几何或已知多面体单元几何（例如像素或体素阵列）的分段常数，我们将在给定固定入射波的情况下建立扰动对远场图的注入性。这是文献中第一个独特的决定性结果，所有现有的结果基本上都使用了无数次测量。