We consider the inverse medium scattering of reconstructing the medium contrast using Born data, including the full aperture, limited-aperture, and multi-frequency data. We propose a class of data-driven basis for these inverse problems based on the generalized prolate spheroidal wave functions and related eigenfunctions. Such data-driven eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the data-driven basis, where the reconstruction formula can also be understood in the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the data-driven basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions, $0<s\le1$. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s\le1$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast.

翻译：我们考虑利用Born数据（包括全孔径、有限孔径和多频率数据）重建介质对比度的逆散射问题。基于广义椭球面波函数及其相关特征函数，我们提出了一类用于这些逆问题的数据驱动基。此类数据驱动特征函数是傅里叶积分算子的特征函数，其显著特性包括：可解析延拓至全空间、具有双重正交性，并在带限函数类中具有完备性。我们首先建立了利用数据驱动基重建对比度的Picard准则，相应的重建公式亦可从数据处理与解析延拓的角度理解。广义椭球面波函数的另一个突出特性在于：圆盘上的数据驱动基同时构成Sturm-Liouville微分算子的基。借助Sturm-Liouville理论，我们估计了$H^s$函数（$0<s\le1$）的谱截断近似的$L^2$逼近误差。由此得到了针对含噪数据的谱截断正则化策略，以及全孔径情形下$H^s$（$0<s\le1$）对比度的显式稳定性估计。在有限孔径和多频率情形中，我们还获得了针对含噪数据的谱截断正则化策略以及一类对比度的稳定性估计。