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 data-driven basis functions 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 from 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. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s<1/2$) 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、有限孔径及多频数据）重建介质对比度的逆散射问题。基于广义长椭球波函数及相关特征函数，我们为这些逆问题提出了数据驱动基函数。此类数据驱动特征函数是傅里叶积分算子的特征函数；它们能够解析延拓至全空间，具有双正交性，且在带限函数类中完备。首先，我们建立了利用数据驱动基重建对比度的Picard准则，其中重建公式亦可从数据处理与解析外推的角度加以理解。与广义长椭球波函数相关的另一显著特点是：圆盘上的数据驱动基同时也是Sturm-Liouville微分算子的一组基。借助Sturm-Liouville理论，我们估算了$H^s$函数谱截断近似的$L^2$逼近误差。这为噪声数据的谱截断正则化策略以及全孔径情形下$H^s$（$0<s<1/2$）对比度的显式稳定性估计提供了理论依据。在有限孔径与多频情形下，我们还获得了噪声数据的谱截断正则化策略以及一类对比度的稳定性估计。