A Direct Sampling Method and Its Integration with Deep Learning for Inverse Scattering Problems with Phaseless Data

We consider in this work an inverse acoustic scattering problem when only phaseless data is available. The inverse problem is highly nonlinear and ill-posed due to the lack of the phase information. Solving inverse scattering problems with phaseless data is important in applications as the collection of physically acceptable phased data is usually difficult and expensive. A novel direct sampling method (DSM) will be developed to effectively estimate the locations and geometric shapes of the unknown scatterers from phaseless data generated by a very limited number of incident waves. With a careful theoretical analysis of the behavior of the index function and some representative numerical examples, the new DSM is shown to be computationally efficient, easy to implement, robust to large noise, and does not require any prior knowledge of the unknown scatterers. Furthermore, to fully exploit the index functions obtained from the DSM, we also propose to integrate the DSM with a deep learning technique (DSM-DL) to achieve high-quality reconstructions. Several challenging and representative numerical experiments are carried out to demonstrate the accuracy and robustness of reconstructions by DSM-DL. The DSM-DL networks trained by phased data are further theoretically and numerically shown to be able to solve problems with phaseless data. Additionally, our numerical experiments also show the DSM-DL can solve inverse scattering problems with mixed types of scatterers, which renders its applications in many important practical scenarios.