Computational Methods for Inverse Scattering Problems with Incomplete Data

Inverse scattering problems are mathematical inverse problems arising from the physical phenomenon of wave—matter interaction. The mathematical problem of solving an inverse scattering problem is corresponding to the recovery of unknown/inaccessible physical objects by the measurement of the wave field scattered by the objects. Applications of these problems include sub-wavelength imaging, nondestructive testing, geophysical exploration, radar and sonar.

This project is concerned with inverse scattering problems with incomplete measurement data. We consider two types of incomplete data that are of significant importance in applications, namely the limited-aperture data and the phaseless data. The data is called limited-aperture if the measured far-field pattern is not available for all incident angles and all observation angles, and is called phaseless if the measured data contains only the amplitude but not the phase information.

For the case of limited-aperture data, we propose two approaches for the computational methods. The first approach is to apply the existing numerical methods on the limited- aperture data directly or a zero extension of the data to full aperture. From the previous works of the PI and preliminary new numerical experiments, we find this simple approach yields surprisingly good results worth further numerical and theoretical studies. The second approach is to first retrieve the data in the unavailable aperture, and then apply the existing numerical methods on the retrieved full-aperture data. One of the method we propose to accomplish this is to consider an approximate analytic continuation through truncated Fourier series expansion and progressive expansion of the aperture. We also consider the case of limited-aperture multi-frequency data, and propose to utilize the frequency-dependent property of the wave to facilitate the retrieval of the missing data.

For the case of phaseless data, we propose to extend the PI’s previous work on the reconstruction of convex polyhedral obstacles and gesture recognition. In particular, we consider the reconstruction of non-convex polygonal obstacles and general polyhedral medium scatterers. Due to the complexity of the ray path in these new problems, we shall not consider our previous approach but resort to optimization methods. On the other hand, the key ideas about the geometrical structure of polyhedrons and the physical optics approximation will still be utilized in the set up and numerical solution of the optimization problem.