Data completion algorithms and their applications in inverse acoustic scattering with limited-aperture backscattering data

Abstract

Limited-aperture data brings great challenge for inverse scattering problems. The limited-aperture problem we are particularly interested in is the limited-aperture “backscattering” problem where both the incident and observation directions are on the same aperture. Other than directly using the limited-aperture data, we introduce two data completion algorithms to recover the full-aperture data. With the reconstructed full-aperture data, we then apply the factorization method and direct sampling method for the object reconstruction. Both of the data completion algorithms, though derived in different ways, share the same key ingredient: the limited-aperture data is related to the full-aperture data via the prolate matrix. In the first algorithm we represent the full-aperture data in the form of double Fourier series, and relate the corresponding Fourier coefficients to the limited-aperture data via two prolate matrices. This fundamental idea and algorithm motivate the second algorithm: for each incident direction, we represent the full-aperture data by the single layer potential, and relate the Fourier coefficients of the density to the limited-aperture data via the prolate matrix. The second algorithm can be seen as a “direction-wise” algorithm. Both of the data completion algorithms are independent of the topological and physical properties of the unknown scatterers. A variety of numerical examples are presented to illustrate the effectiveness and robustness of the data completion algorithms with the sampling methods.