Direct reconstruction for acoustic inverse Born scattering

Abstract

We consider the inverse medium scattering problem for the Helmholtz equation in two dimensions, i.e., the task to recover a compactly supported penetrable two-dimensional scatterer from full knowledge of the associated far field data or, equivalently, the far field operator. Although this problem is uniquely solvable, it is severely ill-posed and nonlinear. In the regime of weak scattering, the Born approximation yields a linearized relation between the contrast and the far field data, thus overcoming the second difficulty. This linear setting allows to build on recent work on linearized electrical impedance tomography, which relies on triangular Zernike decompositions, to derive an explicit reconstruction formula that expresses the expansion coefficients of the contrast in terms of those of the far field data. By choosing the expansion functions appropriately, the resulting system matrix decouples into separate (infinite) triangular systems for the spatial angular frequencies in the contrast. Consequently, each of these systems can be solved independently by performing forward substitutions. Our numerical experiments indicate that this approach, combined with an adequate regularization method, remains effective even when applied to full nonlinear far field data beyond the Born regime.