A multivariate interpolation problem arising from the scattering of waves in layered media

We present recent results on the scattering of plane waves in piecewise constant layered media, introducing a new geometric perspective. It turns out that the classical inverse scattering problem generically decouples into two separate problems. One problem is to recover a line from the magnitudes of the projections onto the line of a set of lattice points. The second problem is to determine which function best interpolates a given set of data points on the integer lattice, where the interpolating function is chosen from among the translates of a known, highly oscillatory function. In applications such as acoustic imaging it is required that this interpolation problem be solved numerically, necessitating that a solution be both stable and computationally efficient.