Is a Curved Flight Path in SAR Better than a Straight One?

Abstract

In the plane, we study the transform $R_\gamma f$ of integrating an unknown function $f$ over circles centered at a given curve $\gamma$. This is a simplified model of synthetic aperture radar (SAR), when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set $WF(f)$. If the visible singularities of $f$ hit $\gamma$ once, we show that $WF(f)$ cannot be recovered; i.e., the artifacts cannot be resolved. If $\gamma=\partial\Omega$ is the boundary of a strictly convex domain $\Omega$, we show that this is still true. On the other hand, in the latter case, if $f$ is known a priori to have singularities in a compact set, then we show that one can recover $WF(f|_\Omega)$, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.