Plamen Stefanov, Gunther Uhlmann
In the plane, we study the transform $R_\gamma f$ of integrating a unknown function $f$ over circles centered at a given curve $\gamma$. This is a simplified model of 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 the "artifacts" cannot be resolved. If $\gamma$ is a closed curve, we show that this is still true. On the other hand, if $f$ is known a priori to have singularities in a compact set, then we show that one can recover $\WF(f)$, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.
View original:
http://arxiv.org/abs/1205.0861
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