We study the problem of recovering the initial data of the two dimensional wave equation from values of its solution on the boundary ∂Ω of a smooth convex bounded domain Ω⊂ℝ2. As a main result we establish back-projection type inversion formulas that recover any initial data with support in Ω modulo an explicitly computed smoothing integral operator KΩ. For circular and elliptical domains the operator KΩis shown to vanish identically and hence we establish exact inversion formulas of the back-projection type in these cases. Similar results are obtained for recovering a function from its mean values over circles with centers on ∂Ω. Both reconstruction problems are, amongst others, essential for the hybrid imaging modalities photoacoustic and thermoacoustic tomography. © 2013 Elsevier Ltd. All rights reserved.
Haltmeier, M. (2013). Inversion of circular means and the wave equation on convex planar domains. Computers and Mathematics with Applications, 65(7), 1025–1036. https://doi.org/10.1016/j.camwa.2013.01.036