A theoretically exact filtered back-propagationformula for single-time wave inversion

We consider single-time wave inversion, that is, recovering the initial pressure $f$ from the wave field $\mathcal{W}_T f = u(\cdot,T)$ at a single time $T$, where $u$ solves the wave equation with sound speed $c$ and initial data $(f,0)$. In two dimensions this is the main reconstruction step of full-field photoacoustic projection imaging, where each camera image is a two-dimensional wave field at a single time instant. An efficient direct method is back-propagation by the Fourier multiplier $2\cos(cT|ξ|)$. It yields exact inversion in odd dimensions but not in two. We give its error in closed form, with bounds, and derive an exact explicit filtered back-propagation $f = 2\mathcal{W}_T \mathcal{M}_K \mathcal{W}_T f$ in analogy to filtered backprojection for the Radon transform.