Recovering a Function from Circular Means or Wave Data on the Boundary of Parabolic Domains

Determining a function from its circular means with restricted centers is crucial for many modern medical imaging and remote sensing applications. Examples include photo- and thermoacoustic tomography, ultrasound imaging, and synthetic aperture radar. In this paper we derive an explicit inversion formula for the case where the centers of the circles of integration are restricted to the boundary of parabolic domains. A similar result is obtained for recovering the initial data of the wave equation from data on the boundary of a parabolic domain. These results show that the universal back-projection formulas, previously known to be exact for half-spaces, discs, and ellipsoids, are also exact for parabolic domains. We further present numerical simulations supporting our theoretical results. There, we also include two previously proposed strategies (dynamic aperture length correction and weight factor correction) accounting for the truncation of the parabola in practical applications.