Microlocally correct photoacoustic reconstructions from spatially reduced data
We investigate the inverse source problem for the wave equation arising in photoacoustic and thermoacoustic tomography. If one assumes that the speed of sound is a known constant, then there are many previously developed inversion formulas for the solution of this problem. These formulas require that the data is measured on a closed surface completely surrounding the object or on certain unbounded surfaces. However, in many practical applications, data can only be measured on a finite open surface. In this talk, we will present a non-iterative approach to this problem that yields an approximate solution under certain geometric restrictions on finite open surfaces. This solution coincides with the true solution microlocally--that is, the error is a smooth function. In practical applications, such reconstructions result in qualitatively correct images. For the case of a circular acquisition surface, our method can be implemented with explicit, asymptotically fast methods. We demonstrate the work of these algorithms in a series of numerical simulations.