Perspectives in Fourier-based image reconstruction in photoacoustic tomography

Photoacoustic tomography (PAT) is an emerging imaging technique with great potential for a wide range of biomedical imaging applications. The reconstruction problem of PAT is an inverse source problem, in which the photoacoustic source of interest is induced by a probing optical wavefield. In this work, we revisit the PAT reconstruction problem from a Fourier perspective. By use of standard analytic techniques from inverse source theory, we derive a mathematical relationship between the pressure wavefield data function and its normal derivative measured on an arbitrary aperture that encloses the object and the three-dimensional Fourier transform of the optical absorption distribution evaluated on concentric spheres. We refer to this relationship as a "Fouriershell identity", which is analogous to the well-known Fourier-slice theorem of X-ray tomography. Potential applications of the Fourier-shell identity are identified and discussed.