Ultrasound modulated bioluminescence tomography with a single optical measurement

Ultrasound modulated bioluminescence tomography is an imaging method which can be formulated as a hybrid inverse source problem. In the regime where light propagation is modeled by a radiative transfer equation, previous approaches to this problem require large numbers of optical measurements (Bal et al 2016 SIAM J. Math. Anal. 48 1332). Here we propose an alternative solution for this inverse problem which requires only a single optical measurement in order to reconstruct the isotropic source. Specifically, we derive two inversion formulae based on Neumann series and Fredholm theory respectively, and prove their convergence under sufficient conditions. The resulting numerical algorithms are implemented and experimented to reconstruct both continuous and discontinuous sources in the presence of noise.