Inversion of the Star Transform

We define the star transform as a generalization of the broken ray transform for image reconstruction in single scattering tomography. Using the star transform provides advantages including possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously. We derive the star transform from physical principles, and derive several computationally efficient algorithms for its inversion. We discuss mathematical properties and analyze numerical stability of inversion, and obtain necessary conditions for stable reconstruction. An approach combining scattered rays and ballistic rays to improve reconstruction is provided, and total variation and L1 regularization are utilized to remove noise. Numerical experiments are carried out to test the algorithms of inversion, the possibility to recover the absorption and the scattering coefficients, and the effect of different regularizations.