Diffusive Optical Tomography in the Bayesian Framework
Computational tomography amounts to reconstructing the scattering coefficient in the radiative transfer equation, the model equation for photon dynamics. All currently available algorithms recover the coefficient well in some cases, but poorly in others, no matter how much data is provided. We give a quantitative explanation to this phenomenon using the Bayesian formulation. Specifically, we study the well-posedness of the inverse radiative transfer equation, and measure the deterioration of the stability as the equation changes regimes. As the scattering effect becomes strong, the radiative transfer equation converges to the diffusion equation, whose inverse problem is severely ill-posed. We show this in the Bayesian setting, using both the Kullback-Leibler divergence and the Hellinger distance. Such stability analysis allows us to assess a priori the value of the measured data in different regimes.