Using the topological derivative for initializing a Markov-chain Monte Carlo reconstruction in fluorescence tomography

Fluorescence tomography aims at the reconstruction of the concentration and life-time of fluorescent inclusions from boundary measurements of light emitted. The underlying ill-posed problem is often solved with gradient descent of Gauss-Newton methods, for example. Unfortunately, these approaches don't allow to assess the quality of the reconstruction (e.g. the variance and covariance of the parameters) and also require the tuning of regularization parameters. We intend to mitigate this drawback by the application of topological derivatives and Markov-chain Monte-Carlo (MCMC) methods for solving the inverse problem. This submission focuses on the topological derivative, which is used for the initialization of the MCMC code. The basic idea is to probe every location inside the domain with an infinitely small fluorescent ball and to estimate the effect of such a perturbation on the residual, which is the difference of the theoretically predicted data to the true measurement. Obviously, the reconstructed inclusions should be placed at locations for which the topological derivative is significantly negative, i.e. where the residual decreases. Previous results show that usual first-order approximations deteriorates for probe inclusions close to the boundary. This seems to be a particular feature of certain inverse problems such as fluorescence tomography or electrical impedance tomography. Fortunately this flaw may be corrected using a few higher-order terms which may be explicitly determined With this extension the topological derivative can be utilized as a one-step method for the determination of the number of inclusions and their approximate locations. This outcome is used as initialization for the MCMC algorithm.