Analysis of the inverse problem associated with diffuse correlation tomography
Keywords: inverse problems, tomography, image reconstruction
AbstractThe aim of this article is to study the mathematical analysis for an inverse problem and its numerical implementation associated with diffuse correlation tomography. The coefficients of the diffusion equation governing the propagation of field autocorrelation through a turbid medium (tissue-like) depend on both the optical and mechanical properties of the medium. Assuming the mechanical property is given by a time independent particle diffusion coefficient ($D_B$), we consider the development of regularized Gauss-Newton algorithm for the recovery of $D_B$ from boundary measurements. We study the existence and uniqueness of the forward problem and also for the Fr\'echet derivative operator which are essential for convergence study. The nonlinear minimization problem associated with the recovery of $D_B$ is locally linearized and solved through a regularized Gauss-Newton algorithm. The conditions to be satisfied for the convergence of the Gauss-Newton algorithm are established. Finally, the method is proven through numerical recovery of $D_B$ from intensity autocorrelation measured at the boundary. Once $D_B$ is obtained one can also recover other mechanical properties.