Diffuse optical tomography through solving a system of quadratic equations without re-estimating the derivatives: the "Frozen-Newton" method

Optical tomography (OT) recovers the cross-sectional distribution of optical parameters inside a highly scattering medium from information contained in measurements that are performed on the boundary of the medium. The image reconstruction problem in OT can be considered as a large-scale optimization problem, in which an appropriately defined objective functional needs to be minimized. Most of earlier work is based on a forward model based iterative image reconstruction (MOBIIR) method. In this method, a Taylor series expansion of the forward propagation operator around the initial estimate, assumed to be close to the actual solution, is terminated at the first order term. The linearized perturbation equation is solved iteratively, re-estimating the first order term (or Jacobian) in each iteration, until a solution is reached. In this work we consider a nonlinear reconstruction problem, which has the second order term (Hessian) in addition to the first order. We show that in OT the Hessian is diagonally dominant and in this work an approximation involving the diagonal terms alone is used to formulate the nonlinear perturbation equation. This is solved using conjugate gradient search (CGS) without re-estimating either the Jacobian or the Hessian, resulting in reconstructions better than the original MOBIIR reconstruction. The computation time in this case is reduced by a factor of three.