Regularization opportunities for the diffuse optical tomography problem

In optical tomography, the radiative properties of the medium under investigation are estimated from information contained in measurements provided by a set of light sources and sensors located on the frontier of the probed medium. Such a non-linear ill-posed inverse problem is usually solved through optimization with the help of gradient-type methods. Since it is well known that such inverse problems are ill-posed, regularization is required. This paper compares three distinct regularization strategies for two different optimization algorithms, namely the damped Gauss-Newton and BFGS algorithms, for the two-dimensional diffuse approximation to the radiative transfer equation in the frequency domain. More specifically, the mesh-based regularization is combined with the Tikhonov regularization in the damped Gauss-Newton algorithm. For the BFGS algorithm, the mesh-based regularization is combined with the Sobolev gradients method. Moreover, a space-dependent Sobolev gradients method is proposed for the first time. The performance of the proposed algorithms are compared by utilizing synthetic data. The deviation factor and correlation coefficient are used to quantitatively compare the final reconstructions. Also, three levels of noise are considered to characterize the behaviour of the proposed methods against measurement noises. Numerical results indicate that the BFGS algorithm outperforms the damped Gauss-Newton in many aspects.