An Optimal Bayesian Estimator for Absorption Coefficient in Diffuse Optical Tomography

Diffuse Optical Tomography (DOT) is a well-known imaging technique for detecting the optical properties of an object in order to detect anomalies, such as diffusive or absorptive targets. More specifically, DOT has many applications in medical imaging including breast cancer screening. It is an affordable and noninvasive method to recover the optical properties of a body using photon density measurements from applying laser sources placed at its surface. Mathematically, the reconstruction of the internal absorption or scattering is a severely ill-posed inverse problem and yields a poor quality image reconstruction. Studying coefficient inverse problems in a stochastic setting has increasingly gained in prominence in the past couple of decades. In this work, we will show convergence and optimality for a Bayesian estimator for the absorption coefficient built from the noisy data obtained in a simplified DOT Model. We establish the rate of convergence of such an estimator in the supremum norm loss and show that it is optimal. We also present numerical experiments in support of our theoretical findings.