3-D volumetric reconstruction of the absorption coefficient was carried out using a modified version of the normalized difference approach proposed by Pei et al.36 At each sampling time , the 3-D absorption coefficient map , discretized over the 3-D mesh, is determined by minimizing, using a nonlinear conjugated-gradient algorithm,37 the following cost function38Display Formula
(9)The first summation in the right side of Eq. (9) is over the volume elements of the 3-D mesh. In Eq. (9), is the total number of sources, is the number of detectors for the ’th source, is the flux measured by the ’th optode given the source at time , is a reference state defined as the mean of the baseline measurements, and and are the model predicted measurements for the reference and estimated absorption coefficient at time . Essentially, for each time sample, the initial guess in the optimization process is the reference state . The conjugate gradient descent algorithm is applied iteratively for steps to compute the estimate . Note that the optimized image from a previous time sample can be used also as the initial guess. However, in this paper, concentration calculations were carried out independently of previous or subsequent samples.