Simultaneous retrieval of optical and geometrical parameters of multilayered turbid media via state-estimation algorithms

Abstract

In the present paper we propose an implementation of the Kalman filter algorithm, which allows simultaneous recovery of the absorption coefficient, the reduced scattering coefficient and the thicknesses of multi-layered turbid media, with the deepest layer taken as semi-infinite. The approach is validated by both Monte Carlo simulations and experiments, showing good results in structures made up of four layers. As it is a Bayesian algorithm, prior knowledge can be included to improve the accuracy of the retrieved unknowns. One of the most promising applications of this approach is the capability of real-time monitoring of living organs by near infrared spectroscopy. In particular, determination of blood perfusion in the adult head is one of the desired goals, allowing continuous control of stroke patients. This demands accurate measurement of the optical properties, especially absorption, of the head layers, from scalp to the cortex.

Implementation of the extended Kalman filter for determining the optical and geometrical properties of turbid layered media by time-resolved single distance measurements

“…As explained above, after we ensure that we can apply the EKF, we must provide the initial distributions in order to apply it. For the parameter distributions of µ a, j , µ s, j and l, we use the same change of variables as in [15], but not for t 0 which we assume that follows a Gaussian distribution, as it can be either positive or negative. We assume that the parameters are independent a priori and, using a 3σ-confidence interval around the mean we obtain the values in Table 1.…”

“…As explained in Ref. [15], the result of the EKF is a multivariate distribution which can be too complex to analyze, in order to study each parameter separately, is is possible to marginalize every parameter by integrating the distribution with respect to all the parameters but the one of interest. This gives us information of the uncertain of the parameter of interest after considering all the possible values of the other ones (according to our Fig.…”

“…The Extended Kalman Filter (EKF) [15,22] is a non-linear, non-optimal generalization of the Kalman Filter which is suitable for problems where the forward model is non-linear. In our specific application we will name x to the vector containing the optical properties µ a and µ s (for the corresponding layer), the geometrical parameter l and t 0 (an optional time shift to account for an imperfect correction of the delay due to different fiber arrangements in IRF and phantom measurements), and y to the measured DTOF.…”

“…Although the output of the EKF is a distribution, we may be interested in particular point estimates which may be useful to analyze particular cases, such as the mode, the median, the mean, etc. In this work we will use the Maximum A Posteriori (MAP) estimate, which corresponds to the mode of the resulting distribution after an appropriate reparametrization [15].…”

“…This algorithm is a modified version of the one presented in Ref. [15], where the recovery of optical properties and thicknesses of a four-layered turbid medium in a multi-distance setup using a frequency-domain approach was performed, through the proposal of a static estimation problem. The main improvements are the use of a single-distance approach, a more straightforward deconvolution process that does not require calibration of the corresponding hyperparameters, and the discard of the calibration process, which implies less complexity and computation times.…”