The resolution power of photon tomography is estimated by calculating the root-mean-square deviation of the photon trajectory from its mean path. Statistical simulation is performed using exact values of the migration probability density. To estimate this quantity we calculate the root- mean-square deviation of photon trajectories from the mean path, the initial and final space-time points of trajectory being fixed. We apply the Monte-Carlo procedure in order to calculate the photon-travel deviation variance. In this paper we confine ourselves to infinite isotropically scattering media.
The aim of the present paper is to deliver a method for exact calculation of the probability of the photon migration in infinite and homogeneous medium scattering photons anisotropically. The phase function is represented as an expansion over spherical harmonics. The migration probability is obtained as an expansion with respect to the number of scatterings, the coefficients being dependent on the ratio distance/time and can be easily calculated from recurrence relations. Up to 30 scatterings are taken into account when computing the migration probabilities, the number of effectively contributing scatterings being essentially dependent on the distance-to-time ratio decreasing when approaching the propagation front. The essential property of the method is its capability to exactly describe migration from the first transmitted photons up to the diffusive ones. The limits of the approximate description of anisotropic scattering as isotropic one with an effective value of the scattering coefficient is analyzed by calculating the best-fit value of the scattering coefficient. The present method is a natural generalization of the earlier suggested theory of photon migration with isotropic scattering.
An integral equation for the probability density of the photon migration is constructed. The medium is assumed to be infinite and scatter photons isotropically. The solution is found as an expansion over the number of scatterings. The series occurs to converge rapidly suppose the travel time is fixed. The diffusion approximation is obtained from the integral equation in the limit of large travel time t and distance r such that r << t. It is shown that the probability density tends to infinity near the front. The density calculated is compared with the diffusion limit and the results of a method using path integrals. The latter results appear to be generally poorer than the diffusion theory ones, especially near the front.
A modification of the Monte Carlo simulation of light propagation in turbid isotropically scattering media is proposed. The modification consists in that, besides simulating free paths, multi-scattering photon migrations are statistically modeled by using the exact values of the transition densities obtained for infinite media. An estimate of the efficiency of the method is given. The method is shown to essentially reduce the computing time when modeling the photon propagation far from the boundaries. The angle distribution for the photon after multi-scattering migration is derived. The advantages and drawbacks of the method suggested are discussed.
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