  The phase function is a key element of a light propagation model for Monte
Carlo (MC) simulation, which is usually fitted with an analytic function with
associated parameters. In recent years, machine learning methods were reported
to estimate the parameters of the phase function of a particular form such as
the Henyey-Greenstein phase function but, to our knowledge, no studies have
been performed to determine the form of the phase function. Here we design a
convolutional neural network to estimate the phase function from a diffuse
optical image without any explicit assumption on the form of the phase
function. Specifically, we use a Gaussian mixture model as an example to
represent the phase function generally and learn the model parameters
accurately. The Gaussian mixture model is selected because it provides the
analytic expression of phase function to facilitate deflection angle sampling
in MC simulation, and does not significantly increase the number of free
parameters. Our proposed method is validated on MC-simulated reflectance images
of typical biological tissues using the Henyey-Greenstein phase function with
different anisotropy factors. The effects of field of view (FOV) and spatial
resolution on the errors are analyzed to optimize the estimation method. The
mean squared error of the phase function is 0.01 and the relative error of the
anisotropy factor is 3.28%.
