Sr. Arridge et M. Schweiger, DIRECT CALCULATION OF THE MOMENTS OF THE DISTRIBUTION OF PHOTON TIME-OF-FLIGHT IN TISSUE WITH A FINITE-ELEMENT METHOD, Applied optics, 34(15), 1995, pp. 2683-2687
Modeling of the full temporal behavior of photons propagating in diffu
sive materials is computationally costly. Rather than deriving intensi
ty as a function of time to fine sampling, we may consider methods tha
t derive a transform of this function. To derive the Fourier transform
involves calculation in the (complex) frequency domain and relates to
intensity-modulated experiments. We consider instead the Mellin trans
form and show that this relates to the moments of the original tempora
l distribution. A derivation of the Mellin transform given the Fourier
transform that permits closed-form derivations of the temporal moment
s for various simple geometries is presented. For general geometries a
finite-element method is presented, and it is demonstrated that the c
omputational cost to produce the nth moment is the same as producing t
he first n temporal samples of the original function.