Light Transport in Tissue

Delta functions

The delta function used in radiative transport calculations differs slightly from the usual form of a delta function. It has the usual properties: it is zero if $\hat\mathbf{s}\ne\hat\mathbf{s}'$

$\begin{displaymath} \delta(\hat\mathbf{s}-\hat\mathbf{s}')=0,\qquad\mathrm{if}\qquad \hat\mathbf{s}\ne\hat\mathbf{s}' \end{displaymath}$

(C18)

and the integral of the delta function with another function equals that function evaluated at $\hat\mathbf{s}'=\hat\mathbf{s}$

$\begin{displaymath} \int_{4\pi}f(\hat\mathbf{s}')\delta(\hat\mathbf{s}-\hat\mathbf{s}')d\omega'=f(\hat\mathbf{s}) \end{displaymath}$

(C19)

The delta function $\delta(\hat\mathbf{s}-\hat\mathbf{s}')$ is two-dimensional and is written as the product of two ordinary Dirac delta functions [7]

$\begin{displaymath} \delta(\hat\mathbf{s}-\hat\mathbf{s}')=\delta(\mu-\mu')\delta(\varphi-\varphi') \end{displaymath}$

(C20)

where $\hat\mathbf{s}$ and $\hat\mathbf{s}'$ are described by the angle pairs $(\theta,\varphi)$ and $(\theta',\varphi')$ on the unit circle and $\mu=\cos\theta$ and $\mu'=\cos\theta'$ . To write Equation (C.20) in terms of $\theta$ instead of $\mu$ then using the property [51] of delta functions that

$\begin{displaymath} \delta(f(x))={\delta(x-x_0)\over\vert f'(x_0)\vert} \end{displaymath}$

(C21)

where f(x) is a function which vanishes only at x₀. This property may be used because the integrals are done over the sphere and consequently, $(\cos\theta'-\cos\theta)$ will vanish at one point. The following relation may be obtained

$\begin{displaymath} \delta(\hat\mathbf{s}-\hat\mathbf{s}')= {\delta(\theta-\theta')\delta(\varphi-\varphi')\over \vert\sin\theta \vert} \end{displaymath}$

(C22)

Equation (C.20) is valid as long as neither $\hat\mathbf{s}$ nor $\hat\mathbf{s}'$ coincide with the z-axis. In this case, the azimuthal coordinate is an ignorable coordinate [51] and the expression for the delta function becomes

$\begin{displaymath} \delta(\hat\mathbf{s}-\hat\mathbf{z})={1\over2\pi}\delta(1-\mu) \end{displaymath}$

(C23)

This ensures that the integral over all angles remains unity. Equations (C.23) and (C.21) relate a solid angle delta function depending only on one parameter with the usual two parameter definition [34]

$\begin{displaymath} \delta(1-\cos\theta)=2\pi\delta(\mu-\mu')\delta(\varphi-\varphi') \end{displaymath}$

(C24)

where $\cos\theta$ is the angle between the direction specified by $(\mu,\varphi)$ and $(\mu',\varphi')$

$\begin{displaymath} \cos\theta=\mu\mu'+\sqrt{1-\mu^2}\sqrt{1-\mu'^2}\cos(\varphi-\varphi') \end{displaymath}$

(C25)

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.