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Light Transport in Tissue The redistribution function The single scattering phase function $p(\theta)$ for a tissue determines the amount of light scattered at an angle $\theta$ from the direction of incidence. The redistribution function determines the fraction of light from a cone of angle u will be scattered into a cone of angle v. It is the transmission operator for a single scattering event. The redistribution function h(u,v) is calculated by averaging the phase function over all possible azimuthal angles for fixed angles u and v, $$h(u,v) = {1\over2\pi}\int_0^{2\pi} p\left[uv+\sqrt{1-u^2}\sqrt{1-v^2}\cos\varphi\right]\,d\varphi$$ (3.24) If the cosine of the angle of incidence or exitance is unity (u=1 or v=1) then the redistribution function is equivalent to the phase function h(1,v)=p(v). In the case of isotropic scattering the redistribution function is a constant $$h(u,v) = {1\over4\pi}$$ (3.25) For Henyey-Greenstein scattering, the redistribution function may be expressed in terms of a complete elliptic integral of the second kind [62] $$h(u,v) = {2\over\pi}{1-g^2\over \sqrt{\alpha+\gamma} (\alpha-\gamma)}E(k)$$ (3.26) where g is the average cosine of the Henyey-Greenstein phase function and $$\alpha=1+g^2-2 g u v \qquad \gamma=2 g \sqrt{1-u^2}\sqrt{1-v^2} \qquad k = \sqrt{2\gamma\over\alpha+\gamma}$$ (3.27) and E(k) is the complete elliptical integral tabulated by Jahnke and Emde [33]. This function may be calculated with numerical routines by Press et al. [49]. The redistribution function for the modified Henyey-Greenstein phase function follows directly from Equations (3.25) and (3.26) $$h(u,v) = \beta + (1-\beta){2\over\pi} {1-g^2\over \sqrt{\alpha+\gamma} (\alpha-\gamma)}E(k)$$ (3.28) Other phase functions require numerical integration of Equation (3.24). If the phase function is highly anisotropic, then the integration over the azimuthal angle is particularly difficult and care must be taken to ensure that the integration is accurate. This is important because errors in the redistribution function enter directly into the reflection and transmission operators for thin layers. These errors will be doubled with each successive addition of layers and small errors will rapidly increase. The normalization of the phase functions provides a check on the accuracy of the quadrature method [19,18]. $${1\over2}\sum_{i=1}^n w_i[h(\mu_i,mu_j)+h(-\mu_i,\mu_j)] =1$$ (3.29) where n is the number of quadrature angles used and the minus sign in the second term represents the ``reflected'' component of the scattered light. If relation (3.29) is not satisfied then the number of quadrature angles should be increased. This is not always practical and a number of phase function renormalization methods have been developed to remedy this problem (see review in[67]).

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