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Light Transport in Tissue Definition of reflection and transmission operators Denote the cosine of the angle that a right circular cone makes with the normal by $\mu$ (Figure 3.1A). Furthermore let the incident radiance, which is a function of incident angle $\mu$, be denoted by $L^+_{\mathrm{incident}}(\mu)$ where the plus sign indicates the downward direction. The total radiance transmitted through the slab at a particular cosine angle $\mu'$ is given by [60] $$L_{\rm transmitted}^+(\mu')= \int_0^1 T(\mu',\mu)L_{incident}^+(\mu) \,2\mu d\mu$$ (3.1) The total radiance reflected by the sample is $$L_{\rm reflected}^-(\mu')= \int_0^1 R(\mu',\mu)L_{incident}^+(\mu) \,2\mu d\mu$$ (3.2) Figure: Geometric description of nomenclature. Figure A illustrates the incident radiance $L^+_{\protect\mathrm{incident}}$ with the cosine of the angle of incidence equal to $\mu _0$. Figure B shows the nomenclature for the upward and downward radiances from each surface of the slab. The operators $T(\mu, \mu_0)$ and $R(\mu, \mu_0)$ may be written in matrix form $$T^{ij}=T(\mu_i,\mu_j) \qquad\qquad R^{ij} = R(\mu_i,\mu_j)$$ (3.3) where the angles $\mu_i$ and $\mu_j$ are chosen according to the particular quadrature scheme desired. The superscripts ij indicate the entry ij in the matrices T and R. The radiances $L^+(\mu)$ and $L^-(\mu)$ correspondingly may be written as vectors $$L^i = L(\mu_i)$$ (3.4) The matrix star multiplication $A \star B$ is defined to directly correspond to an integration similar to those in Equations (3.1) and (3.2) $$A\star B = \int_0^1A(\mu_,\mu')B(\mu',\mu'')\, 2\mu d\mu$$ (3.5) then $$A\star B = \sum_j A^{ij}2\mu_jw_j B^{jk}$$ (3.6) where $\mu_j$ is the jth quadrature angle and wj is its corresponding weight. The identity matrix E is then $$E^{ij}={1\over2\mu_i w_i} \delta_{ij}$$ (3.7) where $\delta_{ij}$ is the usual Kronecker delta. Grant and Hunt have shown that this algebra is a semi-group [21,22] and have proven that all matrix manipulations in Section 3.2 are valid mathematically. It is sometimes useful to consider these matrix ``star multiplications'' as normal matrix multiplications which include a diagonal matrix c $$c_{ij}=2\mu_i w_i \delta_{ij}$$ (3.8) Thus a matrix star multiplication may be written $$A\star B = A\, c\, B$$ (3.9) where the multiplications on the R.H.S. of Equation (3.9) are usual matrix multiplications.

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