Kirchhoff approximation for diffusive waves

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Abstract
Quantitative measurements of diffuse media, in spectroscopic or imaging mode, rely on the generation of appropriate forward solutions, independently of the inversion scheme employed. For complex boundaries, the use of numerical methods is generally preferred due to implementation simplicity, but usually results in great computational needs, especially in three dimensions. Analytical expressions are available, but are limited to simple geometries such as a diffusive slab, a sphere or a cylinder. An analytical approximation, the Kirchhoff approximation, also called the tangent-plane method is presented for the case of diffuse light. Using this approximation, analytical solutions of the diffusion equation for arbitrary boundaries and volumes can be derived. Also, computation time is minimized since no matrix inversion is involved. The accuracy of this approximation is evaluated on comparison with results from a rigorous numerical technique calculated for an arbitrary geometry. Performance of the approximation as a function of the optical properties and the size of the medium is examined and it is demonstrated that the computation time of the direct scattering model is reduced at least by two orders of magnitude.