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The equation above applies when the diffusion coefficient is isotropic; in the case of anisotropic diffusion, D is a symmetric positive definite matrix, and the equation is written (for three dimensional diffusion) as:
In diffusion theory, radiance is taken to be largely isotropic, so only the isotropic and first-order anisotropic terms are used: (, ^,) = =, (,), (^) where n, m are the expansion coefficients. Radiance is expressed with 4 terms: one for n = 0 (the isotropic term) and 3 terms for n = 1 (the anisotropic terms).
Gaussian functions are the Green's function for the (homogeneous and isotropic) diffusion equation (and to the heat equation, which is the same thing), a partial differential equation that describes the time evolution of a mass-density under diffusion.
This can be visualized with an ellipsoid, which is defined by the eigenvectors and eigenvalues of D. The FA of a sphere is 0 since the diffusion is isotropic, and there is equal probability of diffusion in all directions. The eigenvectors and eigenvalues of the Diffusion Tensor give a complete representation of the diffusion process.
Photon diffusion equation is a second order partial differential equation describing the time behavior of photon fluence rate ... is an isotropic source ...
Anisotropic diffusion can be used to remove noise from digital images without blurring edges. With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the heat equation which is equivalent to Gaussian blurring. This is ideal for removing noise but also indiscriminately blurs edges too.
The equation of radiative transfer describes these interactions mathematically. Equations of radiative transfer have application in a wide variety of subjects including optics, astrophysics, atmospheric science, and remote sensing. Analytic solutions to the radiative transfer equation (RTE) exist for simple cases but for more realistic media ...
It is also assumed that the recovery can be modelled by diffusion in two dimensions, that is also both uniform and isotropic. In other words, that diffusion is occurring in a uniform medium so the effective diffusion constant D is the same everywhere, and that the diffusion is isotropic, i.e., occurs at the same rate along all axes in the plane.