where ϕ(r, t) is the
density of the diffusing material at location r and time t and D(ϕ, r) is the collective
diffusion coefficient for density ϕ at location r; and ∇ represents the vector
differential operatordel. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.
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:
The diffusion equation has numerous analytic solutions.[1]
The diffusion equation can be trivially derived from the
continuity equation, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:
where j is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological
Fick's first law, which states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
If drift must be taken into account, the
Fokker–Planck equation provides an appropriate generalization.
The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise.
In discretizing space alone, the
Green's function becomes the
discrete Gaussian kernel, rather than the continuous
Gaussian kernel. In discretizing both time and space, one obtains the
random walk.
Discretization in image processing
The
product rule is used to rewrite the anisotropic tensor diffusion equation, in standard discretization schemes, because direct discretization of the diffusion equation with only first order spatial central differences leads to checkerboard artifacts. The rewritten diffusion equation used in image filtering:
where "tr" denotes the
trace of the 2nd rank
tensor, and superscript "T" denotes
transpose, in which in image filtering D(ϕ, r) are symmetric matrices constructed from the
eigenvectors of the image
structure tensors. The spatial derivatives can then be approximated by two first order and a second order central
finite differences. The resulting diffusion algorithm can be written as an image
convolution with a varying kernel (stencil) of size 3 × 3 in 2D and 3 × 3 × 3 in 3D.