Editing Diffusion equation (section)

== 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:
<math display="block"> \frac{\partial\phi(\mathbf{r},t)}{\partial t} = \nabla\cdot \left[D(\phi,\mathbf{r})\right] \nabla \phi(\mathbf{r},t) + {\rm tr} \Big[ D(\phi,\mathbf{r})\big(\nabla\nabla^\text{T} \phi(\mathbf{r},t)\big)\Big] </math>
where "tr" denotes the [[Trace (linear algebra)|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 tensor]]s. The spatial derivatives can then be approximated by two first order and a second order central [[finite difference]]s. 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.