Anisotropic Diffusion Filter - Intuition Behind Parameters

Question

I need to know what is iteration and divergence in anisotropic diffusion filter technique.

• Isotropic diffusion $$\frac{\partial I(x, y, z)}{\partial t}={\rm div}\left[c\cdot \nabla I\left(x, y, z\right)\right], \quad \text{where } c \text{ is the diffusion coefficient}$$

• Anisotropic diffusion $$\frac{\partial I(x, y, z)}{\partial t}={\rm div}\left[g\left(\left\| I\left(x, y, z\right)\right\|\right)\cdot \nabla I\left(x, y, z\right)\right], \quad \text{where } g \text{ is the anisotropic diffusion coefficient }\textbf{(Edge stopping function)}$$

• Here $t$ refers to iteration, what is this iteration ? How is it related to filtering ? I know that iteration refers to number of rounds but how it is related with filtering ?
• Also explain to me why we use divergence in these equation what is the purpose of divergence.

Answer 1

Divergence is a differential operator, something like derivation. It can be approximated by a filter. It measures if a vector field is a ,,sink'' or a ,,source'' in a given locus.

Answer 2

This paper - The Structure of Images describes quite nicely to get to the diffusion equations from the point of view of a filter. It's about the isotropic case, but anisotropic is really just an extension of that. For anisotropic diffusion see Scale Space and Edge Detection Using Anisotropic Diffusion as Deniz suggests this one is quite good as it also includes discretised versions of the equations.

The basic idea is to solve the heat equation where $t$ is the scale of the image, i.e. the amount of filtering to apply. For anisotropic diffusion the diffusion coefficient $c$ is replaced by $g$ which is dependent on the image gradient and therefore filters preferentially depending on the gradient.

The divergence here really just arises from the definition of the heat equation.