# Anisotropic Diffusion Filter - Intuition Behind Parameters

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.
• You have to look for discretised versions of these equations. See the original paper of Perona and Malik on this. Mar 27, 2014 at 23:54
• One remark, though it is called "Anisotropic Diffusion", today this term is usually reserved to methods which changes the diffusion according to the Structure Tensor of the image. It is more appropriate to call this method Non Linear Diffusion.
– Royi
Oct 25, 2014 at 17:44

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.

• First paper you provided is closed access.
– jojeck
Apr 28, 2014 at 13:00
• Which paper? All seems to be accessible now.
– Royi
Sep 14, 2019 at 21:22

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.