# What is meant by isotropic diffusion? [closed]

I am working on diffusion of medical images, and I have come across the term isotropic diffusion. Can anyone explain it for me?

## closed as too broad by jojek♦, lennon310, MBaz, ThP, Peter K.♦Apr 24 '15 at 22:04

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## 2 Answers

It means the diffusion is at the same rate to all directions, not anisotropic diffusion say more along the direction of some fibers.

• Fibers? What is a fiber? – nbro Apr 8 at 15:56
• @nbro say cotton fibers with water diffusing in a fabric made of the fibers, or nerve tracts in the brain – Olli Niemitalo Apr 8 at 15:59
• But here we are interested in computer vision not cotton production or neuroscience. – nbro Apr 8 at 16:08
• @nbro the context is medical imaging, so the diffusion could be physical. See: diffusion MRI. – Olli Niemitalo Apr 8 at 16:19

As Olli has said, Isotropic means the same in all directions. So the diffusion behaves the same regardless of which direction it is diffusing in.

Heat for example diffuses in an isotropic manner. (Well ... unless you have a strange environment with varying conductivity constants or something ... but that's another story)

The heat equation is

$\frac{\partial u} {\partial t} = k \nabla ^2 u$

Where $U(x,y)$ is heat at point $(x,y)$ and $t$ is time

$k$ is the thermal diffusion constant.

I think it helps to see visually what this looks like

(From wikipedia)

This is actually just like doing Gaussian Blur on an image.

(Gaussian convolution is a solution to the heat equation)

Gaussian blur before and after Note that the whole image gets the same blur

Anisotropic diffusion on the other hand may diffuse differently in different directions.

For example, perona and malik suggested doing diffusion based on the Laplacian (that's the 'edginess' $\nabla^2$) of that part of the image.

This can be used for performing blur only in certain parts of an image.

See here for an image which has had anisotropic diffusion. This has had the effect of removing noise, while preserving the main features of the image.

The equation proposed by Perona and Mailk is

$\frac{\partial U}{\partial t} = c(x,y,t) \nabla ^2 U + \nabla c . \nabla U$

where $c$ is a function such as

$C(||\nabla U||) = e^{-(||\nabla U||/K)^2}$ for example