# Flux Diffusion in ITK (Karl Krissian)

## Contents

# Implementing the flux diffusion in ITK

the current vtk implementation is in the slicer CVS

## Current anisotropic diffusion in ITK

The current anisotropic diffusion implemented in ITK are:

- the Perona and Malik equation
- the curvature diffusion

They are both scalar anisotropic diffusion, without data attachment, and running on an explicit numerical scheme.

## Flux diffusion equation

The Flux diffusion is a matrix flux diffusion:

where u(t) is the evolving image, *u*_{0} = *u*(0) is the initial image, β is a data attachment coefficient and M is the diffusion matrix.

In the case of the flux diffusion, we re-write the vector field

in the basis of the gradient and the principal curvature directions of the smoothed image (in 3D):

with (*e*_{0},*e*_{1},*e*_{2}) corresponding respectively to the smooth gradient, minimal and maximal curvature directions.

## Numerical Scheme

We use a Jacobi or a Gauss-Seidel scheme, where the filtered image is the solution of an equation of the form

*N'***v****= 0****,**

v is a vector representing the image and N is a nxn matrix, n being the total number of voxels. the diagonal part of N is detached from N:

*N*=*P*+*D*,

and the Jacobi numerical scheme consists in iterating

*v*^{n + 1}= -*D*^{ - 1}*P'***v**^{n}**.**