View source for Flux Diffusion in ITK (Karl Krissian)

= Implementing the flux diffusion in ITK =

<br />[[Image:FluxDiffusion.ppt|Image:FluxDiffusion.ppt]]

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:<br />

: [[Image:96d5e1fc81df8538de55e32d4cd6ceb0.png|\frac{\partial u}{\partial t} = div(M \nabla u) + \beta (u_0-u)]],<br />

where u(t) is the evolving image, <span class="texhtml">''u''<sub>0</sub> = ''u''(0)</span> is the initial image, <span class="texhtml">β</span> is a data attachment coefficient and M is the diffusion matrix.

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

: [[Image:12c028e84be1fa5a44661f65fd9fe097.png|F = M \nabla u]],

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

: [[Image:8095d098be3c15f4cdc38e0f46d2f566.png|F = \sum_{i=0}^{2} \phi_i(u_{e_i}) e_i]],

with <span class="texhtml">(''e''<sub>0</sub>,''e''<sub>1</sub>,''e''<sub>2</sub>)</span> 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

: <span class="texhtml">''N''''v'' = 0</span>,

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:

: <span class="texhtml">''N'' = ''P'' + ''D''</span>,

and the Jacobi numerical scheme consists in iterating

: <span class="texhtml">''v''<sup>''n'' + 1</sup> = - ''D''<sup> - 1</sup>''P''''v''<sup>''n''</sup></span>.