[Insight-users] Implementation Finite Element Method (FEM) in ITK

[Jeremy Bournesel]

Thanks Brian.

Are you able to explain the bigger picture?

I have the reference and the target image.

If no grid/mesh is specified a uniform mesh is created.

I assume there are no initial forces. But what happens then?
How are the forces and deformations for every node in every element
calculated?
Is it based on pixel data (histogram matching) close to the nodes?

Based on the local energy at the element level, how is the global energy
calculated?

These are the kind of questions I have, it be really awesome if you can
help.
Thanks,
-J




On Tue, Feb 1, 2011 at 8:17 PM, brian avants <stnava at gmail.com> wrote:

> hi jeremy , cristina
>
> here is a brief latex sketch of the method used in the
> FEMCrankNicolsonSolver --
>
> \text{matrix form parabolic PDE, heat equation type} \\
> M\frac{ \partial U}{\partial t} +  K U  = f \\
> \frac{ M U_t - M  U_{t-1}}{\delta_t} +  K U  = f \\
> \frac{ M U_t - M  U_{t-1} +  \delta_t K U }{\delta_t} = f \\
> \text{ here i make the decision to represent $U$ and $f$ at $t$ rather
> than $t-1$ i believe this is backward Euler}\\
> ( M  + \delta_t K ) U_t  =  M  U_{t-1} + \delta_t f_t \\
> \\
> \text{other choices are possible, e.g. } \\
> \text{ Crank-Nicolson discretization is based on averaging forward and
> backward Euler } \\
> ( M + \alpha \delta_t K ) U_t = (  M - ( 1- \alpha ) \delta_t  K )
> U_{t-1} + f \\
>  f  =  \delta_t ( \alpha f_t + ( 1 - \alpha ) f_{t-1} ) \\
> \\
> \text{if}~ \alpha = 1 \text{ backward Euler }\\
> ( M +  \delta_t K ) U_t = M  U_{t-1} + \delta_t f \\
> \\
> \text{if}~ \alpha = 0 \text{ forward Euler }\\
>  M U_t =(  M - \delta_t  K ) U_{t-1} + \delta_t f_{t-1}
>
> hopefully this is helpful.
>
> brian
>
>
>
>
> On Tue, Feb 1, 2011 at 4:01 PM, Jeremy Bournesel
> <jeremy.bournesel at gmail.com> wrote:
> > Hi,
> > I have a decent background regarding FEM in the engineering domain and
> have
> > started looking at it for registering medical images as well.
> > I ran one of the ITK deformable registration demos, read the Software
> Guide
> > and the ITK Powerpoint Presentations, however there is still some magic
> > going on that I don't completely get.
> > From what I understand ITK is using a uniform grid/mesh (if a custom one
> > wasn't supplied) and then calculates the deformation for every element at
> > the nodal points (iteratively).
> > I couldn't find any information on how the exact process is working
> (besides
> > some high level slides of the type K U = F).
> > It'd be awesome if someone can point me to the right document or explain
> it
> > to me.
> > Thanks,
> > Jeremy
> >
> >
> >
> >
> >
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