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

[brian avants]

jeremy

sorry, i was really responding to a question cristina asked some time
ago and it took me a bit to remember the relevant pieces of math to
properly answer her .

> Are you able to explain the bigger picture?

in fem registration, you want to "deform" an image slowly over time
until the external image forces reach equilibrium with the internal
body forces (the regularization provided by the physical model).  the
parameters of the model control all of the above interactions.

> I have the reference and the target image.
> If no grid/mesh is specified a uniform mesh is created.

yes, i think that's correct.

> I assume there are no initial forces. But what happens then?

there is one fixed boundary condition to make the problem well-posed.
otherwise, no forces.

> How are the forces and deformations for every node in every element
> calculated?

we use standard gaussian integration points to integrate the
derivatives of an image similarity over each element and distribute
these derivatives to the nodes of the elements.

> Is it based on pixel data (histogram matching) close to the nodes?

there are a few options but you've got the basic idea --- you're
trying to match the local intensity.

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

not sure exactly what you mean here, but the elements are placed into
the global linear system that models the whole problem in the physical
space.

b.a.

> 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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