[Insight-users] One more question on FEM registration

[Markus Weigert]

Hi,

today I have a more theoretical question on how
the FEM toolkit solves the displacement problem.

On a introductory paper about the Finite Element Method
(dealing with mechanical problems), I read about two different
methods (called Ritz and Galerkin) to solve the problem of estimating 
the displacements.
Both methods seem to work in a similar way on minimizing the potential
energy formulation by using a function representation for the displacement
components and stress and strain formulations. 
I know from the official itk registration - tutorial ppt, that the ITK FEM toolkit also
uses this functional formulation for the displacements.

I currently try to close the gap from the mechanical problem to the registration
problem, and this causes still some problems in understanding.
I mean, which of the described methods does the ITK implementation use in general
and how are the forces, derived from the similarity metric, related to general
stress and strain?

You will find the script at http://www.ce.memphis.edu/7117/pdf_notes/chapter01.pdf .

Also, I still don't understand how the variational expression for the energy, you'll find it
also in the registration - ppt, fits to the equation for minimum potential energy on page 12
(middle) of the mentioned script.
I hope somebody has the time and patience to help me understanding these basic concepts.

Best regards,
Markus Weigert

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