[ITK] On the deformable model

[Nicolás Barnafi]

Hi everyone,

I have been trying to understand exactly what is happening in the
DeformableRegistration1.h file without much success. The problem is:
according to the software guide, the problem being solved comes from the
variational problem given by

min D[image1, image2; u] + S[u]

where D is just the SSD (or the L2 norm of (Im1 - Im2 o phi), with phi the
unknown displacement field) and S is a linear elastic potential. From here
you get  the euler lagrange equations (asuming some unspecified boundary
ocndition) and solve it using some semi implicit newton-raphson scheme.
This is where it starts getting blurry, because the ITK ppt on deformable
registration first shows an optical flow formulation, which would mean that
the SSD metric isn't really what is being used, and also if I dig deeper in
the code, I find actually a Crank-Nicolson scheme being used, which really
implies some kind of temporality that really does not exist in the
variational formulation. The only hint I have found was in Modersitzki's
book where a fixed point scheme is artificially stabilized:

A(u[k+1]) = f_u[k]

=> u[k+1] + t A(u[k+1]) = t f_u[k] + u[k].

I would want to know what is exactly happening in that example to be able
to validate an example I implemented in python. Thanks for your time.

Best regards



-- 
Nicolás Alejandro Barnafi Wittwer
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