Hello Morphometricians -
I'm having difficulty writing out the thin-plate spline functions fx,
fy, and fz which the displacement of the landmarks in the x, y, and z
directions (following
http://www-asds.doc.ic.ac.uk/~giso/pubs/leedsok/node5.html )
(x, y, z) -> ( fx(x,y,z), fy(x,y,z), fz(x,y,z) )
What I'm thinking of at the moment is that: in order to compute fx, I
drop x (reducing the problem to 2D) so that: (x,y,z) -> (y,z), and solve
the 2D TPS equation. Likewise in order to compute fy and fz, I drop y
and z respectively, so that:
(x,y,z) -> (x,z)
(x,y,z) -> (x,y)
... and proceed as above to obtain the 3D above transformation:
(x, y, z) -> ( fx(x,y,z), fy(x,y,z), fz(x,y,z) )
However, there'll be sets of simultaneous equations to solve - which is
pretty odd because we'll end up with 3 times the number of weights as
expected (one for each dimension). Is this how to to go about the
problem? Anyone care to shed light please?
Many thanks,
- Olumide
PS: (Related questions)
(1) Bookstein
(http://www.mail-archive.com/[email protected]/msg00015.html)
advocates the use of |r| instead of the familiar basis function
r*r*log(r) for the 3D case/problem. Why?
(2) some researchers use variants(?) of the 2D TPS equation some of
which include:
1. r*r*log(r)
2. r*log(r)
3. r*log(r*r) - I think
Again, why?
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