Want to learn Thin Plate Spline

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goodjeff

Posted: Fri Nov 03, 2006 10:41 am

Guest

Dear all:

I used a "smoothing thin plate spline(tps)" for image registration,

and it usually helps to yield good results. But I am not clear about
how on earth the close-form solution is obtained from minimizing its
bending energy E(f), which is:
E(f) = \sum | y - f(x) |^2 + \lambda \int (f_xx^2 + f_xy^2 +
f_yy^2) dxdy

And if I add additional iterms to E(f), how can I calculate f again?
I am also not wondering if there is any guide/principle for deciding
the coefficient \lambda, except for trying by experiments.

Many paper refer to Wahba's book: Spline Models for Observational Data
1990. But it is difficult for me to get one. Can anyone point out
another comprehensive reference? A tutorial is most preferred.

Thanks a lot!
Jeff

student, The Chinese University of Hong Kong

Olumide

Posted: Fri Nov 03, 2006 12:50 pm

Guest

goodjeff wrote:

Quote:

I used a "smoothing thin plate spline(tps)" for image registration,

and it usually helps to yield good results. But I am not clear about
how on earth the close-form solution is obtained from minimizing its
bending energy E(f)...

Refer to this presentation:
http://www-cse.ucsd.edu/classes/fa01/cse291/hhyu-presentation.pdf

This is some sort of summary of another paper:
http://tinyurl.com/y48qfq

Quote:

And if I add additional iterms to E(f), how can I calculate f again?
I am also not wondering if there is any guide/principle for deciding
the coefficient \lambda, except for trying by experiments.

Many paper refer to Wahba's book: Spline Models for Observational Data
1990. But it is difficult for me to get one.

I hate it when they do that Smile

. I've seen the book. Its extreemly
complicated. You'll need a good degree is mathematical statistics in
order to understand any part of it.

goodjeff

Posted: Fri Nov 03, 2006 4:53 pm

Guest

Dear Olumide:

The presentation from UCSD is clear as crystal! Thank you very
much.
So if I want to add other regularization iterms, I have to solve
the new variational problem.

Or, if I can decide the form of transform f, like f(x) = Ax +
B(x)C, then I can replace the f in the energy E, and solve ( dE/dA = 0,
dE/dB = 0, dE/dC = 0) to find f, right?

And it seems that \lambda can only be selected by
cross-validation.

Thanks
Jeff

Olumide wrote:

Quote:

goodjeff wrote:
I used a "smoothing thin plate spline(tps)" for image registration,

and it usually helps to yield good results. But I am not clear about
how on earth the close-form solution is obtained from minimizing its
bending energy E(f)...

Refer to this presentation:
http://www-cse.ucsd.edu/classes/fa01/cse291/hhyu-presentation.pdf

This is some sort of summary of another paper:
http://tinyurl.com/y48qfq

And if I add additional iterms to E(f), how can I calculate f again?
I am also not wondering if there is any guide/principle for deciding
the coefficient \lambda, except for trying by experiments.

Many paper refer to Wahba's book: Spline Models for Observational Data
1990. But it is difficult for me to get one.

I hate it when they do that Smile

. I've seen the book. Its extreemly
complicated. You'll need a good degree is mathematical statistics in
order to understand any part of it.

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