vtkThinPlateSplineMeshWarp

[Tim Hutton]

<inserted stuff down below>

At 17:43 03/02/00 +0100, you wrote:
>At 20:19 02/02/00 -0500, David Gobbi a écrit:
>> > (sorry for the delay, I was in California)
>>
>>I'll be heading there is just a couple weeks -- too bad we missed each
>>other :)
>
>I see, SPIE Medical Imaging. Well, I wish I could go there, but my paper 
>was accepted by SPIE Electronic Imaging first. Too bad :(
>
> > You meant r.log(r) ?
>
>>No!  I meant r, as in r = sqrt(x^2 + y^2 + z^2), simply the distance
>>between two points.
>
>
> > U(r) = |r| : something wrong here I guess.
>
>>U(r) = r = |r| = fabs(r) = sqrt(x^2+y^2+z^2) are all the same thing.
>
>That's what I thought, when I wrote "something wrong here" that meant "I'm 
>a bit confused with the use of the distance instead of the r.log 
>formulation". Sorry for the confusion.
>
>>Well, the 'definitive' answer would be in  F.L. Bookstein _Morphometric
>>Tools for Landmark Data_  but someone else has it out of the library
>>right now.  And maybe each of us should work though the 3D energy
>>integral to prove whether 'our' version has the properties expected.
>>
>>In Bookstein's paper 'Shape and the Information in Medical Images:
>>A Decade of the Morphometric Synthesis,' Computer Vision and Image
>>Understanding 66:97-118, he clearly states at the bottom of page 105:
>>
>>      The formalism of the spline goes over to three-dimensional data
>>      almost unchanged -- the kernel is now |r|, the matrix Q now has
>>      four columns, and the integral in Eq. (4) has six terms instead
>>      of three ...
>>
>>I take this to mean that U(r) = r*r*log(r) only for 2D splines, and
>>that U(r) = |r| for 3D splines.
>
>Hum, that's not obvious for me. I guess he just wanted to say that the 
>distance was the Euclidean one ?
>
>OK, I've got 12 articles in front of me, let me see...
>
>All right, if you do not mind, I'll send you two papers :
>         "Surface interpolation with radial basis functions for medical 
>imaging" (Carr)
>         "Creating surfaces from scaterred data using radial basis 
>functions" (Schaback)
>
>Both are reasoning in R^d, and the first one is a 3D application.
>Could you please have a look at them ?
>I'm quite sure the TPS is really r.log(r). If there is still a doubt, let's 
>send an email to Mr Bookstein, I already did once.

Just been to a very nice talk by the man himself, I asked him about this
very question. Without any doubt, the RBF you need for 3d TPS is |r|. He
explained to me why this was so but I understood very little. :)

Hope this is useful information.

Tim.

---------------------------------------------------------------------------
Tim Hutton, Research Assistant            Email: T.Hutton at eastman.ucl.ac.uk
MINORI Project                          Eternal: T.Hutton at excite.co.uk
Dental and Medical Informatics     http://www.eastman.ucl.ac.uk/~dmi/MINORI
          
Eastman Dental Institute, UCL                    Tel: [+44] (0207) 915 2344
256 Gray's Inn Road, London WC1X 8LD, UK         Fax: [+44] (0207) 915 2303
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