[Insight-users] changing Center of Rotation for MatrixOffsetTransformBase

[Michael Schildt]

Hello Karthik Krishnan!

Your advice help me a lot. Creating the new transform seems to work now. 
It's easier than i thought :)
But i have a question reguarding the fact of R = R'. While it is 
intuitively clear for me that this needs to be the case, i wonder how to 
proove it with mathematics.
Is it something like:
- I have to points p1 and p2, a rotation R and translation T
- v is p2-p1
- (v is independant of absolute position of p1 and p2 but the relative 
postion of p1 and p2 needs to be constant to keep the size and direction 
of v)
- p1' = R*p1+T, p2' = R*p2+T, v' = (p2'-p1') = R*p1-R*p2 = R*v
- so i ask, is there a rotation matrix (with all it constraints) R' != R 
with Rv = R'v for all v in Vectorspace
- O = Rv-R'v = (R-R')v   with O beeing the origin of the coordinate system
- this can only be true for arbitrary v if (R-R') is a zero matrix, 
which means every element rij in R must be equal r'ij in R'
Is this correct?

And is the conclusion correct that a different center of rotation does 
not affect the offset and the matrix just the center itself and 
translation differ?
- F is offset
- F' is new offset
- F = T+C-RC
- F' = T'+C'-RC' with T' = T+(C-C')-R(C-C')
      = T+C-R(C-C')-RC'
      = T+C-RC+RC'-RC'
      = T+C-RC
  F' = F

So more general for rigid transform - for a given matrix and offset a 
infinitive number of centers and corresponding translations can be 
found. But for changing the rigid transform you need a certain 
center/translation pair to update the offset after changing the rotation 
matrix or the translaton vector.

Reguards,
    Michael Schildt


Karthik Krishnan schrieb:
> Michael:
>
> Its simple. Derive it yourself :-)
>
> For a centered transform (with parameters : C = Center, T= 
> Translation. R= Rotation matrix.), given a point 'p'
>
> TransformedPoint = R (p - C)  + C  + T
>
> Now if you want your own Center C', let's find the parameters R' and T'
>
> R (p - C)  + C  + T  = R' (p - C')  + C'  + T'
>
> The result is
>   R' = R
>   T' = T + (C-C') - R (C-C')
>
> ie. You will contruct a new transform with the same rotation matrix, 
> your center and the new translation is given by the above equation.
>
> Hope this helps.
> --
> karthik
>
> On 11/16/07, *Michael Schildt* <michael.schildt at ifn-magdeburg.de 
> <mailto:michael.schildt at ifn-magdeburg.de>> wrote:
>
>     Hello!
>
>     I got a rigid transform of an itk registration with a certain
>     center of
>     rotation. But for further processing i need to use an other center. Is
>     it possible to recompute the matrix of a rigid/affine transform for a
>     different center? Because, just setting a new center will result in a
>     different transform.
>
>     Here are some numbers:
>
>     TransformMatrix = -0.0183916 0.938708 -0.344223 -0.0411616 -0.3447
>     -0.93781 -0.998983 -0.00307907 0.0449783
>     Offset = 5.64609 -44.9436 -0.107986
>     CenterOfRotation = 1.09089 6.36365 -2.33333
>
>     I need to have a new center at [127.5, 127.5, 127.5].
>
>     Any hints are welcome :)
>
>
>     Best reguards,
>         Michael Schildt
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