[Insight-users] Degrees of freedom for affine transform?

[Luis Ibanez]

Hi Motes,

Yes the total number of Degrees of Freedom
for a 3D Affine Transform is  = 12.


The 12 terms come from:

              3 from the 3D Translation

and
          9 from the Matrix that combines
             Rotation, Scaling and Shearing.

Please note that the Rotation, Scaling and
Shearing parameters can't not be fully separated,
so, it is not quite possible to say that there are 3
rotation parameters, 3 scaling ones and 3 rotation
ones.

That said,.... it is known that you could take that
Matrix and use Polar Decomposition:

http://en.wikipedia.org/wiki/Polar_decomposition#Matrix_polar_decomposition
http://www.itk.org/pipermail/insight-users/2006-August/019025.html

in order to express it as the product of an Orthogonal
matrix with a residual matrix. The Orthogonal matrix
will contain the Rotational terms, while the residual
matrix will contain a mixture of the Scaling and Shearing
terms.


      Regards,


            Luis



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On Sun, Oct 25, 2009 at 3:14 PM, motes motes <mort.motes at gmail.com> wrote:
> On page 407 in the itkSoftwareGuide its says that an affine transform has:
>
> (N +1)*N
>
> parameters. So in 3D the total number of parameters (DOF) would be:
> (3+1)*3=12. Is it correct that these 12 parameters come from:
>
> 3 parameter from translation
> 3 parameter from scaling
> 3 parameter from rotation
> 3 parameter from shear
>
> which gives a total of 12 parameters?
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