[Insight-users] Affine transform - decomposition ?

[Miller, James V (Research)]

If I recall correctly, the Graphic Gems series of books (I, II, III, IV, ...) had a number of "sections" on transformation decomposition. Graphics Gems II has two nice sections on pages 320-333
 
Note, itk::AffineTransform is stored as a matrix and a vector (offset).  The vector is the translation component. So you can just call GetOffset() to get the 3 translation parameters.  That leaves 9 parameters to decompose (rotation, scale, skew). Reflections will just be negative scales so we can ignore that for now.  
 
The first approach (Graphics Gems II, pg 320) essentially orthogonalizes the matrix to get the rotation components and whatever is left over from the orthogonalization forms the scaling and skew.
 
The second approach (Graphics Gems II, pg 324) converts the decomposition problem to an eigenvector problem. Let the matrix part of the affine transform be M and look at the eigenvector problem
 
        M*v = a*v
 
where v is an eigenvector of M and a is the associated eigenvalue. The eigenvectors and eigenvalues of M tell us about the axis of rotation, the amount of scaling etc.  The presentation is Graphics Gems II, is geared for determining the parameters of rotation, scaling, skew, etc. when the transformation in question is ONE of rotation, scaling, skew, etc.  In other words, if you know the matrix is just a rotation, this is how to extract the rotation parameters; if you know the transformation is just a scaling, this is how to extract the scaling parameters, etc.  I don't think this section addresses the problem of having a general affine transformation and decomposing it into scalings, rotations, etc....  However, many of the ideas in this section would still apply.  The eigenvectors of the matrix M are invariant to the transformation M (only changing the vectors by a scale factor).  Therefore, the eigenvalues are scale parameters.  However, they are the scale parameters along the associated eigenvectors and not the scalings along the cartesian axes.  But this is the general idea.
 
Jim
 
 

-----Original Message-----
From: insight-users-bounces at itk.org [mailto:insight-users-bounces at itk.org]On Behalf Of Roxana Racz
Sent: Monday, April 25, 2005 6:02 PM
To: insight-users at itk.org
Subject: [Insight-users] Affine transform - decomposition ?



Hello everyone,

I am working on a registration application.
I would like to display the **geometrically meaningful** components using the 12 parameters of the affine transform as returned by the registration routine. 

That is: from the 12 "raw" parameters returned by the registration algorithm employed, I need to be able to monitor the following quantities: 
1.translation(3), 
2.rotation(3), 
3.scaling(3), 
4.skew (?) and 
5.reflection(3?)

--- Does anyone know of a method based on itk? (it seems that vtk offers only the extraction of 1, 2, 3: GetPosition, GetOrientation, GetScale, all defined in vtkTransform.cxx).

--- Could anyone recommend a good reading on the underlying maths for this extractions? 

Thanks in advance,
 
Roxana 

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