[Insight-users] Rigid + Scale transform

[Vicente Grau]

Luis, thanks very much for the information, and thanks to Daniel for the 
useful extension. I have tried it and it works fine for me.
Now I have a question about the way the order of the rotation and 
scaling in the ScaleSkewVersor3DTransform. The way the transformation 
matrix is calculated in the code is:

const TScalarType vx = versor.GetX();
  const TScalarType vy = versor.GetY();
  const TScalarType vz = versor.GetZ();
  const TScalarType vw = versor.GetW();
     
  const TScalarType xx = vx * vx;
  const TScalarType yy = vy * vy;
  const TScalarType zz = vz * vz;
  const TScalarType xy = vx * vy;
  const TScalarType xz = vx * vz;
  const TScalarType xw = vx * vw;
  const TScalarType yz = vy * vz;
  const TScalarType yw = vy * vw;
  const TScalarType zw = vz * vw;

  MatrixType newMatrix;
  newMatrix[0][0] = m_Scale[0] - 2.0 * ( yy + zz );
  newMatrix[1][1] = m_Scale[1] - 2.0 * ( xx + zz );
  newMatrix[2][2] = m_Scale[2] - 2.0 * ( xx + yy );
  newMatrix[0][1] = 2.0 * ( xy - zw );
  newMatrix[0][2] = 2.0 * ( xz + yw );
  newMatrix[1][0] = 2.0 * ( xy + zw );
  newMatrix[1][2] = 2.0 * ( yz - xw );
  newMatrix[2][0] = 2.0 * ( xz - yw );
  newMatrix[2][1] = 2.0 * ( yz + xw );

where I have removed the skew part. I can see this is the expression for 
transforming a quaternion into a rotation matrix, with the addition of 
the scale parameters but I don't know how the composition of the two is 
done here. Actually what I would like to know is if this is equivalent 
to scaling + rotation, or rotation + scaling. But if you could point me 
to a document where these expressions are explained in detail, it would 
be really helpful. Thanks!

Vicente


Luis Ibanez wrote:

>
> Hi Vicente,
>
> Daniel Blezek modified the VersorRigidTransformOptimizer in order to
> make possible to use it for Transforms different from the VersorRigid3D
> Transform.
>
> As long as you arrange your extra transforms parameters to be at the
> end of the parameters array when you call GetParameters() and
> SetParameters(), then this optimizers should work for your purpose.
>
>
>
> Daniel: It took me a while to understand the reason why you extended
>         this optimizer. My apologies for being slow.
>
>
>
>   Regards,
>
>
>      Luis
>
>
>
> ----------------------
> Vicente Grau wrote:
>
>> I am trying to do a 3D registration combining a rigid transform with 
>> independent scaling in the three axes. The closest I've found in the 
>> library is the itkScaleSkewVersor3DTransform, so my idea was to use 
>> this as a base and remove the skew part, either by writing a new 
>> transform class or by introducing a factor with SetScale() that 
>> effectively eliminates this effect. I also thought about modifying 
>> QuaternionRigidTransform to allow for non-unit quaternions. What do 
>> you think is the best way to do this?
>> Furthermore, I guess I will need a special optimizer due to the 
>> parameters forming a non-vector space. Is there an optimizer that 
>> works with the ScaleSkewVersor3DTransform? Or, if not, how should I 
>> modify the existing optimizers (VersorRigid3DTransformOptimizer, I 
>> guess)?
>> Any help will be much appreciated!
>>
>> Vicente Grau
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>>
>>