[Insight-developers] Centered Transforms

[Stephen R. Aylward]

Hi,

Yup - the terms center, offset, and translation aren't my favorite 
either (I didn't pick them, I am following the ITK standard)...it was 
decided quite some time ago to call "translation" the component of the 
transform that defines the shift to be applied after rotation about a 
center of rotation.   The offset vector defines the amount of shift to 
apply after a rotation when a center of rotation has not used.

Center (of rotation) must be maintained since it is not specifically 
being optimized during registration.  That is, it must be stored since 
it isn't being passed.

Furthermore, since center isn't being optimized, rotation can only 
decoupled from shifting during registration if the optimizers update 
translation instead of offset during registration.   So, translation is 
a parameter of the optimizer instead of offset.

So, the optimizers drove the addition of the concepts of center of 
rotation and translation in the transforms.

Having a center of rotation greatly simplifies certain registration 
optimizations compared to image alignment when rotation were limited to 
being about the origin.  ITK began years ago by having registration 
optimization temporarily move the origin of an image to the center of 
the image during registration, and then move the origin back after 
registration, but that really is a hack - the origin really shouldn't 
move during registration...so, we added the concept of a center of 
rotation...

I hope this clarifies things a bit.   I don't like classes that have 
redundant/linked variables that must be kept in sync, but there is no 
way that Bill would let us remove the Get/SetOffset functions :)   And, 
actually, I don't think we should either... :)

So this is the rock and the hard place that we are between...   With 
that in mind...any ideas/suggestions?   I've been working with the 
transforms quite a bit - switching between them within a single 
application and composing them in chains while using center of rotation. 
   This has revealed some inconsistencies in how offset and translation 
are handled (not surprisingly) as well as inconsistencies in their APIs. 
   This lead to the development of the MatrixOffsetTransform base class 
that we are adding to make sure all of the affine, similarity, rigid, 
translation, and rotation transforms handle these concepts consistently. 
   I just want to make sure we all agree on which is the lessor of the 
evils :)

Sorry for the long email...just looking for someone I can share the 
blame with when we ultimately pick a standard... :)   Anyone.... 
Anyone... :)

Thanks,
Stephen


Miller, James V (Research) wrote:
> I don't think I am going to be of much help since I do not 
> understand the difference between an "offset" and a "translation".
> 
> Skimming through the headers, it looks like transforms like the 
> affine transform are defined by equations
> 
> y = Ax + b
> 
> where A is the "matrix" stored in the transform and "b" is the offset.
> 
> To me, it looks like the only things that need to be STORED are the
> matrix and offset.  With the "center" and "translation" being computations
> based on A and b.  Conversely, you could argue that one would want to
> set the center and translation. In doing so, the matrix A and offset b
> would have to be updated. However, I would favor not storing the center and
> translation if there are truly secondary parameters completely defined by
> A and b.
> 
> 
> 
> 
> 
> -----Original Message-----
> From: insight-developers-bounces at itk.org
> [mailto:insight-developers-bounces at itk.org]On Behalf Of Stephen R.
> Aylward
> Sent: Monday, March 07, 2005 1:15 PM
> To: Insight-developers (E-mail)
> Subject: [Insight-developers] Centered Transforms
> 
> 
> 
> When someone updates the center of rotation or matrix for a transform, 
> either the translation or the offset must be implicitly updated for 
> consistency.   This implicit update is necessary since translation and 
> offset are related to one another via the transform matrix and the 
> center of rotation.
> 
> My vote is that the offset should be implicitly updated when the center, 
> matrix, or translation is set by the user; and that translation should 
> only be implicitly updated when the offset is set by the user.
> 
> That is, we assume that most people will set the translation component 
> and not the offset component - this assumption is consistent with the 
> behaviour of our optimizers which use translation as a parameter during 
> optimization, not offset.
> 
> As a result, the following code will be valid:
> 
> Using an Euler3DTransform called transform...
> 
> CenterType center;
> center.Fill(10);
> 
> TranslationType translation;
> translation.Fill(0);
> 
> transform.SetRotation( 1, 2, 3 );
> transform.SetTranslation( translation );
> transform.SetCenter( center );
> 
> OutputPointType transformedPoint = transform.TransformPoint( center );
> if( transformedPoint[0] != center[0] )
>    {
>    itkWarningMacro(<< "Oops.  Transform of center is not the center");
>    }
> 
> This is consistent with most of the current transforms.  Can we accept 
> this as the general "rule" for itk?
> 
> The other options are to (2) update the offset implicitly when center or 
> matrix are updated, or (3) require the user to explicitly call 
> ComputeTranslation, ComputeMatrix, ComputeOffset, and 
> ComputeMatrixParameters (which, for example, updates the versor in a 
> versorTransform when the user sets the matrix directly).  Option 3 
> requires significant changes throughout ITK.   Option 2 is okay, but 
> will require a few changes - probably about the same number as 
> implicitly updating transform - however, I see it as less consistent 
> because of how optimizers update transforms....
> 
> Any opinions / comments?
> 
> Thanks,
> Stephen
> 

-- 
===========================================================
Dr. Stephen R. Aylward
Associate Professor of Radiology
Adjunct Associate Professor of Computer Science and Surgery
http://caddlab.rad.unc.edu
aylward at unc.edu
(919) 966-9695