[Insight-users] to combine two affine transformations

[Luis Ibanez]

Hi Henry,


Given two affine transforms:

      Affine Transform1 = AF1 = ( M1, O1 )
      Affine Transform2 = AF2 = ( M2, O2 )


Applying the transform AF1 to a point P implies doing:

               P' = M1 * P + O1


Applying the transforms AF1 and AF2 in a sequence to
a point P imply doing the following:


               M2 * ( M1 * P + O1 ) + O2

which is equivalent to

       ( M2 * M1 ) * P + (M2 * O1) + O2


So, the composition of AF1 and AF2 is AF3


      AF3 = ( M3, O3 )

where

      M3 = M2 * M1

      O3 = M2 * O1 + O2



How are you applying the transformation AF1, AF2 to your image ?
by using a Resample image filter  ?

Notice that the ResampleImageFilter maps points from the fixed
image to the moving (input) image. This tend to create confusion
in some users.



One thing that you may be doing wrong is the order in
which you apply the transforms. If you use the wrong
order you end up with a translation of

       O3 = M1 * O2 + O1

instead of the one above.





    Regards,



        Luis



-------------
Henry wrote:
> Hi all,
> I am confused by the results of combination of two transformations. I
> need to apply affine transform1(Matrix1, Offset1) first and then
> affine transform2 (Matrix2, Offset2).
> 
> Should the result transform3(Matrix3, Offset3) be like
> 
> Matrix3 = Matrix2*Matrix1
> Matrix3 = Matrix2*Offset1 + Offset2 ?
> 
> I apply the transform3 and compare the image with apply two separate
> (transform1 and then transform2),  and found the rotations are the
> same but the offset/translation of the two results are different.
> Any idea why this happens?
> 
> Thanks!!
> Henry
> 
> ~~~~~~~~~~~~~~~
> //my program looks like this:
> 
>   typedef itk::AffineTransform<double,3> AffineTransformType;
>   AffineTransformType::Pointer affine = AffineTransformType::New();
>   itk::TransformFileReader::Pointer transformreader1;
>   itk::TransformFileReader::Pointer transformreader2;
>   transformreader1 = itk::TransformFileReader::New();
>   transformreader2 = itk::TransformFileReader::New();
>   transformreader1->SetFileName( argv[1] );
>   transformreader2->SetFileName( argv[2] );
> 
>     transformreader1->Update();
>     transformreader2->Update();
> 
> //I read in two .tfm files and do the following:
> 
> 
> 	transform->SetMatrix( affine_read2->GetMatrix()*affine_read1->GetMatrix());
> 	transform->SetOffset(affine_read2->GetMatrix()*affine_read1->GetOffset()
> +affine_read2->GetOffset());
> 
> 
> //and then write the final results
> 
>   itk::TransformFileWriter::Pointer writer;
>   writer = itk::TransformFileWriter::New();
>   writer->SetInput(transform);
>   writer->SetFileName( argv[3] );
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