[Insight-developers] Inverse transforms

[Lydia Ng]

Hi Stephen,

I agree with you it would be nice to address the inconsistencies.

"Inverse mapping" is important in resampling because it avoids the
problem of overlap and holes. At each iteration of image-to-image
registration you are implicitly computing the resampled image (with the
current parameters) - so wouldn't the same issues of overlap occur here
as well?

Just to clarify, currently with the "inverse mapping" scenario, the
equation for the mean squares metric looks something like this:
Sum{over all p in the fixed image region} ( F(p) - M(T(p)) )^2

So for each p, we compute mapped position T(p) using the transform and
then interpolate the moving image value at T(p).

What would be the equivalent equation and computation steps with
"forward mapping"?

The second issue is what happens when the transform is not invertible?
For example, BSplineDeformableTransform is useful for representing a
warp but there is no inverse. In this case, we can't easily flip-flop
from inverse to forward mapping or vice versa.

- Lydia


> -----Original Message-----
> From: Stephen R. Aylward [mailto:aylward at unc . edu]
> Sent: Friday, October 31, 2003 11:42 AM
> To: Insight-developers (E-mail)
> Subject: [Insight-developers] Inverse transforms
>=20
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> Hi,
>=20
> 1) ITK registration actually returns a transform from the fixed image
to
> the moving image
> 2) ITK resample image filter expects a transform from the fixed image
to
> the moving image
> 3) No model-to-image registration process is easily implemented using
a
> fixed-to-moving image transform - the model drives where in the fixed
> image the fit should be calculated.   So, all spatial-object-to-image
> transforms, the landmark-landmark registration stuff we are doing, and
> the ICP method we are implementing apply the transform to the moving
> model/points to quantify their match with the fixed image/points.
> Therefore we have to pass the Inverse transform to the resample image
> filter for them.
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> We have the landmark initialized MI registration app that must
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> 1) register using landmarks
> 2) resample the image using the inverse of that transform
> 3) invert the transform to initialize the MI registration process
> 4) resmaple the image using the non-inverse of that transform
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> This is really inconsistent.   Can be documented by saying most
> image-image transforms are fixed to moving transforms and all spatial
> object transforms are moving to fixed.   I say most image-image
> transforms since we are writing one that internally calculates
features
> and landmarks from the moving image to drive the registration with the
> fixed image.
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> I consider this a huge flaw with no easy answer - I understand why the
> resampling is driven by the fixed image.   However, couldn't the
> resampling take the inverse and all of the rest be done in the correct
> way: tranaform maps moving to fixed?
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> Stephen
>=20
> --
> =
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
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=3D=3D=3D=3D=3D=3D=3D=3D=3D
> 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
>=20
>=20
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