[Insight-users] Vector Gradients and transform jacobians

[Tom Vercauteren]

Hi Luke,

As far as I remember, there is no direct filter to compute the
"gradient" of a vector image. You may however try the following
function:
http://www.itk.org/Doxygen/html/classitk_1_1VectorCentralDifferenceImageFunction.html
It computes the Jacobian of a vector image at a given index.

You may also check the following filter:
http://www.itk.org/Doxygen/html/classitk_1_1DisplacementFieldJacobianDeterminantFilter.html
It computes an image of the Jacobian determinant of a given
displacement field. It should be easy to modify to get the full
Jacobian if you need it.

Also note that if you use VectorCentralDifferenceImageFunction on
displacement field you need to add the identity matrix to get the
Jacobian of the corresponding transformation. See e.g. the comments
in:
http://www.itk.org/Doxygen/html/classitk_1_1DeformationFieldJacobianDeterminantFilter.html
http://www.itk.org/Doxygen/html/classitk_1_1DisplacementFieldJacobianDeterminantFilter.html

Tom

On Thu, Nov 20, 2008 at 4:01 PM, Luke Bloy <luke.bloy at gmail.com> wrote:
> Hi everyone,
>
> I have 2 questions.
>
> 1) Id like to be able to compute the gradient of a vector image. The issue
> is that the gradient filters I've looked at all use convolution with a
> derivative operator, which I'm not positive how to modify to work with
> vector images. So my question is... Is there a gradient filter already
> written for vector images, if not is there a finite difference gradient
> filter for computing gradients?
>
> 2) As part of a registration problem I'd like to incorporate the smoothness
> of my deformation field into cost function to be minimized. The way I'd like
> to do this is to use the Jacobian of transformation in the metric. The
> problem is that the GetJacobian method returns the Jacobian of the
> transformation with respect to the parameters, not with respect to the
> image/spatial coordinates. I'm curious if anyone has done this or might
> suggest a good way for me to implement it.
>
> Thanks,
> Luke
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