[Insight-users] Vector Gradients and transform jacobians

[Luke Bloy]

Tom,

Thanks that is exactly what I was looking for.

-Luke

Tom Vercauteren wrote:
> 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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