Hi Jesper,

That was helpful. I was thinking more along the implementation lines for this. So in order to get the Jacobian determinant from the displacement field is there a closed form equation that I could use ? I have a simple idea in mind to simply subtract the neighboring voxel coordinates in order to get the partial derivative for the Jacobian. Thoughts ?

If there is not a closed form solution for this, then how do we implement it?

Thanks! 
________________________________________
From: FSL - FMRIB's Software Library [[log in to unmask]] on behalf of Jesper Andersson [[log in to unmask]]
Sent: Wednesday, November 29, 2017 5:13 AM
To: [log in to unmask]
Subject: Re: [FSL] Eddy Image Space to Model Space

Hi again,

>
> Referring to your paper "An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging" I was having some difficulty determining what is meant by the "local" Jacobian of the transform. Does local refer to the neighboring voxels or a larger window? If yes, what is the size of this window?

the general for non-linear warps the Jacobian is the matrix





where d_x is the warps in the x-direction, d_y is the warps in the y-direction etc. This is defined for every voxel-centre and you can think of it as a local affine transform for the space of that voxel. It contains information about rotation, scaling and shear of that voxel as a consequence of those warps. The determinant of this matrix tells you how the volume of that voxel has changed. In the case of eddy the warps are all in one direction (the PE-direction) so the matrix simplifies. For example for the case of PE in the y-direction the top row becomes 1 0 0 and the bottom row 0 0 1 and the determinant is given by \frac{\partial d_y}{\partial y}.

Jesper


>
> Thanks!