Hi again Colin,
>
> 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 ?
the finite difference approximation is not brilliant for this, in that the two values you subtract are defined in voxel centres, which means that your derivative will be defined on the junction between the voxels. And that is not where you ideally want it.
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> If there is not a closed form solution for this, then how do we implement it?
Usually there are reasonably simple ways of doing that. For example in eddy the EC field (which is a scaled displacement field in the PE-direction) is represented as a low order polynomial (linear, quadratic or cubic) so there it is trivial to differentiate it analytically. The susceptibility field is represented by the coefficients of a set of cubic B-splines, i.e. you get the field by multiplying each coefficient with the pertinent spline (they are regularly spaced on a 3D grid). So to get a derivative field you simply take your 3D spline and differentiate it in one direction (the PE direction), which is again analytic since these splines are polynomials. You then create the derivative field by using the same coefficients, but now multiplying them with these “derivative splines”.
For the case where I have a field that is just represented by the voxel values (i.e. a measured field) I calculate the interpolating spline coefficients and use the same trick with the “derivative splines”.
Jesper
>
> 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?
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> the general for non-linear warps the Jacobian is the matrix
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> 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}.
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> Jesper
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>
>>
>> Thanks!
