How Does the Jacobian Correct Signal Intensity in MRI Imaging?

[pamparana]

Hello everyone,

I have a small question about Jacobian and volume changes. So, I have a signal model from an imaging system where the signal intensities are preserved (it's an EPI MRI imaging system). So, basically for volume elements or voxels that are smaller than actually intended, the signal would look brighter. For the voxels which are larger than intended, the signal would look lighter because of the distribution of the signal. The signal is distorted normally due to some magnetic field inhomogenieties in the imaging field.

Now, the paper goes on to say that the corrected signal is equal to the distorted signal multiplied by the jacobian of the transformation. I do not understand how multiplying by the Jacobian would result in this signal intensity correction?

Does anyone have any idea about this?

Many thanks,

Luc

 

[slider142]

The Jacobian (the norm of the Jacobian matrix) is the change in volume between the old coordinate system and the new coordinate system.
In simple terms, consider the coordinate transformation from standard Cartesian coordinates whose Jacobian matrix is
$$\left(\begin{array}{ccc}2 & 0 & 0\\0 & 2 & 0\\0 & 0 & 2\end{array}\right)$$
The Jacobian is then 8. Note that in the new coordinate system, the basis vectors are twice as long as the basis vectors in the old coordinate system, so the parallelepiped spanned by the basis vectors has a volume of 1 in the new coordinate system, but a volume of 8 considered against the old coordinate system.
Generalizing this to calculus of nonlinear transformations, when we integrate volumes, we use the Jacobian as a locally linear volume change (called the volume element) in order to correct for changes of coordinates.

 

[pamparana]

Hi slider142,

Thanks for your reply and this clears a lot of doubt. However, I still am struggling to understand why the Jacobian works.

Actually I found an explanation in this presentation:
http://www-astro.physics.ox.ac.uk/~sr/lectures/multiples/Lecture5reallynew.pdf
However, the diagram with the 2D Jacobian confused me a bit.

So, in the example that you gave, each basis vector is scaled by a factor of 2. So, basically the partial derivatives represent the rate at which the function is changing wrt to a function variable in the given direction. Is that correct? This rate of change represents the instantaneous expansion or contraction of the volume at that given location. Have I understood this correctly?

Thanks,
Luc

 

[pamparana]

Sorry. Please remove. Could not figure Tex out...