Confused with the three directions

[ShannonZ]

I'm so confused with the coordinates used in the code and the physical one.
Why did the code set Z-Axis with label 'X'?
According to the paper, this tool aims to design gradient coil for Halbach MRI system, which has a transverse static magnetic field. So I think all the three gradient coils should be along the Z axis.

[pfuchs]

The labels are to conform with the convention of the publication describing the algorithm, where the z-direction is always oriented along the background magnetic field component (B_0), which is usually longitudinal to the scanner for normal MRI scanners, but transverse in the case of halbach magnet configuration.

The gradients all generate a linear field component along the B0 direction, but the linearity is not all in the z-direction, since that would mean you can only encode in the z-direction. So perhaps to clarify, the axes are chosen according to the applied magnetic field, which is directed along the z-direction and the others should follow from a right-handed cartesian orientation. The gradients are then denoted by the direction in which the linear change in field is achieved, although the field component that changes is always the z-directed component.

[ShannonZ]

@pfuchs Would you pls give me the DOI of the publication describing the algorithm? I've googled and found one: https://minerva-access.unimelb.edu.au/handle/11343/281515
But it seems that the code was inconsistanist with this article.
For example,
the gradient shape used in this repo is different with Equation (2) in the article.
In this repo, J(phi,z) of direction z use the same form of that of the other two directions. But in the article, J(phi,z) of z is not the same as that of x or y.
……

[ShannonZ]

the parameter b in halbachXgradient() was set to be a*0.001. Why use such a small value? Is the simulated results useful with the small b?
In my opinion, b should be large enough to keep the target field big enough for use.

[pfuchs]

the gradient shape used in this repo is different with Equation (2) in the article.

I don't completely understand this, if you look at the code in the gradientCalculationV2_1.py, lines 62 is equation 2(right hand) and lines 100 & 143 correspond to equation 2(left hand), the longitudinal and transverse gradients respectively.

So they are different, just like in the paper? Only here x is given as the longitudinal, and z and y as transverse.

the parameter b in halbachXgradient() was set to be a*0.001. Why use such a small value?

for the target field method to work the domain needs to be large enough to give a fine enough frequency space sampling (1/FoV), but the actual physical dimensions of course don't allow for meter-long wire loops, so b is chosen so much smaller to lead to wire loops close to the desired dimensionality of the coil. You can play with this yourself, you'll find that the actual coil geometry is determined by the linear region, but if the field of view is not large enough you'll run into trouble, and the linear region is not exactly the size of the coil (of course), since you'll need a larger coil since at the edges the approximation will fail.

[ShannonZ]

We've developed a new method to design gradient coil. If I use the code in this repo to generate some results for comparision, is the article titled "Gradient Coil Design and Realization for a Halbach-Based MRI System" the right one I should cite?

There are some small differences between the code and the article:

Where did these factors or coefficents (1/k，2/pi) used in IFFT come from? The corresponding formulas in the article do not have these factors.

[pfuchs]

The equation in the code is for the stream function, which is derived from the current density, not the current density itself, and those terms come from the relationship between the two. Perhaps @LUMC-LowFieldMRI can elaborate, since I did not write this code, also, you can find more info on the stream-functions and how they are used here (http://resolver.tudelft.nl/uuid:bb09923a-e4b8-45b9-969e-c614e53cdc66, I just realise this file is under embargo, but you can reach out to the author for a copy perhaps).

Also, you could take a look at a previous version of gradientCalculation.py in commit: bbd514e (https://github.com/LUMC-LowFieldMRI/GradientDesignTool/blob/f591c02c9cc53736e73f53aeeb7e8e081e72a642/gradientCalculation.py)
which separates these steps somewhat, and has some explanation on the current density -> stream-function step.