I would like to adress this question in a simulation context:

let say we have a 3D cartesian kspace acquisition, and we set the subject motion during the acquisition.

how to parameterize the rigid head motion ?

In the MRI field we traditionally use the matrix representation for the rigid motion transform (dor instance for the coregistration task). Reading a bit from other field, it seems that the dual quaternion representation could have some great advantage: it gives us more natural parameters to generate different motion. Using the Chales' theorem, we can represent any rigid motion by a unique motion along a screw, which is then define by the rotation angle (around the screw axis) , the translation distance (paralele to the screw axis), and the screw axis line definition.

how to quantify its severity ?

Well the ground truth, is simply given by the displacement field induced in each voxel (for a given rigid transformation). But this is quite costly to compute and it would be interesting to derive metrics directly from the dual quaternion representation to quantify this severity.

The question is then how to build this metric ... ?

We know it should include the rotation angle and the translation distance, but also the screw axis position.

How to quantify the motion artefact severity induce in the image ?

This is the more difficult part: given we know the motion and the kspace acquisition sheme, the artefact induce will strongly depend on where in the kspace specific motion will occur.

The exact same motion, will induce very different change in the image if it occur at the begining of the kspace (high frequency stride ) or just in the middel. (contrast and object shape perturbation)

so let say we solve the previous question and have at each time point a motion severity (related to total displacement induced in the object) what are the weights to take into acount the kspace position ?