# DCT Based modelling

I have been reading this paper on MRI and I only do have novice level signal processing insight. I could not understand the following model:

The displacement field (i.e. d) is modelled as a linear combination of basis warps i.e.,

$$\mathbf d=[\mathbf B_1 \mathbf B_2 ........ \mathbf B_{m_x m_y m_z}] \mathbf b$$

Where each vector $$\mathbf B_i$$ is an unravelled version of one basis function from a truncated 3D discrete cosine transform and $$m_x, m_y, m_z$$ are the order of transform in the $$x-, y-, z-$$ directions, respectively.

• Hi! Yeah, so this is a bit of a non-trivial text, and I don't think I've encountered the term "basis warp" so far; I'm afraid without you asking a specific question and without a lot of context, we have nothing to answer :( – Marcus Müller Feb 8 '19 at 18:14
• The displacement field given here is a function that gives the geometric distortion. In MRI, one encodes each voxel with a frequency and phase, then you apply spectral analysis in order to recover the voxel-wise magnitude. When you have eddy-current or susceptibility inhomogenity, phases and frequencies deflect and therefore reconstruction is erroneous. So maybe "basis warp" term is related with this warp of phase and frequency of individual voxels – starhd Feb 8 '19 at 18:17
• @MarcusMüller You are right. So let me ask this one: what does each column of B represent? It would be great if explained in a more intuitive way than this paper :) – starhd Feb 8 '19 at 19:15