# From Fourier (k space) to wavelet domain in MRI sensing

In compressed sensing MRI (cSENSE MRI) technology the idea seems to entail sampling from the Fourier domain (k space) in a way that, when transformed to the wavelet domain ("sparsification"), the sparsity is maximized.

The inverse recovery problem becomes exact provided that $\mu$ is small:

$$\mu \left( \mathcal F W^\top \right)=\max_{i,j}\vert \langle W_i, \mathcal F_j\rangle \vert$$

i.e. the dot products of the column of the Fourier transform and wavelet transform are minimal ("mutual incoherence").

The idea seems to stem from this paper by Candes, Romberg and Tao.

In the wavelet domain the coefficients include scale and translation, while in Fourier space the coefficients belong to different frequencies without temporal support.

I would like to confirm that there is indeed a triple step: First (partially) filling in k-space with Fourier coefficients; second, randomly sampling these coefficients; and third, transforming them to wavelet space in each MRI image acquisition (as opposed to being an assumption based on the nature of MRI images).

And if this is the case, how does the step from Fourier to wavelet takes place (a reference would be OK).

Or, contrarily, whether the signal is primarily analyzed as wavelets?

• I don't understand the first bit that involves compressed sensing. Are you asking if it is possible to relate the scale / time "domain" of the wavelet transform with the frequency domain of the Fourier Transform? So, say, given a natural frequency, which wavelet output contains it and vice versa (?).
– A_A
Jul 13 '18 at 8:36
• @A_A I am looking for two things: 1. Confirmation that the signal in c SENSE MRI is captured first as Fourier coefficients, and later transformed into wavelets. And two, find out a reference or explanation as to how the transformation from Fourier to wavelets is carried out. Jul 13 '18 at 11:48