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

Question

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?

Answer 1

The image $I$ is in spatial domain.

The sampled raw data, so-called kspace data $K$ stores the fourier coefficients of $I$: $$ K = FI $$ Here $F$ denotes the Discrete Fourier Transform matrix. $K \in spatial-frequency - domain$. The gradient waveform used in a MRI sequence controls the filling trajectory of k-space data. Comparing to a fully filled k-space, partially filling could be achieved by using a specified gradient waveform to skip some locations in k-space.

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;

The two steps did not exist. A tailored gradient waveform could lead to a direct randomly sampled/filled k-space. $$ y = K_{sparse}= PFI $$ Here $P$ is the sampling pattern/mask.

Immedialtly, we can reconstruct a image form k-space data, $$ I = F^HP^Hy $$ Here $H$ is the matrix inverse operator. The quality of the resultant image would be poor because the problem is highly ill-conditioned. Some regularization method can improve it in a large margin.

Wavelet transformed coefficients matrix of a MRI image is sparse. This is a useful a priori. Consequently, L1 regularization is widely used. Now the problem becomes: $$ \arg min_x \|PFW^Hx-y\|^2 + \alpha\|x\|_1 $$ where $W$ represnets the discret wavelet transform (DWT) matrice and $W^H$ is the inverse DWT. The optimizing problem can be solved easily by a lot of algorithms like ADMM, SDMM, CG, etc. Once you get the optimal $x$, the image can be calculated by $I = W^Hx$.

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).

So, there is no direct step to transform Fourier coef. to wavelet coef.