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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. Or in other words, does the sparsity transformation results from undersampling of k-space, or moving from Fourier to wavelet, or both:

Diagram from Lustig, Donoho & Pauly

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?

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As explained in this webinar by Dr. Jan. W. Casselman, the method involves acquiring the Fourier coefficients as usual, except that in this case the K-space is non-uniformly undersampled.

The next step will indeed necessitate a transformation to wavelet space, although this transformation is not obtained directly from this sparsely populated sampled K-space; rather, there is an intermediate image reconstruction via (inverse) Fourier transformation from the frequency to the image domains. Only then is the wavelet transformation of the image carried out.

In wavelet space, thresholding methods are applied to reduce noise, hoping to impact as little as possible the diagnostic signal. This is in fact the step where the Nyquist-Shannon sampling theorem is circumvented.

In a fourth step, there is a transformation from wavelet space to image space. At this point the image will have less noise.

In a final step, this denoised image is compared to the original image generated directly from the undersampled K-space. This probably makes reference to the LASSO regression iterative minimization problem, expressed in the formula:

$$p = \min_p\left( \sum_{i=1}^{\text{no.coils}} \Vert m_{d,i} - ES_{d,i} p\Vert _2^2 + \lambda_1 \Vert R^{-1/2}p \Vert_2^2 \color{red}{+\lambda_2 \Vert \Psi p\Vert_1} \right)$$

(From the Philips white paper by Liesbeth Geerts-Ossevoort, et al. Compressed SENSE Speed done right. Every time.)

where $\lambda_2$ is a regularization parameter and $\Vert \Psi p\Vert_1$ the $L_1$-norm of the sparsity transform into the wavelet domain. $m_{d,i}$ is the signal measured by a coil after denoising. $ES$ corresponds to the matrix product of the under-sampled K-space and a given coil sensitivity. And $R$ is the "coarse resolution data from the integrated body coil obtained with the SENSE reference scan."

LASSO regression makes sense given the underdetermined (wide matrix) linear systems created by the random undesampling of K-space. LASSO regularization will amount to performing variable selection in the constructed model.

Of note the term $\color{red}{\lambda_2 \Vert \Psi p\Vert_1}$ is the only difference between SENSE and cSENSE: In SENSE it is absent.

It follows that the sparsity condition

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

would be assumed in MRI imaging.

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