[Review] Feature-Image VarNet

Accelerated MRI reconstructions via variational network and feature domain learning

서론

Methods

MR image reconstruction

MR signal ${\bf k}i$(k-space domain) recieved by the _i-th coil is related to the MR image $\bf x$ by the forward problem :

\[{\bf k}_i = {\bf m} \odot {\bf F} ({\bf c}_i \odot {\bf x} ) , \ i=1, \cdots , N \tag{1}\]

• $\odot$ operator : element-wise (Hadamard) multiplication - Does resemble multiplication, but distinct from ‘Matrix’ or ‘Vector’ multiplication
• ${\bf c}_i$ : receive coil sensitivity profile (Image domain)
• $\bf F$ : discretized Fourier transform working on with multichannel images concatenated (DFT operator)
• $\bf m$ : predefined undersampling mask(same shape with kspace domain, works on kspace)

One of the traditional methods (Compressed Sensing) is solving (1) by regularized optimization routine

\[\tilde{\bf x} = \argmin_{\bf x} {1\over 2} \sum_i^N \Vert {\bf m} \odot {\bf F} ({\bf c}_i \odot {\bf x} ) -\tilde {\bf k}_i\Vert + \lambda \mathcal {\textbf Q} \{ \bf x\} \tag{2}\]

• $\mathcal {\textbf Q}$ : Regularizer, $\lambda$ : its weighting factor
• $\tilde {\bf k}_i = \bf m \odot {\bf k}_i$ (tilde) is undersampled measured k-space data

If $\mathcal {\textbf Q}$ differentable, problem can be solved with few gradient descent iteration steps as :

\[{\bf k}^{j+1} = {\bf j}^j - \eta^j {\bf m} \big({\bf k}^j-\tilde {\bf k} \big) + \lambda {\bf F} \ \varepsilon {\partial \mathcal {\textbf Q} \big(\mathcal R ({\bf F }^{-1} {\bf k}^j)\big)\over \partial \bf k} \tag{3a}\]

• where $\varepsilon(\ \cdot \ )$ : expand operator, performs individual coil images with ${\bf c}_i$

\[\varepsilon\{z\} = [{\bf c}_1 \odot z, \cdots , {\bf c}_N \odot z] \tag{3b}\]

• where $\mathcal R (\ \cdot \ )$ : reduce operator, multiplies element-wise the conjugate of ${\bf c}_i$ with coil images and sums over coil chans.
• $\eta^j$ : learning rate of gradient descent (iter $j$)

Finally, individual coil images given from ${\bf x}_i = \vert {\bf F}^{-1} {\bf k}_i\vert$ and coil combined images are sum of sqaured roots of individual coil images

Variational network

Inverse problem remains ‘ill-posed’ for high undersampling rates.