[Review] E2E Variational Network

End-to-End Variational Networks for Accelerated MRI Reconstruction

1 . Introduction

MRI의 느린 acquisition speed와

1. Introduction

The slow acquisition speed of MRI(Magnetic Resonance Imaging) had led us to develop two complementary methods: Parallel Imaging (PI) and Compressed Sensing (CS).

• Parallel Imaging uitilizes multiple receiver coils to acquire multiple views simultaneously, then coil images combined in software.
• Compressed Sensing, acquires only a subset of measurements and utilizes CS to reconstruct the final image from undersampled measurements.

alleviate : to make (suffering, deficiency or a problem) less severe

2. Background and Related Work

2.1 Accelerated MRI acquisition

• k-space : 2d frequency domain space, image can be obtained by applying an inverse 2D Fourier transform $\mathcal F^{-1}$ to the measured k-space samples.

\[{\bf k} = {\mathcal F}(\bf x) + \epsilon \tag{1}\]

where $\bf x \in \Complex^M, \bf k \in \Complex^M$ / $\epsilon$ is measurement noise, $\mathcal F$ is the fourier transform operator.

By introducing sensitivity map, image from multiple coils can be obtained with following:

\[{\bf k} _i = \mathcal F (S_i {\bf x}) + \epsilon_i , i=1,2,\dots, N, \tag{2}\]

where $S_i$ is a complex-valued diagonal matrix that encodes position dependent sensitivity map of i-th coil and $N$ is number of coils. Sensitivity maps are normalized to satisfy :

\[\sum_{i=1}^N S_i^{*} S_i = 1 \tag{3}\]

• Undersampled k-spaced data : $\tilde {\bf k} _i = M{\bf k}_i$, where $M$ is binary mask operator that selects a subset of k-space point. Undersampled k-spaced data results in aliasing artifacts in image domain
• Sensitivity map $S_i$ can be estimated using the central region of k-space corresponding to low frequencies, called the Auto-Calibration Signal (ACS) line which is typically fully sampled.

2.2 Compressed Sensing for Parallel MRI Reconstruction

• Classical compressed sensing methods can be formulated by following :

\[\hat {\bf x} = \argmin_{\bf x} {1 \over 2} \sum_i \Vert M\mathcal F (S_i {\bf x} ) - \tilde {\bf k}_i\Vert ^2 + \lambda \Psi({\bf x}) \tag{4}\] \[\hat {\bf x} = \argmin_{\bf x} {1 \over 2} \sum_i \Vert A({\bf x } ) - \tilde {\bf k}\Vert ^2 + \lambda \Psi({\bf x}) \tag{5}\]

where $\Psi$ is regularization function, $A( \ \cdot  \ ) $ is the linear forward operator (multiplies by sensitivity maps → apply 2D FFT → undersampling data), $\tilde {\bf k}$ vector of masked k-space for all coils


→ Find fully sampled image $\hat {\bf x} $ that minimizes difference between undersampled (given) data

• Solving this optimization problem with iterative gradient descent steps using :

\[{\bf x}^{t+1} = {\bf x}^{t} - \eta^t \big(A^{*}(A({\bf x} - \tilde {\bf k}) + \lambda \Phi ({\bf x}^t) \big) \tag {6}\]

where $\eta^t$ learning rate, $\Phi({\bf x}) = \nabla_{\bf x} \Psi$, $A^{*}$ is the hermitian of forward operator $A$ .

2.3 Deep Learning for Parallel MRI Reconstruction