

% \section{MRI Background}
% \label{sec:ap2}

% \subsection{MRI Acquisition and Reconstruction}

% Assuming a fully-sampled MRI acquisition, the ground truth image $\vec{x} \in \mathbb{C}^{n}$ can be recovered from the fully-sampled multi-coil ($n_c>1$) $k$-space $\vec{y} \in \mathbb{C}^{n\times n_c}$ by applying the inverse Fast Fourier transform (FFT), denoted as $\mathcal{F}^{-1}$, followed by the RSS method:
% % 
% \begin{equation}
% \begin{gathered}
%     \vec{x} = \text{RSS} \circ \mathcal{F}^{-1} (\vec{y})=\left(\sum_{k=1}^{n_c}|\mathcal{F}^{-1}(\vec{y}^k)|^2\right)^{1/2},
%     \label{eq:rss}
% \end{gathered}
% \end{equation}
% % 
% or alternatively, with known sensitivity maps $\vec{S}$, the SENSE operator:
% % 
% \begin{equation}
% \begin{gathered}
%     \vec{x} = | \text{SENSE} \circ \mathcal{F}^{-1} (\vec{y}) |=  \left|\sum_{k=1}^{n_c}{\vec{S}^k}^{*} \mathcal{F}^{-1}(\vec{y}^k)\right|,
%     \label{eq:sense}
% \end{gathered}
% \end{equation}
% % 
% where,
% % 
% \begin{equation}
% \begin{gathered}
%     \vec{x} = \text{RSS} (\vec{z})=\left(\sum_{k=1}^{n_c}|\vec{z}^k|^2\right)^{1/2},\quad \vec{z} \in \mathbb{C}^{n \times n_c} 
% \end{gathered}
% \end{equation}
% % 
% and
% % 
% \begin{equation}
% \begin{gathered}
%     \text{SENSE}(\vec{z}) =  \sum_{k=1}^{n_c}{\vec{S}^k}^{*}\vec{z}^k,\quad \vec{z} \in \mathbb{C}^{n \times n_c}.
% \end{gathered}
% \end{equation}

% \noindent
% In accelerated acquisitions, the fully-sampled $k$-space is subsampled via an operator $\mat{U}_{\mat{M}}$, which selectively retains pixels present in the sampling set $\mat{M}$ and sets others to zero:
% % 
% \begin{equation}
%     \mat{U}_\mat{M}(\vec{y})_j = \vec{y}_j \cdot \mathbb{1}_\mat{M}(j) = \Bigg\{\begin{array}{lr}
%         \vec{y}_j, & \text{if } j\in\mat{M}\\
%         0, & \text{if } j \notin \mat{M}.
%         \end{array}
% \end{equation} 

% \noindent
% The forward problem of the accelerated acquisition is described by:
% % 
% \begin{equation}
%     \Tilde{\vec{y}}_\vec{M} = \mathcal{A}_{\vec{M}, \mat{S}} (\vec{x}) + \vec{e},
%     \label{eq:forward_problem}
% \end{equation}
% % 
% \noindent
% where $\vec{e}\in\mathbb{C}^{n\times n_c}$ represents noise and $\mathcal{A}_{\vec{M}, \mat{S}}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n\times n_c}$ denotes the forward operator which maps an image to individual coil images using the coil sensitivity maps $\mat{S}$ through the expand operator $\mathcal{E}_\mat{S}$, applies the FFT, denoted by $\mathcal{F}$, and applies subsampling via $\mat{U}_{\mat{M}}$:
% % 
% \begin{equation}
%         \mathcal{A}_{\vec{M}, \mat{S}} (\vec{x}) = \mat{U}_{\mat{M}} \circ \mathcal{F} \circ \mathcal{E}_\mat{S}(\vec{x}) = \Big\{\mat{U}_{\mat{M}} \circ \mathcal{F} (\mat{S}^k \vec{x}) \Big \}_{k=1}^{n_c}.
%         \label{eq:forward_op}
% \end{equation}

% \noindent
% The adjoint operator of $\mathcal{A}_{\vec{M}, \mat{S}}$, denoted by $\mathcal{A}^{*}_{\vec{M}, \mat{S}}:\mathbb{C}^{n\times n_c}\rightarrow\mathbb{C}^{n}$, subsamples the input multi-coil data via $\mat{U}_{\mat{M}}$, maps them to the image domain using $\mathcal{F}^{-1}$, and reduces them to a single image using $\mat{S}$ via the reduce operator $\mathcal{R}_\mat{S}$:

% \begin{equation}
%      \mathcal{A}^{*}_{\vec{M}, \mat{S}}(\vec{y}) = \mathcal{R}_\mat{S}\circ \mathcal{F}^{-1} \circ \mat{U}_{\mat{M}} (\vec{y}) =\\
%      \sum_{k=1}^{n_c} {\vec{S}^k}^{*} \mathcal{F}^{-1} \left(\mat{U}_{\mat{M}} (\vec{y}^k) \right).
%      \label{eq:backward_op}
% \end{equation}

% % \subsection{MRI Reconstruction}
% Typically, the process of recovering an image from the subsampled $k$-space measurements $\Tilde{\vec{y}}_{\mat{M}}$ is formulated as a regularized least squares optimization problem:

% \begin{equation}
%     \vec{x}^{*} =  \arg\min_{\vec{x}^{'}}\frac{1}{2}\left|\left| \mathcal{A}_{\vec{M}, \mat{S}}(\vec{x}^{'}) - \Tilde{\vec{y}}_{\mat{M}}\right|\right|_2^2 + \mathcal{G}(\vec{x}^{'}),
% \label{eq:variational_problem}
% \end{equation}

% \noindent
% where $\mathcal{G}: \mathbb{C}^{n}\rightarrow\mathbb{R}$ represents an arbitrary regularization functional that incorporates prior reconstruction information. \Equation{variational_problem} lacks a closed-form solution, and a solution can only be obtained numerically.  

% The operator DC$_\mat{M}$ denotes the data consistency operator, which enforces consistency between the available and predicted measurements and is defined as:
% % 
% \begin{equation}
%     \text{DC}_{\mat{M}}(\vec{w}_1, \, \vec{w}_2) = \mat{U}_{\mat{M}}(\vec{w}_1) + \mat{U}_{\mat{M}^{c}}(\vec{w}_2).
% \end{equation}
