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Namgyun Lee added paragraph_Sparse_MRI_Appendix_A__.tex
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\paragraph{Sparse MRI Appendix A revisited}
Here we provide a detailed derivation of the gradient of each term in the cost function
defined in Equation [A1].
\section{Theory}
\subsection{Image Reconstruction}
%\usepackage{amsmath}
\DeclareMathOperator*{\argmin}{argmin}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\nc}{n_c}
\newcommand{\nf}{n_f}
%\paragraph{Image Reconstruction}
The reconstruction is obtained by solving the following unconstrained convex optimization problem:
\begin{align*}
x = \argmin_m \|\tilde F_\Omega \tilde Sm - y\|_{\ell_2}^2 + \lambda_1\|Vm\|_{\ell_1} + \lambda_2\|\Psi m\|_{\ell_1} + \lambda_3\|T_v m\|_{\ell_1},
\end{align*}
where $\tilde F_\Omega = [\tilde F_0, \ldots, 0; 0, \ldots,\tilde F_{\nc-1}]$ is the undersampled Fourier operator, $\tilde S = [\tilde S_0, \ldots, 0; 0, \ldots,\tilde S_{\nc-1}]$ is the sensitivity matrix, $y = [y_0, \ldots, y_{\nc-1}]^T$ is the acquired 4-D k-space data, $V$ is a 1-D spectral high-pass filter along the saturation frequency dimension, $\Psi$ is a 4-D wavelet transform, and $T_v$ is a 3-D spatial finite-differences transform.
3-D coil sensitivity maps for each 3-D image of a 4-D CEST dataset were separately estimated from a fully sampled region in the center of 3-D k-space using a 3-D extension of the eigenvalue problem approach to sensitivity maps (calibration region: XX, kernel size: XX, $\sigma_{\mathrm{cut-off}}^2 = XX$, threshold: XX) \cite{Uecker_2013}.
