text:
\section{Introduction}\label{Section:introduction}
Magnetic resonance imaging (MRI) serves as an indispensable tool in clinical diagnosis, but suffers from relatively long data acquisition time \cite{2017_review_PI}. To enable fast imaging, parallel imaging and sparse sampling are two featured approaches. The former is armed with multi coils in the equipment while the latter breaks the Nyquist sampling barrier for sparse images.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0pt}
\setlength{\belowcaptionskip}{0pt}
\centering
\includegraphics[width=3.2in]{./eps/Fig1}
\caption{Parallel imaging and reconstruction methods. (a) Diagram of parallel imaging; (b) SENSE; (c) auto-calibration signal; (d) SPIRiT.}
\label{fig_PI_SENSE_SPIRiT}
\end{figure}

In parallel imaging, reduced k-space data are acquired using an array of receiver coils (Fig. \ref{fig_PI_SENSE_SPIRiT} (a)) to accelerate data acquisition. Each receiver coil is more sensitive to the specific area of the object near the coil, which allows acquired data to contain more information than single-coil data do. However, undersampling of k-space data results in aliased images, thus reconstruction methods are required to recover missing data for the sake of providing artifact-free images. These parallel imaging reconstruction approaches could be categorized into two main genres: Image domain methods which base on the assumption that there exists a composite image and images of coils can be obtained by multiplying composite image with corresponding sensitivity maps (Fig. \ref{fig_PI_SENSE_SPIRiT} (b)), such as sensitivity encoding (SENSE) \cite{1999_SENSE}. Another type is the k-space domain method of which lies deeply on the fact that each k-space data points of a given coil can be formulated as a linear combination of the multi-coil signals of its neighboring k-space points (Fig. \ref{fig_PI_SENSE_SPIRiT} (d)), such as generalized autocalibrating partially parallel acquisitions (GRAPPA) \cite{2002_GRAPPA}, iterative self-consistent parallel imaging reconstruction (SPIRiT) \cite{2010_SPIRiT} and so on. These parallel imaging approaches achieve reliable reconstruction and some of them have been applied to clinical MRI scans. However, they need auto-calibration signals (ACS), as shown in Fig. \ref{fig_PI_SENSE_SPIRiT} (c), to estimate the coil sensitivity maps for encoding, such as SENSE, or to estimate the kernels for recovery of the missing k-space data, such as GRAPPA and SPIRiT. Once the number of acquired ACS is limited, the accuracy of estimated sensitivity maps or kernels decreases, resulting in large reconstruction errors \cite{2014_MRM_Lustig}. Therefore, how to properly reconstruct the image under limited ACS is still challenging.%
\begin{figure*}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=6.4in]{./eps/Fig2}
\caption{Illustration of ALOHA. (a) Low-rankness of weighted k-space with wavelet filters; (b) the sequential image reconstruction for two directions. Note: ${{\mathbf{W}}_{=}}$ and ${{\mathbf{W}}_{\perp}}$ denote the weights in the horizontal and vertical direction.}
\label{fig_ALOHA}
\end{figure*}%

In sparse sampling, k-space data are undersampled and then reconstructed with prior constraints. Two common constraints are sparsity and low-rankness. Sparsity approaches seek a sparse representation of an image under pre-constructed \cite{2007_Sparse_MRI, 2007_MRI_TV, 2017_PCS_EMBC, 2016_pFISTA, 2010_xiaobo,2018_MIA_reweighted_sparse_algorithm} or adaptive \cite{2016_TBME_Zhifang, 2016_MIA_Lai, 2011_Ravishankar, 2015_blind_CS_Ravishankar, 2016_Data_driven_Ravishankar, 2014_PANO, 2013_MRI_xiaobo, 2014_TBME_Wang} basis or dictionaries. Low-rankness methods commonly explore the linear correlations among multiple MRI image \cite{2007_PSF_Liang} \cite{2012_TMI_Zhao} \cite{2015_LR_sparse_MRI}, thus, they are primarily suitable for dynamic or higher-dimensional imaging \cite{2014_PLoSOne_Yu, 2016_high_dim_MR_tensor, 2015_MIA_Zhang,2018_MIA_LRTV_algorithm}. With in-depth and comprehensive developments, the low-rankness of k-space data of a single image or multiple images can be achieved \cite{1989_polynomial_Liang, 2014_MRM_Lustig, 2014_LORAKS, 2016_P_LORAKS, 2016_Off_the_grid, 2016_ALOHA}, which has produced promising image reconstructions under more flexible undersampling patterns, e.g. both randomly and uniformly sparse sampling. Rather than treating the whole k-space as a column of a matrix, these new approaches \cite{1989_polynomial_Liang, 2014_MRM_Lustig, 2014_LORAKS, 2016_P_LORAKS, 2016_Off_the_grid, 2016_ALOHA} have transformed the k-space into structured matrices, e.g. Hankel or Toeplitz matrices, to mine the linear correlation.%

Researchers explore the low-rankness of structured matrices from different properties, e.g. coil sensitivities have compact k-space support in SAKE \cite{2014_MRM_Lustig}, the image has limited spatial support or slowly varying phase in LORAKS \cite{2014_LORAKS,2015_AC_LORAKS,2015_W_LORAKS,2016_P_LORAKS}, transform (wavelets or derivatives) domain sparsity in the ALOHA \cite{2016_ALOHA}, Ongie's method \cite{2016_Off_the_grid} and GSLR \cite{2018_Hu_LR_TMI}. These low rank structured matrix approaches obviously improve the image reconstruction and provide flexibility to both uniformly and randomly sparse sampling. Among them, those taking advantage of the low-rank property, derived from the transform domain sparsity, of weighted Hankel matrix in k-space hold the flexibility to retain low-rankness since MRI images can be sparsely represented via various transforms. ALOHA \cite{2016_ALOHA} grew out of this observation and proposed to utilize the low-rankness of weighted k-space data (Fig. 2) in reconstruction. However, ALOHA separately enforces the low-rankness of vertical and horizontal directions (Fig. 2 (b)), which results in ALOHA yielding the suboptimal solution.
Ongie et al. \cite{2016_Off_the_grid} and GSLR \cite{2018_Hu_LR_TMI} proposed to simultaneously enforce the low-rankness of both vertical and horizont,al directions. However, though calibration data, i.e. ACS (Fig. 1 (c)), is common in parallel imaging and these low-rank structured matrix methods \cite{2015_AC_LORAKS, 2016_ALOHA,2016_P_LORAKS,2018_Hu_LR_TMI} are compatible to the parallel imaging as shown in TABLE \ref{Table 1}, none of them take the calibration data into consideration for better image reconstruction. It should be noticed that AC-LORAKS \cite{2015_AC_LORAKS} only used the calibration data for algorithm acceleration but not for performance enhancement.%

\begin{table}[htbp]
  \footnotesize
  \centering
  \caption{Differences among Low Rank Structured Matrix Methods}
  \label{Table 1}
    \begin{tabular}{m{1.9cm}<{\centering}m{1cm}<{\centering}m{1.2cm}<{\centering}m{1.3cm}<{\centering}m{1.4cm}<{\centering}}
    \hline
    Method & Parallel Imaging  & k-space Weighting   & Simultaneous Weighting & Calibration Consistency  \\
    \hline
    SAKE         & \checkmark     & $\times$       & /            & $\times$  \\
    LORAKS family& \checkmark     & \checkmark     & \checkmark   & \checkmark  \\
    GSLR         & \checkmark     & \checkmark     & \checkmark   & $\times$ \\
    ALOHA        & \checkmark     & \checkmark     & $\times$     & $\times$  \\
    \hline
    STDLR-SPIRiT & \checkmark     & \checkmark     & \checkmark   & \checkmark \\
    \hline
    \end{tabular}%
    \begin{tablenotes}[flushleft]
    \footnotesize
    \item Note: k-space weighting means k-space data Hadamard multiplying by weights which are the Fourier transform of the convolved filters in sparse representations. Simultaneous weighting indicates imposing low-rank constraints on horizontal and vertical directions. Calibration consistency means exploiting the data consistency from calibration data. The slash / means the corresponding operation is not needed or implemented yet.
     \end{tablenotes}
\end{table}%

In this work, we develop a k-space domain reconstruction method that jointly explores the low-rankness and calibration data. The two directional sparsity of the target image is simultaneously enforced by minimizing the low-rankness of the two directional weighting of the k-space. Besides, intra- and inter-coil data relationships are enforced in the form of self-consistency over k-space. Overall, it is expected that the new approach can further reduce the image reconstruction errors in parallel MRI.

The remainder of the paper is organized as follows. In Section \ref{Section:relatedWork}, we briefly review the related $\ell_1$-SPIRiT and ALOHA methods. In Section \ref{Section:proposedMethod}, the proposed method and algorithm are described. Section \ref{Section:experimentalResults} demonstrates the reconstruction performance and Section \ref{Section:discussions} discusses the effects of the number of ACS lines, parameter settings, empirical convergence of algorithm and comparison with other state-of-the-art low-rank structured matrix methods. Finally, conclusions will be drawn in Section \ref{Section:conclusion}.%

\section{Related Work} \label{Section:relatedWork}
\subsection{$\ell_1$-SPIRiT}
SPIRiT reconstructs parallel MRI image by enforcing the self-consistency of multi-coil k-space data \cite{2010_SPIRiT}. As shown in Fig. \ref{fig_PI_SENSE_SPIRiT} (d), SPIRiT first estimates the linear relation of the intra- and inter-coil k-space data from a small area of the fully sampled k-space center, which is also called as ACS (Fig. \ref{fig_PI_SENSE_SPIRiT} (c)), and then applies this relationship to the rest of the k-space data. In other word, SPIRiT enforces calibration consistency between every point in k-space and its entire neighborhood across all coils in an operator form:
\begin{equation}\label{(1)}
     \mathbf{X}=\mathcal{G}\mathbf{X},
\end{equation}%
where $\mathbf{X}$ denotes k-space data for all coils, and $\mathcal{G}$ is an operator that convolves the k-space data with a series of calibration kernels that are estimated from the ACS (Fig. \ref{fig_PI_SENSE_SPIRiT} (c))\cite{2010_SPIRiT}. Let $\mathbf{Y}$ be the acquired k-space data with non-acquired positions zero-filled for all coils and $\mathcal{U}$ represents the operator that performs undersampling and zerofilling on non-acquired data points, the data acquisition was given by:
\begin{equation}\label{(2)}
    \mathbf{Y}=\mathcal{U}\mathbf{X}.
\end{equation}%

Then, the SPIRiT reconstruction was formulated as:
\begin{equation}\label{(3)}
	\underset{\mathbf{X}}{\mathop{\min }} \, \left\| \mathcal{G}\mathbf{X}-\mathbf{X} \right\|_{F}^{2} \ \ s.t. \ \ \left\| \mathbf{Y}-\mathcal{U}\mathbf{X} \right\|_{F}^{2}\le \varepsilon,
\end{equation}%
where ${{\left\| \cdot  \right\|}_{F}}$ denotes the Frobenius norm, and $\varepsilon $ is a parameter that constraints the noise in the measurements. It has been observed that SPIRiT outperformed SENSE and GRAPPA with superior image quality \cite{2010_SPIRiT}. To better regularize the image, SPIRiT includes an additional penalty in the objective function and constructed the so-called $\ell_1$-SPIRiT model:
\begin{equation}\label{(4)}
    \resizebox{.9\hsize}{!}{$ \underset{\mathbf{X}}{\mathop{\min }}\,\left\| \mathcal{G}\mathbf{X}-\mathbf{X} \right\|_{F}^{2}+\lambda {{\left\| \Psi {{\mathcal{F}}^{-1}}\mathbf{X} \right\|}_{1}} \ \ s.t.\ \ \left\| \mathbf{Y}-\mathcal{U}\mathbf{X} \right\|_{F}^{2}\le \varepsilon $,}
\end{equation}%
where $\Psi $ denotes the sparse transform, ${{\mathcal{F}}^{-1}}$ denotes the inverse Fourier transform and parameter $\lambda$ balances the $\ell_1$ constraint and calibration consistency.

\subsection{ALOHA}\label{Subsection:proposedMethod}
In ALOHA \cite{2016_ALOHA}, the MRI image ${\mathbf{S}} \in {\mathbb{C}^{M \times N}}$ is assumed to be sparse in transform domain. For instance,
\begin{equation}\label{(5)}
    \Psi {\bf{S}} = {\bf{Z}},
\end{equation}%
where $\Psi $ denotes a sparse transform, such as wavelet, and the sparse signal $\mathbf{Z}$ is modeled as the finite superposition of Dirac functions as
\begin{equation}\label{(6)}
	\mathbf{Z}=\sum\limits_{i=1}^{R}{{{a}_{i}}\delta \left( x-{{x}_{i}},y-{{y}_{i}} \right)},
\end{equation}%
where ${{a}_{i}}$ denotes the amplitude of the $i^{\text{th}}$ Dirac function, ${{x}_{i}}$ and ${{y}_{i}}$ the shifts along the vertical and horizontal axes, and $R$ the number of Dirac functions.

Performing Fourier transform on \eqref{(5)}, and with the convolution theorem of Fourier transform, we obtain the spectral form of $\mathbf{Z}$ as below
\begin{equation}\label{(7)}
    \mathbf{M}=\mathcal{F}\left( \mathbf{Z} \right)=\mathcal{F}\left( \Psi \mathbf{S} \right)=\mathcal{F}\left( \Psi  \right)\odot \mathcal{F}\left( \mathbf{S} \right)=\mathbf{W}\odot \mathcal{F}\left( \mathbf{S} \right),
\end{equation}%
where $\odot $ denotes the Hadamard product and $\mathcal{F}$ denotes the Fourier transform and $\mathbf{W}=\mathcal{F}\left( \Psi  \right)$ denotes the weights obtained from applying Fourier transform to sparse transform filters. (See Fig. \ref{fig_ALOHA} (a) for a schematic illustration).

According to the Theorem 2.1 in \cite{2016_ALOHA}, the weighted k-space $\mathbf{M}$ satisfies the low-rankness of the Hankel matrix $\mathcal{H}\mathbf{M}$ as
\begin{equation}\label{(8)}
	rank\left( \mathcal{H}\mathbf{M} \right)=R.
\end{equation}%
where $\mathcal{H}$ is an operator that converts $\mathbf{M}$ into a block Hankel matrix \cite{1992_matrixPencil_Hua, 2014_LR_Yuejie,2018_TBME_Hengfa,2017_ACCESS_Guo} with a dimension of ${({{k}_{1}}{{k}_{2}})\times ((N-{{k}_{1}}+1)(M-{{k}_{2}}+1))}$, and ${{k}_{1}}$ and ${{k}_{2}}$ are pencil parameters.

Assuming that the image is sparse in the transform domain, then very few nonzeros are presented in $\mathbf{Z}$, leading to an effective $R$ be much smaller than the size of block Hankel matrix $\mathcal{H}\mathbf{M}$. Therefore, $\mathcal{H}\mathbf{M}$ enjoys low-rank properties. Ultimately, the recovery of undersampled k-space data can be regarded as a low rank matrix completion problem \cite{2016_ALOHA} as:
\begin{equation}\label{(9)}
	\underset{\mathbf{M}}{\mathop{\min }}\,{{\left\| \mathcal{H}\mathbf{M} \right\|}_{*}} \ \ s.t.\ \ \mathcal{U}\mathbf{M}=\mathbf{\hat{Y}},
\end{equation}%
where $\mathbf{\hat{Y}}=\mathbf{W}\odot \mathbf{Y}_{s}$, $\mathbf{Y}_{s}$ denotes the acquired k-space of a single coil and ${{\left\| \cdot  \right\|}_{*}}$ denotes the matrix nuclear norm.

Extending the above model to multi-coil data acquisition in parallel MRI, the image ${{\mathbf{S}}_{j}}$ of the $j^{\text{th}}$ coil is represented as:
\begin{equation}\label{(10)}
	{{\mathbf{S}}_{j}}={{\mathbf{C}}_{j}}\odot \mathbf{S}_{comp},\ j=1,2,...,J,
\end{equation}%
where ${{\mathbf{C}}_{j}}$ denotes the sensitivity map of the $j^{\text{th}}$ coil, $\mathbf{S}_{comp}$ the composite image.

A fatter matrix cascaded of Hankel matrices is formed as below
\begin{equation}\label{(11)}
    \resizebox{.8\hsize}{!}{$\mathbf{L}=\left[ \mathcal{H}\left( \mathbf{W}\odot {{\mathbf{X}}_{1}} \right),...,\mathcal{H}\left( \mathbf{W}\odot {{\mathbf{X}}_{j}} \right),...\mathcal{H}\left( \mathbf{W}\odot {{\mathbf{X}}_{J}} \right) \right]$},
\end{equation}%
where ${{\mathbf{X}}_{j}}=\mathcal{F}\left( \mathbf{S}_{j} \right)$ denotes the k-space data of the $j^{\text{th}}$ coil. According to the derivation in \cite{2016_ALOHA}, the matrix $\mathbf{L}$ is low-ranked. Then, the ALOHA for parallel MRI reconstruction \cite{2016_ALOHA} was expressed as:
\begin{equation}\label{(12)}
	\begin{matrix}
  \underset{{{\mathbf{M}}_{j}}}{\mathop{\min }}\,{{\left\| \left[ \mathcal{H}{{\mathbf{M}}_{1}},....\mathcal{H}{{\mathbf{M}}_{J}} \right] \right\|}_{*}}\ , \\
  s.t.\ \mathcal{U}{{\mathbf{M}}_{j}}=\mathbf{W}\odot {{\mathbf{Y}}_{j}},\ j=1,...,J. \\
\end{matrix}
\end{equation}%
where ${{\mathbf{M}}_{j}}=\mathbf{W}\odot {{\mathbf{X}}_{j}}$ and ${{\mathbf{Y}}_{j}}$ denotes the acquired k-space data in $j^{\text{th}}$ coil with non-acquired positions zero-filled.

It's important to note that when applying weighting on 2D k-space data, ALOHA \cite{2016_ALOHA} applies weighting sequentially on each direction. Specifically speaking, it first deals with the optimization problem \eqref{(13)} that enforces weighting along horizontal direction (Fig. \ref{fig_ALOHA} (b)), and then the solution of \eqref{(13)} will be used as an initialization input for the optimization problem \eqref{(14)} which applies weighting along vertical direction (Fig. \ref{fig_ALOHA} (b)).
\begin{equation}\label{(13)}
  \underset{{{\mathbf{M}}^{=}}}{\mathop{\min }}\,{{\left\| \mathcal{H}{{\mathbf{M}}^{=}} \right\|}_{*}} \ \ s.t.\ \ \mathcal{U}{{\mathbf{M}}^{=}}={{\mathbf{W}}_{=}}\odot \mathbf{Y}.
\end{equation}%
\begin{equation}\label{(14)}
  \underset{{{\mathbf{M}}^{\perp}}}{\mathop{\min }}\,{{\left\| \mathcal{H}{{\mathbf{M}}^{\perp}} \right\|}_{*}} \ \ s.t.\ \ \mathcal{U}{{\mathbf{M}}^{\perp}}={{\mathbf{W}}_{\perp}}\odot \mathbf{Y}.
\end{equation}%
where ${{\mathbf{W}}_{=}}$ and ${{\mathbf{W}}_{\perp}}$ denote the weights which are the Fourier transform of filters in the horizontal and vertical directions (Fig. \ref{fig_ALOHA}). Please note that ALOHA replaces the nuclear norm terms in \eqref{(13)} and \eqref{(14)} with the equivalent non-convex terms using the matrix factorizations technique \cite{2004_matrix_factorizations,2017_ACCESS_Guo,2018_TBME_Hengfa} for algorithm acceleration when ALOHA implementation.

\begin{figure}[!h]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=2.8in]{./eps/Fig3}
\caption{Reconstructed single-coil Shepp-Logan phantom using STDLR and ALOHA. (a) Fully sampled Shepp-Logan phantom and 2D random sampling pattern at sampling rate 0.15; (b)-(e) are reconstruction errors using ALOHA and (f) is the reconstruction error using STDLR. (b) (or (c)) is the reconstruction error produced by solely solving sub-problem (\ref{(13)}) (or (\ref{(14)})), respectively; (d) (or (e)) is the reconstruction error by first solving sub-problem (\ref{(13)}) and then (\ref{(14)}) (or first solving (\ref{(14)}) and then (\ref{(13)})) sequentially. Note: The ALOHA with one-level pyramidal decomposition (determined by the filter size) was compared here.}
\label{fig_STDLR_ALOHA}
\end{figure}

The non-convexity of the optimization problems and the sequential processing of the two non-convex problems, however, make ALOHA undergoes the scarce of low rank constraint along both directions simultaneously, resulting in the sub-optimal reconstruction (Fig. \ref{fig_STDLR_ALOHA}).
As shown in Figs. \ref{fig_STDLR_ALOHA} (b) and (c), distinct reconstruction errors are presented when solving (\ref{(13)}) (or (\ref{(14)})) as the first step for sequential reconstruction. These errors may propagate into the subsequent reconstruction and lead to a significant difference in the final reconstructed images (Figs. \ref{fig_STDLR_ALOHA} (d) and (e)). In addition, ALOHA provides obviously different solutions when changing the order of solving two sub-problems (Figs. \ref{fig_STDLR_ALOHA} (d) and (e)), leading to difficulty in choosing the optimal order in practical applications.

Therefore, properly exploring the low-rankness in k-space along horizontal and vertical directions is very important to reduce the image reconstruction errors.

\section{Proposed Method} \label{Section:proposedMethod}
\subsection{Proposed reconstruction model}
In this work, we first propose to simultaneously enforce the low-rankness of k-space both along horizontal and vertical directions as follows:
\begin{equation}\label{(15)}
    \small{\left (\mathbf{STDLR}\right )} \ \ \mathop {\min }\limits_{{{\bf{X}}_s}} {\left\| {{\cal H}{{\bf{W}}_ = }{{\bf{X}}_s}} \right\|_*} + {\left\| {{\cal H}{{\bf{W}}_ \perp }{{\bf{X}}_s}} \right\|_*} + \frac{\lambda }{2}\left\| {{{\bf{Y}}_s} - \mathcal{U}{{\bf{X}}_s}} \right\|_F^2,
\end{equation}%
where $\mathbf{X}_s$ denotes the targeted single-coil k-space data, $\mathbf{Y}_s$ the acquired single-coil k-space data and the nuclear norm term ${{\left\| \cdot  \right\|}_{*}}$ imposes the low-rank constraint on the weighted Hankel structured matrix. We call this model, one of the two proposed models in this work, STDLR for short. Please note that the way we enforce jointly weighting is different from the way Ongie's \cite{2016_Off_the_grid} and GSLR methods \cite{2018_Hu_LR_TMI} did in which they cascaded the two weighting matrices together to enforce low-rank constraint. In our way to enforce simultaneous weighting, the low-rank matrices lie in much smaller signal space expected to alleviate computation complexity. Compared with ALOHA, STDLR imposes a stronger low rank constraint allowing a possible better reconstruction. Reconstructions on Shepp-Logan phantom (Fig. \ref{fig_STDLR_ALOHA}) show that STDLR achieves lower reconstruction error than ALOHA.

A basic way to extend STDLR into parallel MRI is cascading the Hankel matrix of each coil into a fatter matrix \cite{2016_P_LORAKS,2016_ALOHA} as:
\begin{equation}\label{HWXH}
\begin{aligned}
    \tilde{\mathcal{H}}{{\mathcal{W}}_{=}}\mathbf{X} & =\left[ \mathcal{H}{{\mathbf{W}}_{=}}\odot {{\mathbf{X}}_{1}},...,\mathcal{H}{{\mathbf{W}}_{=}}\odot {{\mathbf{X}}_{J}} \right],\\
     \tilde{\mathcal{H}}{{\mathcal{W}}_{\perp}}\mathbf{X} & =\left[ \mathcal{H}{{\mathbf{W}}_{\perp}}\odot {{\mathbf{X}}_{1}},...,\mathcal{H}{{\mathbf{W}}_{\perp}}\odot {{\mathbf{X}}_{J}} \right],\\
\end{aligned}
\end{equation}%
where  $\mathbf{X}=\left[ {{\mathbf{X}}_{1}},{{\mathbf{X}}_{2}},...,{{\mathbf{X}}_{J}} \right]$ denotes the targeted k-space data from all coils and $\mathbf{Y}=\left[ {{\mathbf{Y}}_{1}},{{\mathbf{Y}}_{2}},...,{{\mathbf{Y}}_{J}} \right]$ are acquired k-space data with zero-filling at non-acquired positions, and ${{\mathcal{W}}_{=}}\mathbf{X}=\left[ {{\mathbf{W}}_{=}}\odot{{\mathbf{X}}_{1}},...,{{\mathbf{W}}_{\perp}}\odot {{\mathbf{X}}_{J}} \right]$,
${{\mathcal{W}}_{\perp}}\mathbf{X}=\left[ {{\mathbf{W}}_{\perp}}\odot{{\mathbf{X}}_{1}},...,{{\mathbf{W}}_{\perp}}\odot {{\mathbf{X}}_{J}} \right]$,

Then, a basic STDLR with cascaded Hankel matrices can be modeled as:
\begin{equation}\label{(15-1)}
    \mathop {\min }\limits_{\bf{X}} {\left\| {\tilde {\cal H}{{\cal W}_= }{\bf{X}}} \right\|_*} + {\left\| {\tilde {\cal H}{{\cal W}_ \perp}{\bf{X}}} \right\|_*} + \frac{\lambda }{2}\left\| {{\bf{Y}} - {\cal U}{\bf{X}}} \right\|_F^2.
\end{equation}%
Although the cascaded Hankel matrices were designed to implicitly utilize the correlation among multiple coils, we found this approach still suboptimal (Fig. \ref{fig_STDLR_SPIRiT_advan}(a)).

To reinforce the reconstruction in parallel imaging, our intention is to mine the self-consistency, that are estimated from the calibration data, over multi-coil k-space. In parallel MRI, SPIRiT achieved promising results and suggested a valuable concept that each k-space signal of a specific coil can be formulated as a linear combination of the multi-coil signals of its neighboring k-space points \cite{2002_GRAPPA, 2010_SPIRiT}. Taking this self-consistency property into account, we further exploit the aforementioned proposed STDLR model in \eqref{(15)} to cope with parallel imaging as:
\begin{equation} \label{(16)}
    \begin{aligned}
        \small{\left ( \textbf{STDLR-SPIRiT} \right )} \quad & \mathop {\min }\limits_{\bf{X}} {\left\| {\tilde {\cal H}{{\cal W}_ \bot }{\bf{X}}} \right\|_*} + {\left\| {\tilde {\cal H}{{\cal W}_ = }{\bf{X}}} \right\|_*} + \\
        & \frac{{{\lambda _1}}}{2} \left\| {{\cal G}{\bf{X}} - {\bf{X}}} \right\|_F^2 + \frac{{{\lambda _2}}}{2}\left\| {{\bf{Y}} - {\cal U}{\bf{X}}} \right\|_F^2, \\
    \end{aligned}
\end{equation}%
where ${{\lambda}_{1}}$ and ${{\lambda }_{2}}$ trade off among the low-rank constraint, self-calibration consistency and undersampled data fidelity. The second proposed approach uses the same calibration $\mathcal{G}$ as the SPIRiT. Since the third term is the same as the SPIRiT, we call the second proposed model in \eqref{(16)} as STDLR-SPIRiT. The schematic illustration of STDLR-SPIRiT is depicted in Fig. \ref{fig_flowchart_STDLR_SPIRiT}. The STDLR-SPIRiT holds the advantage over the STDLR with cascaded Hankel matrices on reducing reconstruction errors at edges. The reason is that STDLR-SPIRiT utilizes the local linear relations existed in intra- and inter-coils that have been explicitly estimated from ACS data (Figs. \ref{fig_STDLR_SPIRiT_advan}(a) and (b)). Besides, although with the same ACS strategy, STDLR-SPIRiT is particularly useful when the ACS data are limited, compared with the state-of-the-art sparsity-based SPIRiT \cite{2010_SPIRiT} as shown in Fig. \ref{fig_STDLR_SPIRiT_advan} (d). These observations imply that both the low-rankness and estimated self-consistency improve the reconstruction in the proposed STDLR-SPIRiT.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig4}
\caption{The flowchart of the proposed method. Note: ${{\mathbf{W}}_{=}}$ and ${{\mathbf{W}}_{\perp}}$ denote the weights in the horizontal and vertical direction.}
\label{fig_flowchart_STDLR_SPIRiT}
\end{figure}
\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=2.8in]{./eps/Fig5}
\caption{Reconstruction merits of 4-coil brain image using STDLR-SPIRiT. (a)-(b) are the reconstruction errors of STDLR with cascaded Hankel matrices and STDLR-SPIRiT; (c) a square root of sum of squares image of fully sampled data; (d) is the
reconstruction errors versus number of ACS lines by $\ell_1$-SPIRiT and STDLR-SPIRiT. All reconstructions in this experiment are conducted under 1D Gaussian Cartesian undersampling pattern with a fixed sampling rate of 0.34.}
\label{fig_STDLR_SPIRiT_advan}
\end{figure}

To tackle \eqref{(16)}, time-consuming singular value decomposition (SVD) is hardly avoided since singular value thresholding is utilized for nuclear norm minimization \cite{2010_SVT, 2017_ACCESS_Guo, 2018_TBME_Hengfa}. In our case, the size of low-rank matrix cascaded of Hankel matrices $(J{{k}_{1}}{{k}_{2}})\times ((N-{{k}_{1}}+1)(M-{{k}_{2}}+1))$ increases dramatically after pencil parameters ${{k}_{1}},\ {{k}_{2}}$ increase, resulting in a sharp increase in SVD calculation time. For example, the dimension of $\tilde{\mathcal{H}}{{\mathcal{W}}_{=}}\mathbf{X}$ reaches $54756\times 2116$ for a 4-coil $256\times 256$ image with pencil parameters ${{k}_{1}}=23$, ${{k}_{2}}=23$. Performing one-time SVD on $\tilde{\mathcal{H}}{{\mathcal{W}}_{=}}\mathbf{X}$ needs 12.25 second on a CentOS 7 computation server with two Intel Xeon CPUs of 3.5GHz and 112GB RAM. Thus, we seek an SVD-free algorithm to reduce the computation time. Here, we employ matrix factorization technique \cite{2010_SVT, 2017_ACCESS_Guo, 2018_TBME_Hengfa}, leading to a dramatic reduction of computation time. The SVD-free algorithm is based on the relationship \cite{2004_matrix_factorizations} shown below:
\begin{equation}\label{(17)}
    {{\left\| \mathbf{A} \right\|}_{*}}=\underset{\mathbf{P},\mathbf{Q}}{\mathop{\min }}\,\frac{1}{2}\left( \left\| \mathbf{P} \right\|_{F}^{2}+\left\| \mathbf{Q} \right\|_{F}^{2} \right)\  \ s.t.\ \ \mathbf{P}{{\mathbf{Q}}^{H}}=\mathbf{A},
\end{equation}%
where $\mathbf{P}$ and $\mathbf{Q}$ denote two factorized matrices, and the upper subscript $H$ denotes the Hermitian transpose of a complex matrix. Hence, \eqref{(16)} can be reformulated as matrix factorization constraint:
\begin{equation}\label{(18)}
\begin{aligned}
        \underset{\mathbf{X},\mathbf{P},\mathbf{Q}}{\mathop{\min }} & \frac{1}{2}\sum\limits_{i}{\left( \left\| {{\mathbf{P}}_{i}} \right\|_{F}^{2} + \left\| {{\mathbf{Q}}_{i}} \right\|_{F}^{2} \right)}+\frac{{{\lambda }_{1}}}{2}\left\| \mathcal{G}\mathbf{X} - \mathbf{X} \right\|_{F}^{2} + \\
        & \frac{{{\lambda }_{2}}}{2}\left\| \mathbf{Y} - \mathcal{U}\mathbf{X} \right\|_{F}^{2} \ \ s.t. \ \  {{\mathbf{P}}_{i}}\mathbf{Q}_{i}^{H}=\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}\mathbf{X}. \\
\end{aligned}
\end{equation}%
where $i$ denotes $\perp$ or $=$, i.e. vertical and horizontal directions.

In summary, STDLR-SPIRiT in \eqref{(16)} is the final model of this work. And \eqref{(18)} is the accelerated version of the proposed STDLR-SPIRiT model to avoid the time-consuming SVD computation.

\subsection{SVD-free low rank reconstruction algorithm}
\begin{figure*}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=6.4in]{./eps/Fig6}
\caption{Reconstruction results and errors of a phantom using the Cartesian pattern with a sampling rate of 0.27. (a) An SSOS image of fully sampled data; (b-e) SSOS images of reconstructed results by GRAPPA, $\ell_1$-SPIRiT, ALOHA and STDLR-SPIRiT, respectively; (f) the Cartesian undersampling pattern; (g-j) the reconstruction error distribution corresponding to the above methods.}
\label{fig_results_phantom}
\end{figure*}
Here, we adopt the Alternating Direction Method of Multiplier (ADMM) \cite{2011_ADMM} to deal with the proposed model in \eqref{(18)}. It should be noticed that the non-convex optimization problem \eqref{(18)} can be successfully solved by ADMM \cite{2016_ALOHA,2018_TBME_Hengfa,2017_ACCESS_Guo}.
The augmented Lagrangian form of \eqref{(18)} is
\begin{equation}\label{(19)}
    \begin{aligned}
         & \underset{{{\mathbf{D}}_{i}}}{\mathop{\max }} \ \underset{\mathbf{X},{{\mathbf{P}}_{i}},{{\mathbf{Q}}_{i}}}{\mathop{\min }} \frac{{{\lambda }_{1}}}{2}\left\| \mathcal{G}\mathbf{X}-\mathbf{X} \right\|_{F}^{2} + \frac{{{\lambda }_{2}}}{2}\left\| \mathbf{Y}-\mathcal{U}\mathbf{X} \right\|_{F}^{2} +\\
         & \frac{1}{2}\sum\limits_{i} \! {\left( \! \left\| {{\mathbf{P}}_{i}} \right\|_{F}^{2}+\left\| {{\mathbf{Q}}_{i}} \right\|_{F}^{2} \! \right)} \!\! + \!\! \sum\limits_{i} \!{\left\langle \! {{\mathbf{D}}_{i}},\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}\mathbf{X}\!-\!{{\mathbf{P}}_{i}}\mathbf{Q}_{i}^{H} \! \right\rangle} \\
         & + \sum\limits_{i}{\frac{{{\beta }_{i}}}{2}\left\| \tilde{\mathcal{H}}{{\mathcal{W}}_{i}}\mathbf{X}-{{\mathbf{P}}_{i}}\mathbf{Q}_{i}^{H} \right\|_{F}^{2}}, \\
    \end{aligned}
\end{equation}%
where ${{\mathbf{D}}_{i}}$ denotes the Lagrangian multiplier, $\left\langle \cdot ,\cdot  \right\rangle $ the inner product in the Hilbert space of complex matrices, defined by $\left\langle \mathbf{A},\mathbf{B} \right\rangle =\Re \left\langle \mathbf{A}\left( : \right),\mathbf{B}\left( : \right) \right\rangle =\Re \left( trace\left( {{\mathbf{A}}^{*}}\mathbf{B} \right) \right)$, $\Re$ denotes the real part, and ${{\beta }_{i}}$ denotes the penalty parameter for the $i^{\text{th}}$ direction. The solution of \eqref{(19)} can be obtained through alternatively solving sub-problems below:
\begin{equation}\label{sub-pro-1}
\begin{medsize}
    \begin{aligned}
        {{\mathbf{X}}^{\left( k+1 \right)}} & = \arg \underset{\mathbf{X}}{\mathop{\min }} \frac{{{\lambda }_{1}}}{2} \left\| \mathcal{G}\mathbf{X} - \mathbf{X} \right\|_{F}^{2} + \sum\limits_{i} \left\langle \mathbf{D}_{i}^{\left( k \right)},\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}\mathbf{X} - \mathbf{P}_{i}^{\left( k \right)} \right. \left. \mathbf{Q}_{i}^{\left( k \right)H}  \right\rangle \\
        & + \frac{{{\lambda }_{2}}}{2} \left\| \mathbf{Y}-\mathcal{U}\mathbf{X} \right\|_{F}^{2} + \sum\limits_{i} {\frac{{{\beta }_{i}}}{2} \left\|  \tilde{\mathcal{H}}{{\mathcal{W}}_{i}}\mathbf{X} - \mathbf{P}_{i}^{\left( k \right)}\mathbf{Q}_{i}^{\left( k \right)H} \right\|_{F}^{2} } ,\\

    \end{aligned}
\end{medsize}
\end{equation}%
\begin{equation}\label{sub-pro-2}
\begin{medsize}
    \begin{aligned}
        \mathbf{P}_{i}^{\left( k+1 \right)} = & \arg \underset{{{\mathbf{P}}_{i}}}{\mathop{\min }} \left\langle \mathbf{D}_{i}^{\left( k \right)},\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}-{{\mathbf{P}}_{i}}\mathbf{Q}_{i}^{\left( k \right)H} \right\rangle + \\
        &  \frac{1}{2}\left\| {{\mathbf{P}}_{i}} \right\|_{F}^{2} + \frac{{{\beta }_{i}}}{2}\left\| \tilde{\mathcal{H}}\mathcal{W}{{}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}-{{\mathbf{P}}_{i}}\mathbf{Q}_{i}^{\left( k \right)H} \right\|_{F}^{2}, \\
    \end{aligned}
\end{medsize}
\end{equation}%
\begin{equation}\label{sub-pro-3}
\begin{medsize}
    \begin{aligned}
        \mathbf{Q}_{i}^{\left( k+1 \right)} = & \arg \underset{{{\mathbf{Q}}_{i}}}{\mathop{\min }} \left\langle \mathbf{D}_{i}^{\left( k \right)},\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}-\mathbf{P}_{i}^{\left( k+1 \right)}\mathbf{Q}_{i}^{H} \right\rangle + \\
        &  \frac{1}{2}\left\| {{\mathbf{Q}}_{i}} \right\|_{F}^{2} + \frac{{{\beta }_{i}}}{2}\left\| \tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}-\mathbf{P}_{i}^{\left( k+1 \right)}\mathbf{Q}_{i}^{H} \right\|_{F}^{2}, \\
    \end{aligned}
\end{medsize}
\end{equation}%
\begin{equation}\label{sub-pro-4}
\begin{medsize}
    \mathbf{D}_{i}^{\left( k+1 \right)}=\mathbf{D}_{i}^{\left( k \right)}+{{\tau }_{i}}\left( \tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}-\mathbf{P}_{i}^{\left( k+1 \right)}{{\mathbf{Q}}_{i}}^{\left( k+1 \right)H} \right).
\end{medsize}
\end{equation}%
where ${{\tau}_{i}}$ is the step size of the $i^{\text{th}}$ direction and is set as 1 for all $i$. For fixed $\mathbf{P}_{i}^{\left( k \right)}$, $\mathbf{Q}_{i}^{\left( k \right)}$ and $\mathbf{D}_{i}^{\left( k
\right)}$, ${{\mathbf{X}}^{\left( k+1 \right)}}$ has a close-form solution as
\begin{equation}\label{(21)}
\begin{medsize}
    \begin{aligned}
        & {{\mathbf{X}}^{\left( k+1 \right)}}=\bigg( {{\lambda }_{1}}\left( {{\mathcal{G}}^{*}}\mathcal{G}-{{\mathcal{G}}^{*}}-\mathcal{G}+\mathbf{I} \right)+\sum\limits_{i}{{{\beta }_{i}}\mathcal{W}_{i}^{*}{{{\tilde{\mathcal{H}}}}^{*}}\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}} + \\
        &  {{\lambda }_{2}}{{\mathcal{U}}^{*}}\mathcal{U}  \bigg) ^{-1}  \left[ {{\lambda }_{2}}{{\mathcal{U}}^{*}}\mathbf{Y}+\sum\limits_{i}{{{\beta }_{i}}\mathcal{W}_{i}^{*}{{{\tilde{\mathcal{H}}}}^{*}} \left(  \mathbf{P}_{i}^{\left( k \right)}\mathbf{Q}_{i}^{\left( k \right)H}-\frac{\mathbf{D}_{i}^{\left( k \right)}}{{{\beta }_{i}}} \right)} \right] , \\
    \end{aligned}
\end{medsize}
\end{equation}%
where the upper subscript $*$ denotes the adjoint operator.
For fixed ${{\mathbf{X}}^{\left( k+1 \right)}}$, $\mathbf{Q}_{i}^{\left( k \right)}$ and $\mathbf{D}_{i}^{\left( k \right)}$, $\mathbf{P}_{i}^{\left( k+1 \right)}$ is obtained by
\begin{equation}\label{(22)}
\begin{medsize}
\begin{aligned}
	\mathbf{P}_{i}^{\left( k+1 \right)} \!\!=\!\! \left( \! {{\beta }_{i}}\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}}\!+\!\mathbf{D}_{i}^{\left( k \right)} \! \right)
    \! \mathbf{Q}_{i}^{\left( k \right)} \! {{\left( \! \mathbf{I}\!+\!{{\beta }_{i}}\mathbf{Q}_{i}^{\left( k \right)H}\mathbf{Q}_{i}^{\left( k \right)} \! \right)}^{-1}} \!.
\end{aligned}
\end{medsize}
\end{equation}
For fixed ${{\mathbf{X}}^{\left( k+1 \right)}}$, $\mathbf{P}_{i}^{\left( k+1 \right)}$ and $\mathbf{D}_{i}^{\left( k \right)}$, $\mathbf{Q}_{i}^{\left( k+1 \right)}$ can be solved by
\begin{equation}\label{(23)}
\begin{medsize}
\begin{aligned}
	\mathbf{Q}_{i}^{\left( k+1 \right)} = & {{\left( {{\beta }_{i}}\tilde{\mathcal{H}}{{\mathcal{W}}_{i}}{{\mathbf{X}}^{\left( k+1 \right)}} + \mathbf{D}_{i}^{\left( k \right)}  \right)}^{*}} \\
    & \mathbf{P}_{i}^{\left( k+1 \right)} {{\left( \mathbf{I} + {{\beta }_{i}}\mathbf{P}_{i}^{\left( k+1 \right)H}\mathbf{P}_{i}^{\left( k+1 \right)} \right)}^{-1}} . \\
\end{aligned}
\end{medsize}
\end{equation}
The numerical algorithm is summarized in Algorithm \ref{alg:1}.
\begin{algorithm}[htb]
\footnotesize
\caption{MRI image reconstruction with STDLR-SPIRiT}\label{alg:1}
\hspace*{0.02in}{\bf{Input:}}  $\mathbf{Y}$, $\mathcal{U}$, $\mathcal{G}$, ${{\lambda }_{1}}$, ${{\lambda }_{2}}$, ${{\tau }_{i}}$, ${{\beta }_{i}}$ .\\
\hspace*{0.02in}{\bf{Initialization:}} ${{\mathbf{P}}_{i}}$ and ${{\mathbf{Q}}_{i}}$ are initialized as random matrix, ${{\mathbf{D}}_{i}}=\mathbf{1}$, and $k=1$.\\
\hspace*{0.02in}{\bf{Output:}} ${\mathbf{X}}$.
\begin{algorithmic}[1]
\WHILE{ $k\le 100$ and $  \left\| {{\mathbf{X}}^{\left( k+1 \right)}}-{{\mathbf{X}}^{\left( k \right)}}\  \right\|_{F}^{2}\ /\left\| {{\mathbf{X}}^{\left( k \right)}}\  \right\|_{F}^{2}\ge {{10}^{-6}}$ }
\STATE
Update $\mathbf{X}$ by solving equation \eqref{(21)};
\STATE
Update ${{\mathbf{P}}_{i}}$ by using \eqref{(22)};
\STATE
Update ${{\mathbf{Q}}_{i}}$ by using \eqref{(23)};
\STATE
Update multiplier ${{\mathbf{D}}_{i}}$ by using \eqref{sub-pro-4};
\STATE $ k = k + 1 $;
\ENDWHILE
\end{algorithmic}
\end{algorithm}

\begin{table}[htbp]
\footnotesize
  \centering
  \caption{Differences among Compared Methods.}\label{Table 2}
    \begin{tabular}{m{1.95cm}<{\centering}m{1.4cm}<{\centering}m{1cm}<{\centering}m{1.5cm}<{\centering}m{0.9cm}<{\centering}}
    \toprule
    Method & Calibration consistency & Image sparsity & k-space low-rankness & Wavelet \\
    \midrule
    GRAPPA             & \checkmark     & $\times$     & $\times$       & / \\
    $\ell_1$-SPIRiT    & \checkmark     & \checkmark   & $\times$       & db4 \\
    ALOHA              & $\times$       & $\times$     & \checkmark     & Haar \\
    STDLR-SPIRiT       & \checkmark     & $\times$     & \checkmark     & Haar \\
    \bottomrule
    \end{tabular}%
        \begin{tablenotes}[flushleft]
         \footnotesize
         \item Note: If the constraint, such as calibration consistency, image sparsity and k-space low-rankness is checked of the method row, it means the corresponding constraint was enforced directly in the approach.
         Image sparsity means imposing sparse assumption in transform domain. The k-space low-rankness means imposing low rank constraint in k-space domain. Wavelet denotes the type of wavelet used in sparsifying transform or k-space weighting.
     \end{tablenotes}
\end{table}%
\section{Experimental Results}\label{Section:experimentalResults}
In this section, the reconstruction performance of the proposed method is evaluated on phantom and \textit{in vivo} MRI data. Cartesian sampling with random phase encoding, pseudo radial sampling and 2D random sampling data are retrospectively constructed from fully sampled data. The ACS signals of sampling patterns are optimized for kernel estimation. The proposed method, STDLR-SPIRiT, is compared with three state-of-the-art reconstruction methods including GRAPPA, $\ell_1$-SPIRiT and ALOHA. Both GRAPPA and $\ell_1$-SPIRiT are typical parallel imaging reconstruction methods using kernel estimation. In addition, the $\ell_1$-SPIRiT exploits the sparsity of images under wavelets transform. ALOHA with cascaded Hankel matrices in (\ref{(12)}) is chosen for comparison since it utilizes the low-rankness of weighted k-space of multi-coil data. Differences among these methods are summarized in Table \ref{Table 2}.
\begin{figure*}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=6.4in]{./eps/Fig7}
\caption{Reconstruction results and errors of human brain using the Cartesian pattern with a sampling rate of 0.34. (a) An SSOS image of fully sampled data; (b-e) SSOS images of reconstructed results by GRAPPA, $\ell_1$-SPIRiT, ALOHA and STDLR-SPIRiT, respectively; (f) the Cartesian undersampling pattern; (g-j) the reconstruction error distribution corresponding to reconstructed image above them.}
\label{fig_results_brain_typical}
\end{figure*}

The implementation of GRAPPA and $\ell_1$-SPIRiT was shared at Dr. Michael Lustig's website \cite{code_SPIRiT} while ALOHA was shared at Dr. Jong Chul Ye' website \cite{code_ALOHA}. Parameters of all the compared methods are optimized to obtain the lowest RLNE (See Supplementary Material for details). All computation procedures run on a CentOS 7 computation server with two Intel Xeon CPUs of 3.5 GHz and 112 GB RAM. For 4-coil parallel MRI, the reconstruction time of GRAPPA, $\ell_1$-SPIRiT, ALOHA, and the proposed STDLR-SPIRiT are 26.8 seconds, 22.5 seconds, 458.3 seconds, and 1090.3 seconds, respectively.

In all experiments, the reconstructed multi-coil images are combined by a square root of sum of squares (SSOS), and the error distributions of multi coils were combined into single-coil difference image with SSOS.

We adopt relative $\ell_2$ norm error (RLNE) and mean structure similarity index measure (MSSIM) \cite{2004_TIP_Zhou} as objective criteria to quantify the reconstruction performance. The RLNE is defined as
\begin{equation}\label{(24)}
  \text{RLNE}=\frac{{{\left\| \mathbf{x}-\mathbf{\hat{x}} \right\|}_{2}}}{{{\left\| \mathbf{x} \right\|}_{2}}},
\end{equation}%
where $\mathbf{x}$ denotes the column stacked fully sampled k-space data and  $\mathbf{\hat{x}}$ the column stacked reconstructed k-space data. A lower RLNE demonstrates higher consistency between the fully sampled image and the reconstructed image. The mean measure of structural similarity (MSSIM) on two images $\mathbf{A}$ and $\mathbf{B}$ is calculated through
\begin{equation}\label{(25)}
    \resizebox{.8\hsize}{!}{$ \text{MSSIM}\left( \mathbf{A},\mathbf{B} \right) = \frac{1}{M} \sum\limits_{i=1}^{M}{\frac{\left( 2{{\mu }_{{{\mathbf{a}}_{i}}}}{{\mu }_{{{\mathbf{b}}_{i}}}} + {{C}_{1}} \right) \left( 2{{\sigma }_{{{\mathbf{a}}_{i}}{{\mathbf{b}}_{i}}}} + {{C}_{2}} \right)}{\left( \mu _{{{\mathbf{a}}_{i}}}^{2} + \mu _{{{\mathbf{b}}_{i}}}^{2} + {{C}_{1}} \right) \left( \sigma _{{{\mathbf{a}}_{i}}}^{2} + \sigma _{{{\mathbf{b}}_{i}}}^{2} + {{C}_{1}} \right)}} $},
\end{equation}%
where $\mathbf{A}$ and $\mathbf{B}$ represent fully sampled and reconstructed SSOS images, respectively; ${{\mu }_{{{\mathbf{a}}_{i}}}}$ , ${{\mu }_{{{\mathbf{b}}_{i}}}}$, ${{\sigma }_{{{\mathbf{a}}_{i}}}}$, ${{\sigma }_{{{\mathbf{b}}_{i}}}}$ and ${{\sigma }_{{{\mathbf{a}}_{i}}{{\mathbf{b}}_{i}}}}$ respectively denote the means, standard deviations and covariance of the local window ${{\mathbf{a}}_{i}}$ and ${{\mathbf{b}}_{i}}$; $\mathbf{M}$ the number of local windows. Constants ${{C}_{1}}$ and ${{C}_{2}}$ are introduced to avoid the case when the denominator multiplied by $\mu _{{{\mathbf{a}}_{i}}}^{2}+\mu _{{{\mathbf{b}}_{i}}}^{2}$ and $\sigma _{{{\mathbf{a}}_{i}}}^{2}+\sigma _{{{\mathbf{b}}_{i}}}^{2}$ is close to zero. A higher MSSIM indicates higher detail preservation in reconstruction.

\subsection{Experiments on Phantom Data}
The fully sampled Phantom data (Fig. \ref{fig_results_phantom} (a)) were acquired from a 3T SIEMENS Trio whole-body scanner (Siemens Healthcare, Erlangen, Germany) equipped with a 32-coil head coil, using 2D T2-weighted turbo spin echo sequence (matrix size = $384\times384$, TR/TE = 2000 ms /9.7 ms, FOV = 230 mm$\times$187 mm, slice thickness = 5 mm). The acquired data of 32 coils were compressed into eight virtual coils \cite{2013_ESPIRIT}.
Reconstructions of phantom data are depicted in Fig. \ref{fig_results_phantom}. GRAPPA (Fig. \ref{fig_results_phantom} (b)) and ALOHA (Fig. \ref{fig_results_phantom} (d)) reconstructed images exhibit obvious ringing artifacts in both the white background region and gray objects. $\ell_1$-SPIRiT (Fig. \ref{fig_results_phantom} (c)) produces good reconstructed image with better artifacts suppression. But, with closer inspection, visible noise of the left-bottom oblique sectors can be found. In comparison, STDLR-SPIRiT preserves image resolution and sharped edges (Fig. \ref{fig_results_phantom} (e)). Also, the difference images and quality metric in Table \ref{Table_3} serve to demonstrate STDLR-SPIRiT possessing the lowest reconstruction error (Fig. \ref{fig_results_phantom} (j)) and highest MMSIM.

\begin{table}[htbp]
  \footnotesize
  \centering
  \caption{RLNE/MSSIM FOR PHANTOM AND HUMAN BRAIN DATA RECONSTRUCTIONS.}
  \label{Table_3}
    \begin{tabular}{m{0.6cm}<{\centering}m{1.2cm}<{\centering}m{1.3cm}<{\centering}m{1cm}<{\centering}m{2cm}<{\centering}}
    \toprule
    \multicolumn{1}{c}{Images} & GRAPPA & $\ell_1$-SPIRiT & ALOHA & STDLR-SPIRiT \\
    \midrule
    \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 6}}  & \multicolumn{1}{c}{0.0862} & \multicolumn{1}{c}{0.0692} & \multicolumn{1}{c}{0.0827} & \multicolumn{1}{c}{\textbf{0.0470}} \\
          & /0.8951 & /0.9674 & /0.9512 & \textbf{/0.9831} \\
    \midrule
    \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 7}}  & \multicolumn{1}{c}{0.1335} & \multicolumn{1}{c}{0.0866} & \multicolumn{1}{c}{0.1117} & \multicolumn{1}{c}{\textbf{0.0735}} \\
          & /0.9609 & /0.9868 & /0.9791 & \textbf{/0.9919} \\
    \midrule
    \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 8}}  & \multicolumn{1}{c}{0.2726} & \multicolumn{1}{c}{0.1891} & \multicolumn{1}{c}{0.1840} & \multicolumn{1}{c}{\textbf{0.1187}} \\
          & /0.9067 & /0.9485 & /0.9523 & \textbf{/0.9777} \\
    \midrule
    \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 9}} & \multicolumn{1}{c}{0.1127} & \multicolumn{1}{c}{0.0848} & \multicolumn{1}{c}{0.0997} & \multicolumn{1}{c}{\textbf{0.0744}} \\
          & /0.9411 & /0.9755 & /0.9723 & \textbf{/0.9833} \\
    \bottomrule
    \end{tabular}%
\end{table}%

\subsection{Experiments on Brain Imaging Data}
Three brain datasets are used in experiments here. The first brain dataset shown in Fig. \ref{fig_results_brain_typical} and Fig. \ref{fig_results_brainData} (a) were two slices acquired from a healthy volunteer using the 2D $T_2$-weighted turbo spin echo sequence (matrix size = 256 $\times$ 256, TR/TE=6100 ms/99 ms, field of view = 220 mm $\times$ 220 mm, slice thickness = 3mm). They were obtained from a 3T SIEMENS Trio whole-body scanner (Siemens Healthcare, Erlangen, Germany) equipped with a 32-coil head coil. Four virtual coils were compressed from the acquired data of 32 coils \cite{2013_ESPIRIT}. The second brain dataset shown in Fig. \ref{fig_results_brain_typical_2} and Fig. \ref{fig_results_brainData} (b) were two slices acquired from a healthy volunteer on a 1.5T Philips MRI scanner (Philips Healthcare Best, the Netherlands) equipped with an 8-coil head coil using the 2D $T_1$-weighted fast-field-echo sequence (matrix size = 256 $\times$ 256, TR/TE = 1700 ms/390 ms, FOV = 230 mm$\times$230 mm, slice thickness = 5 mm). The third brain dataset depicted in Fig. \ref{fig_results_brainData} (c) were acquired from a 3T GE MRI scanner (GE Healthcare, Waukesha, Wis, USA) equipped with a 12-coil head coil using the 2D $T_1$-weighted SPGR sequences (matrix size = 256 $\times$ 256, TR/TE = 400 ms/9 ms, FOV = 240 mm $\times$ 240 mm, slice thickness = 6 mm.).

Serious ringing artifacts remain in the reconstructions by GRAPPA (Fig. \ref{fig_results_brain_typical} (b)) and ALOHA (Fig. \ref{fig_results_brain_typical} (d)). The $\ell_1$-SPIRiT produces the visually similar reconstructed image (Fig. \ref{fig_results_brain_typical} (c)) as STDLR-SPIRiT does. Both methods show great capability of artifacts suppression and present reliable reconstructions.
For better visualization of their differences, we offer zoom-in images of reconstructions. From the zoom-in images, we could observe relatively stronger noise in $\ell_1$-SPIRiT reconstruct image and to some extend, blur of details.
Nevertheless STDLR-SPIRiT restores image with better signal to noise ratio and fine details preservation (Fig. \ref{fig_results_brain_typical} (e)). In addition, the error images illustrate that the proposed STDLR-SPIRiT achieves the lowest reconstruction error (Fig. \ref{fig_results_brain_typical} (j)). Objective criteria in Table \ref{Table_3} indicates that the proposed method produces the lowest reconstruction error and the highest structure similarity to the fully sampled image. Evaluation criteria are consistent with visual inspections mentioned above.
\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig8}
\caption{The reconstruction results of T2-weighted brain image versus different sampling rates under Cartesian undersampling pattern. (a) and (b) display the variation trends of RLNE and MSSIM respectively; (c) the Cartesian undersampling pattern with a sampling rate of 0.24; (d) an SSOS image of fully sampled data; (e-h) SSOS images of reconstructed results by GRAPPA, $\ell_1$-SPIRiT, ALOHA and STDLR-SPIRiT. Note: The fully sampled images in Figs. \ref{fig_results_brain_typical} and \ref{fig_results_diff_patterns_RLNE} are the same.}
\label{fig_results_diff_patterns_RLNE}
\end{figure}
\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig9}
\caption{Reconstructed brain images using pseudo radial pattern with sampling rate of 0.20. (a) An SSOS image of fully sampled data; (b, c, e, f) reconstruction errors introduced by GRAPPA, $\ell_1$-SPIRiT, ALOHA and STDLR-SPIRiT; (d) pseudo radial undersampling pattern; (g) and (h) display the  RLNE and MSSIM versus sampling rate, respectively.}
\label{fig_results_brain_typical_2}
\end{figure}

As shown in Figs. \ref{fig_results_diff_patterns_RLNE} (a) and (b), with the increase of sampling rates, the growth of MSSIM and decline of RLNE are observed in all methods. Moreover, the STDLR-SPIRiT outperforms other methods under all sampling rates in terms of RLNE and SSIM. Particularly, even at a relatively low sampling rate (0.24), the proposed method still provides reliable reconstructions (Fig. \ref{fig_results_diff_patterns_RLNE} (h)) whereas GRRAPA and ALOHA images carry distinct artifacts. $\ell_1$-SPIRiT produces an image with nice artifacts suppression (Fig. \ref{fig_results_diff_patterns_RLNE} (f)), nonetheless the image is blurred (the zoom-in image) and bear incorrect contrast reconstruction of the withe matter in some areas, for instance, the mottled white matters around the image center. By contrast, STDLR-SPIRiT retains promising image resolution and fine details as the fully sampled image does (Fig. \ref{fig_results_diff_patterns_RLNE} (h)).

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig10}
\caption{More brain images and undersampling patterns. (a-c) Three different brain images; (d) the Cartesian sampling pattern of sampling rate 0.34; (e) the pseudo radial sampling pattern of sampling rate 0.20; (f) the 2D random sampling pattern of sampling rate 0.18.}
\label{fig_results_brainData}
\end{figure}
An experiment with the pseudo radial sampling pattern was conducted, and results were shown in Fig. \ref{fig_results_brain_typical_2}. All tested approaches provide good reconstructions with relatively small error. The overall error of GRAPPA reconstruction seems to be the biggest one, but its error inside the skull has nice random distribution, which would be preferred in applications. $\ell_1$-SPIRiT, ALOHA and STDLR-SPIRiT errors have reached the promising low level, assuring reliable reconstructions. It is worthy of mentioning that STDLR-SPIRiT outperforms compared methods interns of RLNE and MSSIM, and obtains the smallest reconstruction error, which serves as another demonstration for the state-of-the-art performance of the proposed method. Other undersampling patterns and brain images in Fig. \ref{fig_results_brainData} are also used for reconstruction for further validation and the RLNE and MSSIM metrics are summarized in Table \ref{Table 4}. Superior metrics provided by the proposed method imply better reconstructed images.

\begin{table}[htbp]
\footnotesize
\begin{center}
  \centering
  \caption{RLNE/MSSIM Results for Brains in Fig. \ref{fig_results_brainData} Using Different Samplings.}
  \label{Table 4}
    \begin{tabular}{m{0.8cm}<{\centering}m{0.8cm}<{\centering}m{0.8cm}<{\centering}m{1.3cm}<{\centering}m{0.8cm}<{\centering}m{1.0cm}<{\centering}}
    \toprule
    \multicolumn{1}{c}{Images} & \multicolumn{1}{c}{Pattern} & GRAPPA & $\ell_1$-SPIRiT & ALOHA & STDLR-SPIRiT \\
    \midrule
    \multicolumn{1}{c}{\multirow{6}[6]{*}{Fig.10(a)}} & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(d)}} & \multicolumn{1}{c}{0.1399} & \multicolumn{1}{c}{0.1110} & \multicolumn{1}{c}{0.1232} & \multicolumn{1}{c}{\textbf{0.0904}} \\
          &       & /0.9626 & /0.9799 & /0.9754 & \textbf{/0.9883} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(e)}} & \multicolumn{1}{c}{0.1222} & \multicolumn{1}{c}{0.1013} & \multicolumn{1}{c}{0.1370} & \multicolumn{1}{c}{\textbf{0.0879}} \\
          &       & /0.9704 & /0.9798 & /0.9722 & \textbf{/0.9869} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(f)}} & \multicolumn{1}{c}{0.1540} & \multicolumn{1}{c}{0.1013} & \multicolumn{1}{c}{0.1103} & \multicolumn{1}{c}{\textbf{0.0832}} \\
          &       & /0.9523 & /0.9788 & /0.9764 & \textbf{/0.9878} \\
    \midrule
    \multicolumn{1}{c}{\multirow{6}[6]{*}{Fig.10(b)}} & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(d)}} & \multicolumn{1}{c}{0.1854} & \multicolumn{1}{c}{0.1115} & \multicolumn{1}{c}{0.1132} & \multicolumn{1}{c}{\textbf{0.0903}} \\
          &       & /0.8665 & /0.9518 & /0.9490 & \textbf{/0.9682} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(e)}} & \multicolumn{1}{c}{0.1260} & \multicolumn{1}{c}{0.1005} & \multicolumn{1}{c}{0.1103} & \multicolumn{1}{c}{\textbf{0.0846}} \\
          &       & /0.9259 & /0.9632 & /0.9617 & \textbf{/0.9754} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(f)}} & \multicolumn{1}{c}{0.1706} & \multicolumn{1}{c}{0.1018} & \multicolumn{1}{c}{0.1011} & \multicolumn{1}{c}{\textbf{0.0851}} \\
          &       & /0.8693 & /0.9574 & /0.9584 & \textbf{/0.9725} \\
    \midrule
    \multicolumn{1}{c}{\multirow{6}[6]{*}{Fig.10(c)}} & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(d)}} & \multicolumn{1}{c}{0.1306} & \multicolumn{1}{c}{0.0830} & \multicolumn{1}{c}{0.0916} & \multicolumn{1}{c}{\textbf{0.0779}} \\
          &       & /0.9362 & /0.9883 & /0.9830 & \textbf{/0.9902} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(e)}} & \multicolumn{1}{c}{0.1161} & \multicolumn{1}{c}{0.0864} & \multicolumn{1}{c}{0.1073} & \multicolumn{1}{c}{\textbf{0.0822}} \\
          &       & /0.9624 & /0.9880 & /0.9827 & \textbf{/0.9905} \\
\cmidrule{2-6}          & \multicolumn{1}{c}{\multirow{2}[2]{*}{Fig. 10(f)}} & \multicolumn{1}{c}{0.1269} & \multicolumn{1}{c}{0.0859} & \multicolumn{1}{c}{0.0952} & \multicolumn{1}{c}{\textbf{0.0807}} \\
          &       & /0.9437 & /0.9876 & /0.9851 & \textbf{/0.9908} \\
    \bottomrule
    \end{tabular}%
    \end{center}
\end{table}%

\section{Discussions}\label{Section:discussions}
\subsection{Discussion on ACS}
The proposed STDLR-SPIRiT is robust to the number of ACS lines. As shown in Fig. \ref{fig_results_ACS_RLNE} (a), the reconstruction errors - RLNE - of STDLR-SPIRiT the smallest among all methods. The SPIRiT monotonically reduces reconstruction errors when the number of ACS lines increases. When ACS signals ($>14$ lines) are enough, the $\ell_1$-SPIRiT achieves much lower error than GRAPPA and ALOHA. These observations indicate that the SPIRiT effectively mines the self-consistency of k-space via ACS, resulting in improved image reconstruction. However, with a decrease of ACS lines ($<14$ lines), the accuracy of kernels estimated in the $\ell_1$-SPIRiT would decrease, leading to a higher RLNE than that of ALOHA. This observation suggests that low-rankness in ALOHA is a very valuable property to regularize image reconstructions and may alleviate the influence of the decline of ACS numbers. The proposed STDLR-SPIRiT inherits advantages of SPIRiT and ALOHA, leading to the lowest reconstruction error under all the number of ACS lines.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig11}
\caption{The variation trends of RLNE (a) and MSSIM (b) versus the number of ACS lines. This experiment is carried out with a series of Gaussian Cartesian sampling patterns but all at the same sampling rate of 0.34. The difference between each sampling pattern is the number of ACS lines. Note: all experiments are conducted on the brain data shown in Fig. \ref{fig_results_brain_typical} (a).}
\label{fig_results_ACS_RLNE}
\end{figure}

\subsection{Discussion on Parameter Settings}
In this subsection, the effect of parameters setting in STDLR-SPIRiT will be discussed, including regularization parameters ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$ as well as pencil parameters ${{k}_{1}}$ and ${{k}_{2}}$.
The brain image in Fig. \ref{fig_results_brain_typical} (a) and the sampling pattern in Fig. \ref{fig_results_brain_typical} (f) were used to perform reconstructions. Typical settings are ${{\lambda }_{1}}={{10}^{4}}$, ${{\lambda}_{2}}={{10}^{6}}$ and ${{k}_{1}}={{k}_{2}}=23$. When one parameter is analyzed, other parameters are set as the typical values.

The reconstruction errors versus different values of regularization parameters are depicted in Fig. \ref{fig_results_parameter_setting} (a). The selection of ${{\lambda}_{1}}$ and ${{\lambda}_{2}}$ relies on noise level and the amount of ACS data for kernel estimation. As can be seen in Fig. \ref{fig_results_parameter_setting} (a), under fixed noise level and sampling pattern, there exists a wide range of ${{\lambda}_{1}}$ (${{10}^{1}}\le {{\lambda}_{1}}\le {{10}^{5}}$) and ${{\lambda}_{2}}$ (${{10}^{4}}\le {{\lambda}_{2}}\le {{10}^{7}}$) leading to relatively low reconstruction errors. Too small or too large values of ${{\lambda}_{1}}$ and ${{\lambda}_{2}}$ produce higher reconstruction errors. When the amount of data for SPIRiT kernel estimation are not adequate, we suggest choosing a relatively small value for ${{\lambda}_{1}}$ to alleviate the effect of less accurate kernel estimation. In this paper, we set ${{\lambda }_{1}}={{10}^{4}}$ and ${{\lambda}_{2}}={{10}^{6}}$ for all experiments.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.2in]{./eps/Fig12}
\caption{Variation trends of reconstruction errors by STDLR-SPIRiT with different parameter settings. (a) The RLNE versus regularization parameters; (b-c) the RLNE and MSSIM versus pencil parameters. Note: all experiments are conducted on the brain data shown in Fig. \ref{fig_results_brain_typical} (a) under Gaussian Cartesian undersampling pattern shown in Fig. \ref{fig_results_brain_typical} (f).}
\label{fig_results_parameter_setting}
\end{figure}
The values of ${{k}_{1}}$ and ${{k}_{2}}$ determine the size of the block Hankel matrix, which in turn affect the rank of the matrix. The rank of block Hankel matrix is related to the sparsity level of transform domain. When ${{k}_{1}}$ and ${{k}_{2}}$ are too small (for example, less than $13$ here), the rank of the block Hankel matrix is smaller than the sparsity level, leading to larger reconstruction errors, as shown in Figs. \ref{fig_results_parameter_setting} (b) and (c). When ${{k}_{1}}$ and ${{k}_{2}}$ are bigger than the sparsity level of the transform domain, the reconstruction errors are not sensitive to ${{k}_{1}}$ and ${{k}_{2}}$ values. However, big values of ${{k}_{1}}$ and ${{k}_{2}}$ cost large computation load. Therefore, there is a tradeoff between reconstruction quality and computation load, we empirically set ${{k}_{1}}$ and ${{k}_{2}}$ in the range from 21 to 25.

\subsection{Algorithm Convergence}
We adopted ADMM \cite{2011_ADMM} to solve the non-convex STDLR-SPIRiT model in \ref{(16)}.  It has been shown that ADMM can be used for successfully solving non-convex optimization problems \cite{2016_ALOHA,2018_TBME_Hengfa,2017_ACCESS_Guo}. Empirically, we observed that RLNE decreases to a relatively small value as iteration increases (e.g. the point C depicted in \ref{fig_convergence}(a)), and then stabilizes in a small RLNE level. For example, as the iteration rises (from point A to C), proposed approach achieving increasingly better reconstruction, and eventual provides reliable reconstruction (\ref{fig_convergence}(c-d)).

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig13}
\caption{Empirical convergence results. (a) is the empirical convergence of STDLR-SPIRiT; (b-e) are reconstructions of STDLR-SPIRiT at A to D iterations depicted in (a), respectively.}
\label{fig_convergence}
\end{figure}

\subsection{Comparison with Other State-of-the-art Methods}
In this sub-section, we compare the performance of the proposed method with other state-of-the-art low-rank structured matrix approaches AC-LORAKS(S) \cite{2015_AC_LORAKS} and GSLR \cite{2018_Hu_LR_TMI} on two data set.

\begin{figure}[htbp]
\setlength{\abovecaptionskip}{0.cm}
\setlength{\belowcaptionskip}{-0.cm}
\centering
\includegraphics[width=3.4in]{./eps/Fig14}
\caption{Reconstructions of multi-coils brain images. (a) An SSOS image of fully sampled image; (b-d) SSOS images of reconstructed images by AC-LORAKS(S), GSLR and STDLR-SPIRiT; (e) the pseudo radial undersampling pattern with a sampling rate of 0.20; (f-h) the reconstruction error distribution corresponding to the above methods; (i) an SSOS image of the fully sampled image; (j-l) SSOS images of reconstructed images by AC-LORAKS(S), GSLR and STDLR-SPIRiT; (m) the 2D random undersampling pattern with a sampling rate 0.18; (n-p) the reconstruction error distribution corresponding to the above methods. Note: The first tested image and the corresponding pseudo pattern were from Fig. \ref{fig_results_brain_typical_2}. The second tested image and the corresponding 2D random pattern were from Fig. \ref{fig_results_brainData}.}
\label{fig_results_otherMethods}
\end{figure}
Here we choose AC-LORAKS(S) instead of other LORAKS variants since AC-LORAKS and the proposed method both exploit the calibration data, but, please note AC-LORAKS uses it for only algorithm acceleration. The letter 'S' in the bracket of AC-LORAKS(S) denotes the applying of phase constraint in AC-LORAKS. It has been shown that LORAKS with phase constraint produces the best result compared with other constraints \cite{2014_LORAKS}, such as, support constraint. Regarding of GSLR, it cascades the weighting matrices on the horizontal and vertical directions into a fatty low-rank matrix, which increases the computational complexity, and suggests that imposing simultaneous weighting on multi-coils images would be difficult because the resulting matrix would be of vast dimension. Note that the GSLR code shared by the author reconstructs multi-coils images channel-by-channel, that is to say, GSLR makes no use of the correlations among coils. This indicates GSLR reconstruction may be inferior to that of STDLR-SPIRiT.

Reconstructions of the tested images were presented in Fig. \ref{fig_results_otherMethods}, in which AC-LORAKS(S), GSLR and STDLR-SPIRiT all provide images with nice artifacts suppression. However, apparent noise could be inspected in AC-LORAKS(S) reconstruction (Figs. \ref{fig_results_otherMethods}(b) and (j)), but this may not be a bad thing because it allows for a better visualization of the image. In addition, blurring of image can be observed in GSLR reconstruction (Figs. \ref{fig_results_otherMethods}(g) and (k)). Whilst STDLR-SPIRiT produces promising reconstruction with satisfying noise level and resolution (Figs. \ref{fig_results_otherMethods}(d) and (i)). The error images shown in Figs. \ref{fig_results_otherMethods} (f-h) and (n-p) reveals STDLR-SPIRiT embracing the lowest structure error compared with AC-LORAKS(S) and GSLR. The GSLR yielding obvious error compared with the proposed method verifies the aforementioned discussion about GSLR.

\section{Conclusion}\label{Section:conclusion}
A simultaneous two-directional low-rank structured Hankel matrix reconstruction approach, STDLR-SPIRiT, is proposed for accelerated data acquisition in parallel MRI. The proposed approach simultaneously utilizes the self-consistency of k-space data and the low-rankness of the weighted k-space data, which yields a better exploitation of correlation among inter- and intra-coils. The STDLR-SPIRiT provides robust reconstruction with a limited number of auto-calibration signals. Experiments on phantom and brain MRI demonstrate that the proposed approach achieves superior performance in artifacts suppression and edge preservation than the state-of-the-art methods. Although a singular value decomposition-free algorithm is derived to reduce the computation load, faster algorithms are still expected in the future \cite{2014_LORAKS,2016_Off_the_grid}. Besides, the low rank Hankel matrix is highly related to exponential functions, and latest reconstruction methods of multi-dimensional exponentials \cite{2018_Vandermonde_Jiaxi, 2017_TSP_Jiaxi} may improve the Hankel-based MRI reconstruction or be extended into higher dimensional MRI in fast imaging.

\section*{Acknowledgments}
Xiaobo Qu is grateful to Prof. Chun Yuan for hosting his visit at University of Washington. This work was supported in part by National Key R\&D Program of China (2017YFC0108703), National Natural Science Foundation of China (61571380, 61871341, 61811530021, U1632274, 61672335 and 61671399), Natural Science Foundation of Fujian Province of China (2018J06018), Fundamental Research Funds for the Central Universities (20720180056), Science and Technology Program of Xiamen (3502Z20183053), and China Scholarship Council (201806315010).

\begin{table}[htbp]
  \footnotesize
  \centering
  \caption{PARAMETERS FOR GRAPPA.}
    \begin{tabular}{m{0.8cm}<{\centering}m{1.4cm}<{\centering}m{1.3cm}<{\centering}m{2.3cm}<{\centering}}
    \toprule
    \multicolumn{1}{c}{Images} & Pattern & Kernel size & Kernel calibration parameters \\
    \midrule
    \multicolumn{1}{c}{Fig. 6} & /     & $9 \times 9$   & 0.03 \\
    \midrule
    \multicolumn{1}{c}{Fig. 7} & /     & $9 \times 9$   & 0.05 \\
    \midrule
    \multicolumn{1}{c}{Fig. 8} & /     & $9 \times 9$   & 0.05 \\
    \midrule
    \multicolumn{1}{c}{Fig. 9} & /     & $5 \times 5$   & 0.10 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (a)}} & Fig. 10 (d) & $9 \times 9$   & 0.05 \\
    \cmidrule{2-4}          & Fig. 10 (e) & $9 \times 9$   & 0.10 \\
    \cmidrule{2-4}          & Fig. 10 (f) & $7 \times 7$   & 0.10 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (b)}} & Fig. 10 (d) & $5 \times 5$   & 0.03 \\
    \cmidrule{2-4}          & Fig. 10 (e) & $5 \times 5$   & 0.10 \\
    \cmidrule{2-4}          & Fig. 10 (f) & $5 \times 5$   & 0.03 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (c)}} & Fig. 10 (d) & $9 \times 9$   & 0.10 \\
    \cmidrule{2-4}          & Fig. 10 (e) & $5 \times 5$   & 0.10 \\
    \cmidrule{2-4}          & Fig. 10 (f) & $5 \times 5$   & 0.10 \\
    \bottomrule
    \end{tabular}%
  \label{Table 5}%
\end{table}%

\begin{table}[htbp]
  \footnotesize
  \centering
  \caption{PARAMETERS FOR $\ell_1$-SPIRIT.}
    \begin{tabular}{m{0.8cm}<{\centering}m{1.4cm}<{\centering}m{0.8cm}<{\centering}m{1.5cm}<{\centering}m{1.5cm}<{\centering}}
    \toprule
    \multicolumn{1}{c}{Images} & Sampling pattern & Kernel size & Kernel calibration parameter & Wavelet regularization parameter \\
    \midrule
    \multicolumn{1}{c}{Fig. 6} & /     & $9 \times 9$   & 0.0010 & 0.0010 \\
    \midrule
    \multicolumn{1}{c}{Fig. 7} & /     & $7 \times 7$   & 0.0030 & 0.0015 \\
    \midrule
    \multicolumn{1}{c}{Fig. 8} & /     & $7 \times 7$   & 0.0030 & 0.0015 \\
    \midrule
    \multicolumn{1}{c}{Fig. 9} & /     & $5 \times 5$   & 0.0030 & 0.0005 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (a)}} & Fig. 10 (d) & $5 \times 5$   & 0.0050 & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (e) & $7 \times 7$   & 0.0050 & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (f) & $5 \times 5$   & 0.0010 & 0.0015 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (b)}} & Fig. 10 (d) & $7 \times 7$   & 0.0010 & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (e) & $5 \times 5$   & 0.0050 & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (f) & $5 \times 5$   & 0.0010 & 0.0015 \\
    \midrule
    \multicolumn{1}{c}{\multirow{3}[6]{*}{Fig. 10 (c)}} & Fig. 10 (d) & $5 \times 5$   & 0.0030 & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (e) & $7 \times 7$   & 0.0300  & 0.0015 \\
    \cmidrule{2-5}          & Fig. 10 (f) & $5 \times 5$   & 0.0050  & 0.0015 \\
    \bottomrule
    \end{tabular}%
  \label{Table 6}%
\end{table}%

\section*{Supplementary Material}\label{SI}
We ran reconstruction experiments using compared approaches with a series of combinations of model-specific parameters subjecting to each tested data, and the parameters of each method allowing the lowest RLNEs were chosen. The detail parameter settings of GRAPPA and $\ell_1$-SPIRiT are listed in TABLE \ref{Table 5} and TABLE \ref{Table 6}. As for ALOHA, two levels of pyramidal decomposition are adopted with LMaFit tolerances $10^{-1}$, $10^{-2}$. For our proposed method, we set ${k_1} = {k_2} = 23$, ${\lambda _1} = {10^4}$ and ${\lambda _2} = {10^6}$ for all experiments.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


I am on page 145 of the book Women in Clothes! Hooray. If you feel so inclined, you can read my entire interview here. Thank you Sheila, Leanne, and Heidi for inluding me.
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And this is my sweet Lea reading my teeny tiny contribution.


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Featured Image Of The Week
The Freedmen's Hospital was founded in 1862 in Washington, DC. It was the first hospital of its kind to aid in the medical treatment of former slaves. Later it became the major hospital for the African American community in Washington, D.C. The hospital was founded on the grounds of Camp Barker at 13th and R Streets in Northwest Washington. It remained in that location until a new building was completed in 1909 at Bryant and 6th Street. Through much of its history the hospital was managed by the U.S. government.
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text:
The Trump-Putin bromance reshaping the world
From Russia with love: Trump said "there was no collusion at all" to sway the election. © Getty
Was the summit a good sign? Donald Trump's meeting with Vladimir Putin symbolised a new form of competitive, personality-led diplomacy that could upend the established world order.
"Our relationship has never been worse than it is now. However, that changed as of about four hours ago." That was President Donald Trump's extraordinary message as he stood beside Russia's president, Vladimir Putin, following a controversial summit in Helsinki yesterday.
The agenda for the meeting had been vague, in line with Trump's off-the-cuff approach to diplomacy. In two hours of one-on-one talks they discussed Syria, denuclearisation, and their shared desire to improve ties between the US and Russia. Putin declared the Cold War "a thing of the past".
Trump was under pressure to confront Putin over Russia's alleged meddling in the 2016 presidential election, for which 12 Russian intelligence officers were indicted by the US Justice Department just last week. Trump, however, said he believed Putin's "powerful" denial and did not see "any reason why" Moscow would be responsible for the attack.
His comments have been met with disbelief by the US and its allies, who perceive the US president as willing to turn against his own security officials for the sake of smoother relations with Russia.
Those Western allies, whose ties with Moscow have collapsed since the annexation of Crimea, will be watching nervously.
On Sunday, in a shock to the established order, Trump called the EU a "foe" for "what they do to us in trade". Famed as a deal-maker, Trump appears keen to put economics ahead of traditional ties, describing Putin as a "competitor" rather than an enemy.
Trump has also rejected NATO and the UN in favour of carrying out diplomacy through high-profile summits with strongman leaders such as Kim Jong-un and Xi Jinping.
Yesterday, the president of the European Council, Donald Tusk, declared that "the architecture of the world is changing" and he called on major powers to work together to "prevent conflict and chaos".
Meanwhile, Russian politician Valentina Matvienko said the summit was of "huge significance… for global stability."
Was the summit a good sign?
Not at all, say some. Trump is doing Putin's work for him, sowing division in the West and destabilising international structures. His habit of leaning towards strongman leaders over those elected democratically is a very bad sign. Furthermore, Trump puts more faith in Putin, an autocrat with a history of human rights abuses, than his own intelligence services.
Yes, argue others. The US and Russia hold 90% of the world's nuclear weapons, so improved relations can only be a good thing. International institutions are inefficient and the traditional diplomacy of to-ing and fro-ing doesn't get anything done. Real change is best achieved when nations pursue their interests directly.
Is this summit good or bad for the world?
Should the US honour its historical alliances?
Watch a video of Trump's press conference with Putin. Looking at their body language and listening to what they say, write a paragraph on who you think came out on top.
Research Trump's interactions with Russia, China and North Korea, and compare them with his interactions with NATO and the EU. Write an essay on the statement: "Trump's approach to diplomacy is the way forward."
"The supreme art of war is to subdue the enemy without fighting."
Donald Trump and Vladimir Putin met at Finland's presidential palace in Helsinki yesterday. Putin arrived almost an hour late, in what has been widely seen as a power play. During their first interaction, Trump appeared to wink at Putin. Politicians in the US have rushed to condemn Trump for siding with Russia against the FBI. Paul Ryan, the speaker of the House, called on the president to recognise that "Russia is not our ally."
Whether the summit will lead to any concrete outcomes. The agenda for the meeting was vague and the two sides did not make any firm agreements. While Trump and Putin may have better relations, Trump's refusal to criticise Putin over the Russian hacking scandal has caused huge division in the US. It is unclear how this will play out.
A time of political hostility between Western powers and the Soviet Union. It started in 1947 and lasted until the latter's collapse in 1991. The term "cold" is used because there was no large-scale fighting between the two sides, but tensions led many to fear the outbreak of nuclear war.
To formally accuse or charge with a crime. The Russians are accused of hacking into the emails of the Democratic Party — whose candidate was Hillary Clinton — in order to spread damaging information and boost the Trump campaign.
Annexation of Crimea
In 2014, pro-Russian activists and disguised troops took control of Crimea, a self-governing republic that had been part of Ukraine since 1954.
In trade
The EU has imposed trade tariffs on some imports from the US in response to the Trump administration's decision to introduce heavy taxes on aluminium and steel from the EU.
NATO and the UN
Trump has pulled the US out of the United National Human Rights Council. He also called NATO, a military alliance of western nations, "obsolete".
Autocrat
A ruler who has absolute power.
Key moments from the Trump-Putin press conference. Guardian News — YouTube. (3:09)
"Trump sides with Russia against the FBI at Helsinki summit." BBC News . (900 words)
Trump's summit with Putin condemned as "treasonous". The Guardian . (1,100 words)
Trump says "we're all to blame" for poor US-Russia relations. CNN . (600 words)
"IMF warns Trump trade war could cost global economy $430 billion." The Guardian . (700 words)
Gideon Rachman assesses Trump's attitude to foreign policy. The Financial Times . (900 words) (Paywall)
Golf, Queen, demos, Chequers: Trump's in town


Company directory - T 'n T SERVICES, INC. through to T @ COATING PTE. LTD.
This is an alphabetical listing of UK companies showing T 'n T SERVICES, INC. through to T @ COATING PTE. LTD. . This listing is for companies whose name starts with the character T. These companies are on the company registrar and therefore registered with the UK Government organisation, Companies House. This list covers private limited companies, public limited companies, limited liability partnerships and registered overseas organisations. Sole traders will not be listed here. Why is this useful?
T 'n T SERVICES, INC. FAIRFAX, VA T 'n T SERVICES, INC. was formed 28 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N T, INC. BRADENTON, FL T 'N T, INC. was formed 63 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N T, INC. , FL T 'N T, INC. was formed 49 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' A CHARTERS LLC SUITE 220 HOUSTON, Texas T 'N' A CHARTERS LLC was formed 4 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' A ENTERPRISES, LLC LODI, CA T 'N' A ENTERPRISES, LLC was formed 15 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' C TRUCKING CARSON CITY, NV T 'N' C TRUCKING was formed 20 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' F DESIGNS LIMITED CO. MEATH, T 'N' F DESIGNS was formed 10 years ago. We do not have information about what type of business this was. The business is now dissolved and no longer trading.
T 'N' L ADVENTURES INC. Central City, CO T 'N' L ADVENTURES INC. was formed 21 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T CONCRETE CONSTRUCTION, INC. KENT, WA T 'N' T CONCRETE CONSTRUCTION, INC. was formed 22 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T CUSTOM TILE, LLC NEW SMYRNA BEACH, FL T 'N' T CUSTOM TILE, LLC was formed 10 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'n' T Enterprises Inc. Littleton, CO T 'n' T Enterprises Inc. was formed 39 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T GROWERS, INC. LAKE WORTH, FL T 'N' T GROWERS, INC. was formed 33 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T INVESTMENTS, LLC EVERETT, WA T 'N' T INVESTMENTS, LLC was formed 14 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T INDUSTRIES, INC. CLEARWATER, FL T 'N' T INDUSTRIES, INC. was formed 33 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T REAL ESTATE EXCHANGORS, INC. COCOA BEACH, FL T 'N' T REAL ESTATE EXCHANGORS, INC. was formed 33 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T TRANSLATIONS AND INTERPRETING, CORPORATION CAPE CORAL, FL T 'N' T TRANSLATIONS AND INTERPRETING, CORPORATION was formed 11 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T 'N' T WINES LIMITED ROMSEY, HAMPSHIRE T 'N' T WINES was formed 8 years ago. Before closing-down, its business was 'Retail sale of beverages in specialised stores'. T 'N' T WINES was dissolved on the 2016-12-20 and the company has ceased trading.
T "LC" FORT WALTON BEACH, FL T "LC" was formed 14 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T "N" J INC. ALTHA, FL T "N" J INC. was formed 20 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T "N" S HOME INSPECTIONS, LLC APPLE VALLEY, CA T "N" S HOME INSPECTIONS, LLC was formed 7 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T "n" T Arvada, CO T "n" T was formed 12 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T "S AFFORDABLE LAWN CARE & PRESSURE WASHING SERVICE INC LAKE WALES, FL T "S AFFORDABLE LAWN CARE & PRESSURE WASHING SERVICE INC was formed 2 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.
T (2013) LIMITED BIRMINGHAM, WEST MIDLANDS T (2013) was formed 7 years ago. Before closing-down, its business was 'Take-away food shops and mobile food stands'. T (2013) was dissolved on the 2015-09-29 and the company has ceased trading.
T (2688) LIMITED BIRMINGHAM, WEST MIDLANDS T (2688) was formed 6 years ago. Before closing-down, its business was 'Take-away food shops and mobile food stands'. T (2688) was dissolved on the 2015-04-28 and the company has ceased trading.
T (280) LTD BIRMINGHAM, WEST MIDLANDS T (280) was formed 8 years ago. Before closing-down, its business was 'Take-away food shops and mobile food stands'. T (280) was dissolved on the 2015-04-07 and the company has ceased trading.
T (3018) LIMITED BIRMINGHAM, WEST MIDLANDS T (3018) was formed 5 years ago. The company did not declare the nature of its business prior to the business closing. T (3018) was dissolved on the 2015-09-01 and the company has ceased trading.
T (360) LIMITED BIRMINGHAM, WEST MIDLANDS T (360) was formed 4 years ago. The company did not declare the nature of its business prior to the business closing. T (360) was dissolved on the 2016-02-02 and the company has ceased trading.
T (368) LIMITED WADEBRIDGE, CORNWALL T (368) was formed 6 years ago. Before closing-down, its business was 'Take-away food shops and mobile food stands'. T (368) was dissolved on the 2017-12-12 and the company has ceased trading.
T (383) LIMITED BIRMINGHAM, ENGLAND T (383) was formed 5 years ago. We do not have information about what type of business this was. T (383) was dissolved on the 2016-05-17 and the company has ceased trading.
T (390) LIMITED BIRMINGHAM, WEST MIDLANDS T (390) was formed 5 years ago. We do not have information about what type of business this was. T (390) was dissolved on the 2016-05-17 and the company has ceased trading.
T (3998) LIMITED BIRMINGHAM, WEST MIDLANDS T (3998) was formed 5 years ago. We do not have information about what type of business this was. T (3998) was dissolved on the 2016-04-12 and the company has ceased trading.
T (6338) LIMITED BIRMINGHAM, WEST MIDLANDS T (6338) was formed 5 years ago. We do not have information about what type of business this was. T (6338) was dissolved on the 2016-04-19 and the company has ceased trading.
T (8866) LIMITED BIRMINGHAM, WEST MIDLANDS T (8866) was formed 4 years ago. The company did not declare the nature of its business prior to the business closing. T (8866) was dissolved on the 2016-01-05 and the company has ceased trading.
T (983) LIMITED STOKE-ON-TRENT, ENGLAND T (983) was formed 5 years ago. The company did not declare the nature of its business prior to the business closing. T (983) was dissolved on the 2015-06-09 and the company has ceased trading.
T @ BAILEY INDOOR-OUTDOOR LIVING LIMITED BOLTON, LANCASHIRE T @ BAILEY INDOOR-OUTDOOR LIVING was formed 1 years ago. We do not have information about what type of business this is. The business is active however there is a proposal to strike off the company register - possibly for failing to comply with filing rules.
T @ C CHASON ENTERPRISES, LLC TALLAHASSEE, FL T @ C CHASON ENTERPRISES, LLC was formed 15 years ago. We do not have information about what type of business this is. The company seems to be no longer trading or is non compliant.


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Obituary- Alice Angelina Veronda
Private family services are planned for Alice Angelina Veronda, who died Sept. 17, 2001, at Barton Hospital Skilled Nursing and Rehabilitation Facility after a brief illness. She was 85.
Mrs. Veronda was born July 20, 1916, at Los Gatos, Calif., to prominent Petaluma, Calif., civic and business leader Hiram Hopkins and his wife Elvira. She graduated from Petaluma High School and attended the then College of the Pacific at Stockton, Calif., before marrying and moving to Hawaii. In the small plantation community of Honokaa on the island of Hawaii Mrs. Veronda began her teaching career when the only teacher in a one-room schoolhouse fell ill and she was asked to replace him.
She returned in the 1940s to her home town, where she was co-founder of an innovative cooperative nursery school which drew on parents as instructors and aides. Her teaching position was in the Cotati Cooperative Nursery School. For more than two decades Mrs. Veronda was a frequent visitor to South Lake Tahoe, where her daughter and son-in-law, Lona and Fred Roberts, have lived since 1978. She had been a resident of South Lake Tahoe for the past 14 months, after giving up her homes in Cotati, Calif., and Kaiua, Hawaii.
Mrs. Veronda's ability to see beauty in the smallest detail of any environment was a talent that served her well when Alzheimer's Disease forced her to literally live "in the moment." She was a retired teacher who found beauty even in a world dimmed by the disease. The caring and humane staff of the Skilled Nursing Facility helped her enjoy a secure and pleasant life even though she was confined to a wheelchair by arthritis.
Survivors include her son and daughter-in-law, Walter and Jane Wright of Honolulu, Hawaii; daughter and son-in-law, Lona and Fred Roberts of South Lake Tahoe; grandchildren, Kathy Wright and David Wright, both of Seattle, Lori Roberts of Oroville, Calif., and James and Joseph Roberts of Los Angeles; and great-grandson, Henry Fanning of Seattle.
The family will conduct private services Sept. 22 at the Roberts home in South Lake Tahoe.
Donations in Mrs. Veronda's memory may be made, for research, to the Alzheimer's Association, 919 North Michigan Ave., Suite 1100, Chicago, Ill. 60611-1676, and, for programs, to Alzheimer's Association Northern Nevada Chapter, P.O. Box 6362, Reno, NV 89513.