text:
\section{Introduction}
\label{sec::intro}
\input{intro}

\section{Methodology}
\label{sec::method}
\input{method}

\section{Experiments}
\label{sec::experiments}
\input{experiments}

\section{Conclusions}
\label{sec::conclusions}
\input{conclusion}

\section*{Acknowledgments}
We thank Dr. Michael McCann at Los Alamos National Laboratory, and Shijun Liang and Siddhant Gautam at Michigan State University for helpful discussions. We thank all reviewers for their constructive comments on our initial draft.

\bibliographystyle{alpha}

\subsection{Implementation Details}
\paragraph{\texttt{MNet}{} structure}
The \texttt{MNet}{} mask predictor starts with a double-convolution block that is followed by four downsampling blocks, which are used in the classic Unet, and ends with four fully-connected layers. The input to the \texttt{MNet}{} is the initially collected appropriately zero-padded limited (low-frequency) $k$-space information with real and imaginary parts separated into two channels, and the output is the sampling decision with respect to unobserved (higher)
frequencies. The design of \texttt{MNet}{} consists of two parts: the encoding convolutional layers that emulate the encoder part in the Unet structure, and fully connected feedforward layers which originated in classification networks.
\begin{figure*}[!ht]
  \centering
  \includegraphics[width=\textwidth]{figures/mnet.pdf}
  \caption{Structure of \texttt{MNet}{} used in training. The kernel of 2D convolution is of size $3\times 3$, stride and dilation of sizes $1\times 1$ and no padding. The Double-Conv layers maintain the size of the image for its input and output. The kernel size of the MaxPool layer is $3\times3$, and the kernel size of the AvgPool layer is $2\times 2$.}
  \label{fig::mnet_structure}
\end{figure*}

\paragraph{Reconstructors} For Unet reconstructors, we adopt a standard 8-block Unet architecture following~\cite{2D_unet}. When carrying out the \texttt{mask-backward}{} training, both the input to and output from the reconstructor are single-channel real-valued images (to correspond with magnitude of reconstructions).
For the input, we take the magnitude of the inverse FFT initial reconstruction.
In the separate training process for individual reconstructors, the input images have 2 channels (for a complex-valued image, the real and imaginary parts are separated into 2 channels), which led to slightly better performance,
and the outputs are single-channel real-valued images. The Unet reconstructor contains four downsampling blocks and four upsampling blocks, each consisting of two $3\times3$ convolutions separated by ReLU and instance normalization. We let the first downsampling block to have 64 channels which are expanded from the input 1 or 2 channels. The Unet used in the \texttt{mask-backward}{} process has the residual structure with skip connections, while the Unet used in the separate training cases does not.

We also briefly review the idea of the MoDL reconstructor~\cite{JMoDL_20}. The MoDL reconstructor consists of a neural-network denoiser and a data-consistency-enforcing block. The image reconstruction problem the MoDL reconstructor tackles is of the form $\widehat{\mathbf{x}} = \mathrm{arg}\,\min_{\mathbf{x}} \|\mathbf{y} - \mathbf{A}\mathbf{x}\|_\mathrm{F}^2 + \|\mathbf{x} - \mathcal{D}_{\Phi}(\mathbf{x})\|_\mathrm{F}^2$, where $\mathcal{D}_\Phi(\cdot)$ is a neural-network denoiser. The two successive or alternating steps in implementing the MoDL reconstructor are specifically $\mathbf{x}_{n+1} = (\mathbf{A}^\mathrm{H}\mathbf{A} + \mathbf{I})^{-1} (\mathbf{z}_n + \mathbf{A}^\mathrm{H}\mathbf{y})$ (the data-consistency-enforcing block) and $\mathbf{z}_{n+1}=\mathcal{D}_{\Phi}(\mathbf{x}_{n+1})$ (the denoising step), where $\mathbf{A}^\mathrm{H}$ is the Hermitian of operator $\mathbf{A}$. One typically uses the conjugate gradient (CG) algorithm to compute the inverse in the data-consistency-enforcing block. In the implementation of MoDL in this work, we set the MoDL reconstructor to have 4 MoDL blocks owing to the consideration of computation complexity in the hardware and the error tolerance of the CG algorithm is set to be $5\times 10^{-5}$. The neural-network denoiser used in the MoDL reconstructor is a Unet as previously described with 64 channels in the first downsampling block without residual structure.

\paragraph{Dataset} We used the single-coil data in the NYU FastMRI dataset~\cite{fastMRI_dataset_18,fastMRI_journal} for training and testing. In each file, we selected the middle 6 slices containing most prominent image patterns
in the knee scan, thus making up a training dataset with 12649 images, and 1287 images were used for validation and 1300 images were used for testing (we follow the setup of~\cite{RL_sampling_20} and split the original fastMRI validation set into a new validation set and a test set with the same amount of volumes). Each slice has the dimension $320\times 320$, and so does its corresponding $k$-space.

\paragraph{Baselines} With the same sampling ratio and low-frequency base (initial) observations, We benchmark our adaptive sampling \texttt{MNet}{} framework against the recent LOUPE sampler + reconstructor scheme, a random sampler, an equi-distance sampler, and an energy-density based sampler~\cite{energy_sampling}. For the energy-density based sampler, we constructed the sampling probability distribution as follows. With respect to each image, we compute the energy of each row of its $k$-space and normalize the energy vector for that image. We then take the average of all normalized energy vectors for each image in the training dataset as the energy-based sampling probability distribution. We observed that the sampling probabilities from the energy-based density heavily concentrate in the low-frequency regime.
We compare the performance of different methods for 4$\times$ and 8$\times$ acceleration of $k$-space sampling, respectively. For a fair comparison to the random and equi-distance samplers, corresponding reconstructors are trained for both these cases.

\paragraph{Hyper-parameters} We used the normalized $\ell_2$ loss and the structural similarity index measure (SSIM) for measuring the image reconstruction quality. SSIM is computed using a window size of $11\times 11$ and hyperparameters $k_1=0.01$, $k_2=0.03$.  When training a reconstructor (Unet or MoDL) in a separate manner (with respect to inputs sampled with \texttt{MNet}{} masks, random masks and equi-distance masks), the loss function is set as $\frac{\| \mathbf{x}_{\textrm{recon}} - \mathbf{x}_{\textrm{gt}}\|_2}{\|\mathbf{x}_{\textrm{gt}}\|_2} - 5\cdot \textrm{SSIM}(\mathbf{x}_{\textrm{recon}}, \mathbf{x}_{\textrm{gt}})$, which leads to better image quality metrics in our experiments compared to using solely the normalized $\ell_2$ loss. The reconstruction loss function of the LOUPE model includes the unnormalized $\ell_2$ error to be consistent with the original work.

For the alternating training framework, recall that we solve the optimization problem~\eqref{eqn::mb_loss} to provide labels (adaptive masks with respect to particular input) to train \texttt{MNet}. We set the consistency parameter $\lambda=5\times10^{-4}$.  In practice, it is necessary to adjust the choice of the sparsity
control parameter $\alpha$ with respect to different image inputs in the training process,
as an improper choice of $\alpha$ may let the \texttt{mask-backward}{} protocol to output degenerate masks (masks for multiple different objects being the same) or
not improve the initial masks.
Hence, we pre-specify an $\alpha$ grid and use a common $\alpha_0 = 2\times 10^{-5}$ as the initial setting with respect to each batch in the training set. If the returned masks from the \texttt{mask-backward}{} protocol fail to demonstrate sufficient adaptivity with respect to the given image batch or do not get updated from the input, we check the sampling ratio of the failed refined masks. If the failed refined masks have higher (lower) sampling ratio than the targeted value before they go through the sampling-budget control, we increase (decrease) $\alpha$ to the next value in the pre-assigned $\alpha$ grid and let the image batch to go through the \texttt{mask-backward}{} again. For $4\times$ acceleration of $k$-space, the $\alpha$-grid is chosen as a geometric sequence from $10^{-5.7}$ to $10^{-3.9}$ with quotient $10^{0.2}$, and for $8\times$ acceleration of $k$-space, it is a geometric sequence from $10^{-5.01}$ to $10^{-3.61}$ with quotient $10^{0.2}$.

We use the RMSprop optimizer in PyTorch through all the training instances. In the alternating training framework, for each batch, we carry out $T=20 $ steps of iteration inside the \texttt{mask-backward}{} process to generate labels of adaptive masks with the learning rate $5\times 10^{-4}$ for Unet and $5\times 10^{-3}$ for mask characterization parameters $\boldsymbol{\xi}$. The learning rate for \texttt{MNet}{} is $5\times10^{-4}$, and 40 steps of update are made to \texttt{MNet}{} weights with respect to one batch of adaptive mask labels output by \texttt{mask-backward}{}. We run the alternating training framework for 10 epochs and obtain the corresponding \texttt{MNet}{} and Unet-co-trained.

When training separate reconstructors (both Unet and MoDL), we used an initial learning rate $10^{-5}$ and the \texttt{ReduceLROnPlateau} scheduler in PyTorch to regulate the learning rate with patience 5, reduce factor 0.8 and minimal learning rate $10^{-6}$. Separate training of all reconstructors uses 40 epochs for each reconstructor.

When training the LOUPE framework, we use the learning rate $10^{-4}$ to update the mask parameters and $10^{-5}$ to update the Unet in the LOUPE framework. The slope parameter is set as 1. The LOUPE model is trained for 40 epochs. These settings lead to good performance of the LOUPE masks.

Regarding the sampling setting, in the 8$\times$ acceleration case, we set the base low-frequency initially observed lines to 8 rows, and remaining sampling budget for high-frequency information as 32 rows; and in the 4$\times$ acceleration case, the base low-frequency observation budget is 16 rows and the remaining sampling budget for high-frequency information is 64 rows.

\begin{figure}[!ht]
    \centering
    \begin{subfigure}[b]{.6\linewidth}
        \centering\captionsetup{width=\linewidth}
        \includegraphics[width=\linewidth]{figures/8fold_mnet_mask_examples.pdf}
        \label{fig:8x_mnet_mask_examples}
        \caption{8$\times$ acceleration masks (second row) predicted by \texttt{MNet}{} for different training slices (first row): Base low-frequency information contains the central 8 rows, and the sampling budget for high-frequency information is 32 rows.}
    \end{subfigure}%
    \vfill
    \begin{subfigure}[b]{.6\linewidth}
        \centering\captionsetup{width=\linewidth}
        \includegraphics[width=\linewidth]{figures/4fold_mnet_mask_examples.pdf}
        \label{fig:4x_mnet_mask_examples}
        \caption{4$\times$ acceleration masks (second row) predicted by \texttt{MNet}{} for different training slices (first row). Base low-frequency information contains the central 16 rows, and the sampling budget for high-frequency information is 64 rows.}
    \end{subfigure}
   \caption{Adaptively predicted masks from \texttt{MNet}{} for slice examples in the single-coil FastMRI knee dataset.}
  \label{fig::mnet_mask_examples}
\end{figure}

\subsection{Results}
In Figure~\ref{fig::mnet_mask_examples}, we visualize the adaptive masks output from the adaptive sampler \texttt{MNet}{} trained by Algorithm~\ref{alg::alternating_training}. Knee images in Figure~\ref{fig::mnet_mask_examples} are the ground truth images. The \texttt{MNet}{} sampler sees the low-frequency information collected in $k$-space and outputs the corresponding full subsampling patterns for sampling high-frequency information in a single pass, which are shown for each image and are object-adaptive.

Figure~\ref{fig::8fold_recon_examples} is an example showing the reconstructed images by several different reconstructors based on information collected from $k$-space according to different masks. The different corresponding masks are shown in Figure~\ref{fig::8fold_mask_cmp}. We consider the following combinations of sampler-reconstructor pairs: \texttt{MNet}{} and co-trained Unet (Unet-CO), \texttt{MNet}{} and follow-up separately-trained Unet (Unet-SEP), \texttt{MNet}{} and a follow-up separately-trained MoDL reconstructor (\texttt{MNet}{}-MoDL), LOUPE co-trained sampler and reconstructor, random sampling mask and separately-trained Unet (rand.-Unet), equi-distance mask and separately-trained Unet (equidist.-Unet), and energy density-based probabilistic mask and separately-trained Unet (prob.-Unet). We note the SSIM and NMSE values above the shown images.

\begin{figure*}[!ht]
  \centering
    \includegraphics[width=.8\textwidth]{figures_new/8fold_recon_examples_new.pdf}
  \caption{Reconstruction results from various combinations of sampler-reconstructor pairs with 8$\times$ acceleration of $k$-space.}
  \label{fig::8fold_recon_examples}
\end{figure*}

One should note that the adaptive \texttt{MNet}{} mask shown in Figure~\ref{fig::8fold_mask_cmp} is for the ground truth object in Figure~\ref{fig::8fold_recon_examples}, while the underlying probabilistic parametrization of LOUPE mask in Figure~\ref{fig::8fold_mask_cmp} is fixed after the training process completes with respect to the training dataset. The random mask is re-generated for each different input during the training and testing process, so the random mask used in the sampling process can differ from one object to another. The equi-distance mask is fixed with respect to the entire training and testing dataset given the pre-assigned amount of low frequencies observed and the amount of sampling budget for remaining high frequencies.

\begin{figure*}[!ht]
  \centering
  \includegraphics[width=.75\textwidth]{figures_new/8fold_masks_new.pdf}
  \caption{Comparison of different masks with respect to the object in Figure~\ref{fig::8fold_recon_examples} with $8\times$ acceleration.}
  \label{fig::8fold_mask_cmp}
\end{figure*}

The complete reconstruction accuracy comparison for 8$\times$ acceleration is shown in Figure~\ref{fig::8fold_accuracy} using box plots. We consider four accuracy criteria to characterize reconstruction quality compared to the ground truth image: relative $\ell_1$ error (normalized mean absolute error, NMAE), normalized mean square error (NMSE), structural similarity index (SSIM), and high frequency error norm (HFEN) \cite{sai2011dlmri}.
\begin{figure*}[!ht]
  \centering
  \includegraphics[width=.9\textwidth]{figures_new/accuracy_8fold.pdf}
  \caption{Comparison via box plots of reconstruction accuracy of different sampling-reconstructor combinations with $8\times$ acceleration of $k$-space. The triangle-shaped dot inside each box is the mean of the data in the box. Relative $\ell_1$ error is defined as $\frac{\|\mathbf{x}_{\textrm{recon}} - \mathbf{x}_{\textrm{gt}}\|_1}{\|\mathbf{x}_{\textrm{gt}}\|_1}$ for $\mathbf{x}_{\textrm{recon}}$ denoting the reconstruction and $\mathbf{x}_{\textrm{gt}}$ the ground truth image.}
  \vspace{0.1in}
  \label{fig::8fold_accuracy}
\end{figure*}

\begin{table}[!ht]
\centering
\begin{tabular}{ lccccc }
\toprule
Method & NMAE & NMSE & HFEN & SSIM \\%  & PSNR\\
\midrule
\texttt{MNet}{}-Unet-CO & 0.1778 & 0.0415 & 0.4150 & 0.6603 \\% & 27.3470  \\
\texttt{MNet}{}-Unet-SEP & 0.1949 & 0.0529 & 0.4274 & \textbf{0.6965}\\% & 26.9146 \\
\texttt{MNet}{}-MoDL & \textbf{0.1701} & \textbf{0.0379} & \textbf{0.4087} & 0.6662 \\% & \textbf{27.9473}\\
LOUPE & 0.1879 & 0.0451 & 0.4330 & 0.6180\\% & 26.8275\\
prob.-Unet & 0.2249 & 0.0817 & 0.4899 & 0.6429\\
rand.-Unet & 0.2512 & 0.1124 & 0.5735 & 0.6237\\% & 24.7051 \\
equidist.-Unet & 0.2403 & 0.0984 & 0.5418 & 0.6359\\% & 24.2393\\
\bottomrule
\end{tabular}
\caption{Reconstruction accuracy comparisons at 8$\times$ acceleration ratio between various sampler-reconstructor combinations. The values in the table are mean values of each box in Figure~\ref{fig::8fold_accuracy}.}
\label{table::8x_result}
\end{table}
Figure \ref{fig::8fold_accuracy} shows that advantages of different combination of samplers and reconstructors under different criteria. The \texttt{MNet}-MoDL pair outperformed other combinations in terms of global accuracy (relative $\ell_1$ error and NMSE) and reconstruction quality of high frequency portions (HFEN), while the \texttt{MNet}-Unet-SEP pair showed better performance in terms of feature characterization (SSIM). Although separately trained sophisticated unrolled-network reconstructor (MoDL) demonstrates higher reconstruction accuracy in multiple criteria, we highlight the comparable performance from the \texttt{MNet}-Unet-CO pair due to the convenience of directly using a co-trained reconstructor without initiating a separate training effort. Our object-adaptive sampling approach through \texttt{MNet}{} outperforms the LOUPE method, and both learning-based approaches are consistently better than the baseline methods that conduct random sampling and equi-distance sampling in $k$-space, respectively.

The performances of the proposed sampler-reconstructor pairs characterized by NMSE and SSIM in this work are on par with the results reported in the FastMRI leaderboard.
Few outliers in the boxplots indicates better stability of co-trained Unet reconstructor and separately trained MoDL reconstructor compared to the separately trained Unet-reconstructor.

One aspect of comparing these samplers is reproducibility. We point out that after the training is concluded, the \texttt{MNet}{} mask prediction is deterministic and has reproducibility guarantee, while LOUPE methods and other probability-based samplers give varying output due to their stochastic nature.
Our results indicate the importance of an object-adaptive sampler compared to population-adaptive samplers such as LOUPE and the energy density-based masks. Moreover, we observed that combining a random sampler with a sophisticated reconstructor MoDL does not bring as good reconstruction results as our method; i.e., making both the sampler and reconstructor adaptive offers benefits.

\begin{figure*}[!ht]
  \centering
    \includegraphics[width=.9\textwidth]{figures_new/4fold_recon_examples_new.pdf}
    \caption{Reconstruction results from various combinations of sampler-reconstructor pairs with 4$\times$ acceleration of $k$-space.}
  \label{fig::4fold_recon_examples}
  \vspace{0.1in}
\end{figure*}

\begin{figure*}[!ht]
  \centering
  \includegraphics[width=.75\textwidth]{figures_new/4fold_mask_cmp_new.pdf}
  \caption{Comparison of different masks with respect to the object in Figure~\ref{fig::4fold_recon_examples} at $4\times$ acceleration.}
  \label{fig::4fold_mask_cmp}
\end{figure*}

\begin{table}[!ht]
\centering
\begin{tabular}{ lccccc }
\toprule
Method & NMAE & NMSE & HFEN & SSIM \\% & PSNR\\
\midrule
\texttt{MNet}{}-Unet-CO & 0.1480 & 0.0294 & 0.2673 & 0.7510 \\% & 29.3977  \\
\texttt{MNet}{}-Unet-SEP  & 0.1654 & 0.0378 & 0.3113 & \textbf{0.7660} \\% & 28.5163 \\
\texttt{MNet}{}-MoDL & \textbf{0.1437} & \textbf{0.0274} & \textbf{0.2663} & 0.7584 \\% & \textbf{29.7685}\\
LOUPE & 0.1540 & 0.0304 & 0.3356 & 0.7381 \\% & 28.8016\\
prob.-Unet & 0.1776 & 0.0449 & 0.3747 & 0.7520 \\
rand.-Unet & 0.2026 & 0.0646 & 0.4448 & 0.7348 \\% & 26.7310 \\
equidist.-Unet & 0.1897 & 0.0545 & 0.4132 & 0.7658 \\% & 26.3538\\
\bottomrule
\end{tabular}
\caption{Reconstruction accuracy comparisons at 4$\times$ acceleration for various sampler-reconstructor combinations. The values in the table are mean values of each box in Figure~\ref{fig::4fold_accuracy}.}
\label{table::4x_result}
\end{table}

\begin{figure*}[!ht]
  \centering
  \includegraphics[width=.8\textwidth]{figures_new/4fold_accuracy_new.pdf}
  \caption{Comparison of reconstruction accuracy of different sampling-reconstructor combinations with $4\times$ acceleration of $k$-space.}
  \label{fig::4fold_accuracy}
\end{figure*}

Figures~\ref{fig::4fold_recon_examples}, \ref{fig::4fold_mask_cmp}, and  \ref{fig::4fold_accuracy} show the reconstruction examples, adaptive or compared mask examples, and reconstruction accuracy comparison results, respectively for the 4$\times$ acceleration case. The performance trend between methods in the 4$\times$ acceleration setting is similar to that for the 8$\times$ acceleration case discussed above. Tables \ref{table::8x_result} and \ref{table::4x_result} show the mean reconstruction accuracy comparison between various sampler-reconstructor combinations.

We observe more concentration of the adaptive mask in the region of lower frequencies in the 4$\times$ acceleration setting.
We have also experimented on another 4$\times$ acceleration setting where we start with 8 lowest frequencies and sample 72 high frequencies. The mask output by \texttt{MNet}{} has similar patterns as in Figure \ref{fig::4fold_mask_cmp}.

\subsection{Background}

Early approaches to improve the sampling efficiency
without degrading image quality in MRI include
careful design of the pulse sequence and $k$-space trajectory~\cite{POSER2018,Bitar_06_MRI_review}, partial or half Fourier methods~\cite{Liang_Book}, and hardware-based acceleration techniques such as parallel data acquisition or parallel MRI (p-MRI)~\cite{pMRI-Survey}.
Despite the availability of several receiver coils, p-MRI is limited to acceleration of typically three to four fold due to increased noise and imperfect artifact correction at higher accelerations.
Recent methods such as compressed sensing (CS)~\cite{emmanuel2004robust}
have leveraged additional structure of priors on the target images to perform reconstruction. MR images
\begin{edits}
\del{are often structured, e.g., they}
\end{edits}
can often be approximated as piecewise smooth functions with relatively few sharp edges,
\begin{edits}
\del{. This type of structure}
\end{edits}
which leads to sparsity in the wavelet domain, as piecewise smooth functions are compressible when represented in the wavelet basis. The MR image reconstruction problem from subsampled $k$-space measurements often takes the form of a regularized optimization problem: \begin{equation}

     \min_\mathbf{x} \,\frac{1}{2}\|\mathbf{A} \mathbf{x} - \mathbf{y}\|^2 + \alpha\mathcal{R}(\mathbf{x}),
    \label{reginversion}
\end{equation}
where $\mathbf{x}\in\mathbb{C}^{m\times n}$ is the underlying MR image to recover,
$\mathbf{y}$
denotes the partial measurements,
$\mathbf{A}$ is a linear MRI measurement operator,
$\mathcal{R}$ is a regularizer, and $\alpha$ is a regularization parameter that controls the trade-off between the fidelity to the $k$-space measurements and alignment with structure imposed by the regularizer $\mathcal{R}$.
When representing an MR image as sparse in the wavelet domain, with $\Phi$ as the wavelet transform, and adding the total variation penalty ($\|\cdot\|_{\mathrm{TV}}$),
the above image reconstruction problem is often formulated as~\cite{lustig2007sparse} \begin{equation}
\label{eqn::classic_img_recon_opt_prob}
\widehat{\mathbf{x}} = \arg  \min_{\mathbf{x}}\,
 \alpha\|\Phi\mathbf{x}\|_1 + \beta \|\mathbf{x}\|_{\mathrm{TV}} + \frac{1}{2}\|\mathbf{A} \mathbf{x} - \mathbf{y} \|_2^2.
\end{equation}
In this work, we focus on this sampling-reconstruction problem in the single-coil setting, where $\mathbf{A} := \mathbf{M} \mathcal{F}$. Here $\mathcal{F}$ is the Fourier transform operator and $\mathbf{M}$ is a subsampling operator. We discuss how our proposed method readily extends to the multi-coil setting in the methodology section.

\begin{edits}
\del{Compressive sensing \add{(CS)} theory reveals that when the measurement operation \add{applied on a signal} is sufficiently incoherent with respect to the \add{sparsifying transform}, one can exactly recover the
underlying image from significantly fewer
samples \cite{FoucartRauhut_CS_Book}.}
\end{edits}

In compressed sensing theory, after fixing a subsampling operator $\mathbf{A}$ which, together with the
sparsifying
transform, has the restricted isometry property (RIP), a non-linear reconstruction method via convex optimization can give an approximation of high quality to the ground truth signal~\cite{CSMRIreview}. Verifying that the RIP holds for a general matrix is an NP hard problem~\cite{Tillmann_NPhard_CS}, and thus early results tend to show that random matrices as subsampling operators satisfy RIP properties with high probability~\cite{Baraniuk_08_RIP_randMats}. A possible $k$-space undersampling scheme in MRI is variable density random sampling~\cite{lustig2007sparse}, where a higher probability is allocated for sampling at lower frequencies than higher frequencies. This sampling scheme accounts for the fact that most of the energy in MR images is concentrated close to the center of $k$-space. In this classic MRI protocol, measurements are collected sequentially using subsampling patterns that are chosen a priori, and an iterative solver such as the proximal gradient descent method~\cite{Huang_PPD} or the alternating direction method of multipliers~\cite{boyd_distributed_2011} is used as the reconstruction method
for the regularized optimization problem.
Some works have focused on appropriate selection of random sampling patterns for CS MRI~\cite{vasanwala2011,adcock:13:btc,adchanpoonrom13}.

Data-driven approaches can elevate the performance of image reconstruction methods by designing data-specific regularizers $\mathcal{R}$ or by introducing deep learning tools as learned reconstructors.
For regularizer design, one may replace the $\ell_1$ penalty for the wavelet coefficients and the total variation penalty in~\eqref{eqn::classic_img_recon_opt_prob} with a learning-based sparsity penalty where a sparsifying transform (or dictionary) is learned in an unsupervised fashion based on a limited number of unpaired clean image patches~\cite{xu_12_ldx} or is learned jointly with image reconstruction~\cite{wensailukebres19}.
\begin{edits}
\del{Alternatively, plug-and-play regularization replaces the proximal step in the typical iterative optimization process for~\eqref{reginversion} by a denoising neural network (or generic denoiser) that essentially functions as an implicit data-based regularizer~\cite{plug_and_play_vbw_13}.
}
\end{edits}
For deep learning based approaches for reconstruction, examples of direct deployment of feed-forward neural networks as denoisers include the U-Net architecture-based networks~\cite{2D_unet} and tandem networks that are trained as generator and discriminator in the GAN framework~\cite{GAN_denoiser_19}. Another approach
\begin{edits}
\del{related deep learning-inspired approach for reconstruction}
\end{edits}
involves unrolling iterative algorithms and learning the regularization parameters in a supervised manner from paired training sets.
\begin{edits}
\del{Each learnable block in the chain of the unrolling strategy is fulfilled by a network to simulate a certain optimization operation~\cite{admm-net2016,NeumannNet_19}.}
\end{edits}
We refer readers to~\cite{Huang_MBIR_review_20,reconreviewsaijeffjong2020,ongie_deep_2020} for detailed reviews of learning-based reconstruction methods.

\subsection{Contributions}
In this work, we propose a neural-network based adaptive sampler, \texttt{MNet}{} that generates object-specific sampling patterns in a single-pass based on limited low-frequency information from varying input objects.
\begin{edits}
\del{In this work, we propose an adaptive sampler, \texttt{MNet}, as a neural network that takes limited low-frequency information as input per scan or frame and outputs corresponding desired undersampling patterns adaptive to different input objects in a single pass. }
\end{edits}
The main advantage of the proposed sampling-pattern predictor \texttt{MNet}{} is that it outputs adaptive masks with respect to each different input object, and thus the generated sampling pattern varies from case to case. This property gives our sampler more potential compared to several earlier proposed samplers.
\begin{edits}
\del{, as the mask output from our sampler is not only data-driven but also object-specific. }
\end{edits}
Moreover, compared to recent sequential-decision-type samplers~\cite{RL_sampling_20}, our sampling approach determines at once the entirety of the sampling pattern in the high frequency regime, thus reducing the processing time for practical deployment.
While the proposed approach is applied for static MRI sampling in this work, it can be readily used for dynamic MRI sampling (e.g.\ predicting the sampling pattern for a following frame based on low-frequency samples of previous frames).
\begin{edits}
\del{, where variations could also be used to enable faster implementation.}
\end{edits}

The sampling network is trained jointly with a parametric reconstructor to optimize the eventual quality of image reconstruction. We propose an alternating training framework to update the parameters of the sampler network and those of the reconstruction network. The key component in this training framework is the internal label generation mechanism, referred to as \texttt{mask-backward}{}, which efficiently generates object-specific sampling patterns with respect to each fully sampled $k$-space data point. This training framework makes training an \texttt{MNet}{} possible without resorting to computationally expensive and approximate greedy methods~\cite{greedy_sampling_mri_18} to supply training labels.

Our numerical experiments show the superior performance of the \texttt{MNet}{} framework with respect to several alternative schemes for undersampled single-coil acquisitions based on the public FastMRI~\cite{fastMRI_dataset_18,fastMRI_journal} dataset.
\begin{edits}
\del{In this work, we develop an adaptive sampler that generates object-specific sampling patterns based on highly limited measurements from varying input objects.
We propose to leverage neural networks to generate undersampling patterns in a single pass using highly undersampled initial acquisitions as input. The sampling network is trained jointly with a reconstructor to optimize eventual image reconstruction quality. Such training of neural networks to predict sampling patterns is essentially a supervised task.
We propose an end-to-end training procedure that takes in fully sampled $k$-space data
and outputs corresponding desired undersampling patterns and image reconstructions. To ignite the process of training a network for sampling pattern or mask prediction, we use an alternating framework that operates between finding object-adaptive binary masks (and a corresponding reconstructor)
exploiting ground truth $k$-space data,
and updating a network for predicting such sampling masks.
}
\end{edits}

\subsection{Connection to Bilevel and Mixed-integer Optimization Problems}
The drawback of the conventional compressive sensing approaches~\cite{lustig2007sparse,adchanpoonrom13} to accelerate MRI scans
is that the measurement operator does not adapt to different objects and thus may not result in optimal image reconstructions in general.

An alternative formulation, first explored in~\cite{saiembc2011},
is in the form of a bilevel optimization problem:
\begin{align}
\label{prob::bilevel_opt}
    &\min_{\boldsymbol{\theta}} \sum_{i=1}^N \frac{1}{2}\|\widehat{\mathbf{x}}_i(\boldsymbol{\theta}) - \mathbf{x}_i^{\textrm{gt}}\|_{2}^{2} \nonumber\\
    &\textrm{s.t. } \widehat{\mathbf{x}}_i( \boldsymbol{\theta} ) = \mathrm{arg}\,\min_\mathbf{x} \frac{1}{2} \|\mathbf{y}_i^{\boldsymbol{\theta}} -\mathbf{A}_{\boldsymbol{\theta}}\mathbf{x}\|_2^2 + \lambda \mathcal{R}(\mathbf{x}) \quad \forall \,i,
\end{align}
where $\boldsymbol{\theta}$ denotes the underlying parameters of the measurement/sensing operator $\mathbf{A}_{\boldsymbol{\theta}}$; $\mathbf{x}_i^{\mathrm{gt}}$ are the ground truth images reconstructed from raw fully-sampled $k$-space data $\mathbf{y}_i^{\mathrm{gt}}$; $\mathbf{y}_i^{\boldsymbol{\theta}}$ are the corresponding measurements from applying the measurement operator $\mathbf{A}_{\boldsymbol{\theta}}$; and $\mathcal{R}(\mathbf{x})$ is the regularization term for the lower level optimization problem. The goal is to ultimately generate
artifact-free
images from partial measurement vectors $\{\mathbf{y}_i\}$
as embodied in the upper level optimization problem. This particular formulation accommodates variation in the measurement operator, thus rendering the operator to be adaptive to a training dataset.
In the MRI setting, the parameters $\boldsymbol{\theta}$ capture the sampling patterns (\textit{masks}) we want neural networks to predict. A major challenge in tackling the bilevel optimization formulation is the \textit{implicit} dependence of the upper-level objective on the parameters $\boldsymbol{\theta}$. It is not clear in either theoretical or practical aspects how to compute the (sub)gradients of the upper-level objective with respect to the pattern parameters. To address this issue, we use a neural network as the solver for the lower-level problem, which essentially renders the `$\textrm{arg} \min$' in formulation \eqref{prob::bilevel_opt} as $\widehat{\mathbf{x}}_i(\boldsymbol{\theta}) = \textrm{Network}(\mathbf{y}_i^{\boldsymbol{\theta}})$, and therefore the gradient computation and back-propagation from the upper-level loss function to the sampling parameters $\boldsymbol{\theta}$ become possible.

In this work, we focus on Cartesian undersampling of $k$-space.
In this case, we can write $\mathbf{A}_{\boldsymbol{\theta}} =\mathrm{diag}(\mathbf{M})\mathcal{F}$, where $\mathrm{diag}(\mathbf{M})$ is a binary diagonal matrix, and a $1$ in a certain entry of $\mathbf{M}$ indicates the corresponding row in $k$-space is observed. Namely, the parameters $\boldsymbol{\theta}$ in the bilevel formulation \eqref{prob::bilevel_opt} are the binary elements in $\mathbf{M}$. A practitioner can pre-set a sampling budget $b$, which is reflected as $\sum_t M_{t} = b$. With a sampling budget constraint, the optimization problem \eqref{prob::bilevel_opt} assumes the flavor of integer programming:
\begin{align}
\label{prob::milp}
    &\min_{\mathbf{M}} \sum_{i=1}^N \frac{1}{2}\|\widehat{\mathbf{x}}_i(\mathbf{M}) - \mathbf{x}_i^{\textrm{gt}}\|_{2}^{2} \nonumber\\
    &\textrm{s.t. }
    \begin{dcases}
    \widehat{\mathbf{x}}_i( \mathbf{M} ) = \mathrm{arg}\,\min_\mathbf{x} \frac{1}{2} \|\mathbf{y}_i^{\mathbf{M}}-\mathrm{diag}(\mathbf{M})\mathcal{F}\mathbf{x}\|_2^2 + \lambda \mathcal{R}(\mathbf{x}) \quad\forall \,i,&\\
    \sum_{t=1}^m M_{t}= b & \\
    M_{t}\in\{0,1\} \quad \forall \,t\in[m].&
    \end{dcases}
\end{align}
There has been work devoted to leveraging neural networks to construct solvers for mixed-integer programming (MIP). In~\cite{NN_for_MIP_20}, a mixed-integer program is represented as a bipartite graph (nodes are variables and constraints of the MIP, and edges connect variables and their corresponding constraints). One neural network is trained to predict integer values in the solution to cater for the integer constraints in a generative style, and another one is trained to learn the policy of branching. Classic solvers are used to tackle sub-problems of the original MIP to provide training data for each network.

The target of predicting a sampling pattern echoes the integer-valued constraint in the optimization problem~\eqref{prob::milp}. We address the integer-valued constraint in our approach through a binarization step, which can be considered as a post-processing action applied on the output of the proposed mask-predicting \texttt{MNet}{}. While the way we deploy the network for mask prediction does not constitute directly a solver for the MIP, it gives a potential new direction of using networks for integer-constraint problems.

\subsection{Prior Art in Sampling Design}
We now briefly review several existing sampling design strategies in the literature.
An early work optimizes $k$-space trajectories using an information gain criterion~\cite{seeger2010}. Another first work~\cite{saiembc2011} directly optimizes a $k$-space undersampling pattern based on reconstruction error-type loss on a training set.
With recent success of deep learning methods, there has been growing interest in the topic of adaptive sampling. A classical non-parametric sampling adaptation method for~\eqref{prob::milp} is the greedy approach~\cite{greedy_sampling_mri_18}, which chooses one $k$-space line or phase encode
to add into the sampled set at each step depending on which line in combination with those already added so far leads to the best reconstruction loss (upper level loss). The greedy approach terminates when the sampling budget is reached. The learned sampling mask is population-adaptive (learned on a dataset) but does not vary with different objects (i.e., not object-adaptive).
The greedy sampling process
can be accelerated by considering random subsets of all candidate high frequencies as cohort and selecting multiple high frequencies into the sampling set at one step.

LOUPE is a parametric model that characterizes the probability of sampling each pixel or row/column in the frequency domain or $k$-space with underlying parameters that are learned simultaneously with the parameters of a reconstructor~\cite{Loupe_20}. The drawback of this approach is that the learned sampling pattern is generic with respect to the training set (population-adaptive) rather than being adaptive to each individual object's characteristics. A similar work~\cite{Sherry_sampling_mri_20} also assumes that the training data are representative enough for new data acquisitions and learns a fixed undersampling pattern.
This latter approach combines the sampling and image reconstruction problem as a bilevel optimization problem, parametrizes the sampling pattern via probability variables to optimize, and resorts to iterative optimization solvers to find the solution.

J(oint)-MoDL~\cite{JMoDL_20} assumes the sampling pattern to be rows/columns sampled in $k$-space and
learns a common undersampling pattern or mask along with a neural network reconstructor from a training set.
Another way of parametrizing sampling patterns is to assign each frequency a logit as the natural logarithm of the unnormalized class probability so that a categorical distribution with respect to all frequency candidates can be accordingly constructed by applying a softmax function on the associated logits, which enables gradient back-propagation, and the output mask contains the frequencies with top-M weights~\cite{generative_samp_20}. PILOT parametrizes the sampling patterns as a matrix to coordinate with the non-uniform FFT measurements, and adds in an additional round of optimization to enforce hardware measurement constraints~\cite{Pilot_mri_2019}.
The BJORK method parametrizes sampling patterns as linear combinations of quadratic B-spline kernels, and attempts to reduce the number of parameters needed to characterize sampling patterns through a scheduling plan that gradually reduces the ratio between the total number of $k$-space samples and the number of B-spline interpolation kernels~\cite{BJORK_21}.
The approach can handle non-Cartesian sampling but does not learn an object-adaptive sampler.

Sequential decision processes have also been exploited as a path to build samplers. One attempt in the reinforcement learning style formulates the sampling problem as a partially observable Markov decision process and uses policy gradient and double deep Q-network to build the sampling policy~\cite{RL_sampling_20}. Another attempt builds a neural network as the sampler to be repeatedly applied in the sampling process and trained simultaneously with the reconstructor~\cite{Bouman_samp_21}.
\begin{edits}
\del{The sequential methods consume additional time for each prediction/decision (idle scanner time) that can introduce artifacts in real-time imaging.}
\end{edits}

\subsection{Structure of this Paper}
The rest of this paper is organized as follows. Section~\ref{sec::method} elaborates the design of the \texttt{mask-backward}{} training framework. Section~\ref{sec::experiments} discusses the architecture of \texttt{MNet}{}, the implementation details and experimental results. We summarize our findings and potential new directions for future work in section \ref{sec::conclusions}.

\subsection{Why do We Need the \texttt{mask-backward}{} Routine?}
We aim to train a neural network that can generate a subsampling pattern for $k$-space based on limited initial low-frequency measurements. An immediate ensuing challenge for training such a network in the supervised style is the lack of labels denoting ideal/best subsampling patterns.
One plausible option is
to generate object-specific masks
for each training image using the greedy algorithm~\cite{greedy_sampling_mri_18} for~\eqref{prob::milp}
and use them as labels. However, the greedy algorithm's labels are not necessarily optimal (i.e., global minimizers in~\eqref{prob::milp}) and
obtaining such labels for a large number of training images (from fully-sampled $k$-space) is computationally expensive or infeasible.

Another natural strategy to circumvent the difficulty of obtaining adaptive mask labels is an end-to-end pipeline that is comprised of an encoder, which maps limited (e.g., low-frequency) information in $k$-space to intermediate adaptive masks, and a decoder that attempts to reconstruct the artifact-free image from the $k$-space information sampled according to the adaptive masks from the encoder. After training of the end-to-end pipeline is concluded, one extracts the encoder part from the pipeline, which can function as a desired mask prediction network. We choose not to proceed with this obvious end-to-end approach owing to two folds of consideration.
The first concern is that binarizing intermediate masks to meet sampling budget target of the mask prediction network is incompatible with gradient-based methods.
While the computation techniques can still get around the hard binarization thresholding in gradient computation and back-propagation, there are not binary labels in the end-to-end training framework to supervise the mask prediction network, which may incur unexpected behavior in its performance.
The second concern is that a deep end-to-end training procedure without intermediate intervention can cause vanishing gradients, which results in slow progress in training. In addition, an end-to-end approach makes it harder to track the performance of each component and thus has less interpretability for the outcome of each component in this framework.
We alleviate these issues by introducing a split variable that represents the output of \texttt{MNet}{} and
satisfies the explicit mask constraints in~\eqref{prob::milp}. A penalty is then introduced to measure the fidelity of the \texttt{MNet}{} output to the split binary variables/labels.

\subsection{Proposed Training Scheme}
\begin{figure*}[!t]
  \centering
  \includegraphics[width=\textwidth]{figures_new/pipeline.pdf}
  \caption{Diagram of the proposed alternating training framework.}
  \label{fig::pipline}
\end{figure*}
We propose a \texttt{mask-backward}{} inferring method (see Subroutine~\ref{alg::mb}) to obtain binary label masks to feed the supervised training process of the mask-predicting network (\texttt{MNet}). The key component of the \texttt{mask-backward}{} method is a parametric reconstructor that serves as a bridge connecting mask parameters and ground truth images.
With the mask $\mathbf{M}$ as a parameter to tune, we first compute a zero-filled image reconstruction via inverse Fourier transform (\texttt{IFFT}) of the observed low-frequency $k$-space data with zeros filling the unobserved $k$-space.
The reconstructor receives this crude initial reconstruction and refines it to be free of aliasing artifacts.
The loss function in~\eqref{eqn::mb_loss} compares the refined image with the ground truth image using specific reconstruction quality metrics,
and is optimized with respect to object-adaptive sampling masks and the reconstructor.
We mainly use a standard 8-block U-Net as the reconstructor with additional residual connections between the downsampling and upsampling blocks. Later, we consider other reconstructors in combination with $\texttt{MNet}$.

\begin{algorithm}[!t]
\caption{Subroutine: \texttt{mask-backward}{}}
\label{alg::mb}
\begin{algorithmic}[1]
\Require Ground truth image $\mathbf{x}^\star$, an initial mask $\mathbf{M}_\textrm{init}$, a reconstructor $U$ with parameters $\Theta$, maximal iteration steps $T$, sparsity control parameter $\alpha$, consistency control parameter $\lambda$, sampling budget $b$
\State Initialize $\mathbf{M}(\boldsymbol{\xi}_0) = \mathbf{M}_\textrm{init}$
\For{iteration count $t<T$}
    \State Sample information in $\mathcal{F}[\mathbf{x}^\star]$ according to binarized $\mathbf{M}(\boldsymbol{\xi}_t)$ to obtain adaptive observations $\mathbf{y}_t$.
    \State Compute the zero-filled \texttt{IFFT} images $\mathbf{x}^{\texttt{IFFT}}_t =\mathcal{F}^{-1}[\mathbf{y}_t]$.
    \State Compute the refined image $\mathbf{x}^{\textrm{recon}}_t = U_{\Theta_t}(\mathbf{x}^{\texttt{IFFT}}_t)$.
    \State Evaluate the loss function in~\eqref{eqn::mb_loss}, compute the gradient with respect to $\boldsymbol{\xi}_t$ and parameters $\Theta$ of the reconstructor $U$, make a step of update to obtain $\boldsymbol{\xi}_{t+1}$ and $\Theta_{t+1}$. \Comment{See Remark 2 for implementation details.}

\EndFor
\State Apply sampling budget control $b$ and binarize $\mathbf{M}(\boldsymbol{\xi}_T)$ to obtain $\mathbf{M}_\textrm{out}$.
\State \textbf{return} $\mathbf{M}_\textrm{out}$, image reconstructor $U_{\Theta_T}$ \Comment{This subroutine can readily handle a batch of images.}

\end{algorithmic}
\end{algorithm}

In order to train the \texttt{MNet}{} with guidance from the \texttt{mask-backward}{} inferring process, we use the following alternating training framework. Let $\mathbf{z}_i$ denote the initially observed low-frequency information of the ground truth image $\mathbf{x}_i^\star$, $\Gamma$ denotes the parametrization of the \texttt{MNet}{}, and $\Theta$ denotes the parametrization of the reconstructor \texttt{Recon}. The optimization problems involved in this training framework are the following that are solved in an alternating manner (Fig.~\ref{fig::pipline}): \begin{align}
&\texttt{mask-backward}:\nonumber\\
&\begin{aligned}
    \label{eqn::mb_loss}
    \min_{\{\mathbf{M}_{\textrm{b},i}\},\Theta}\, \sum_{i} &\bigg( \underbrace{\frac{\| \texttt{Recon}_{\Theta}\big(\mathcal{F}^{-1} \mathrm{diag} (\mathbf{M}_{\textrm{b},i}) \mathcal{F}\mathbf{x}_i^\star \big) - \mathbf{x}_i^\star \|_2}{\| \mathbf{x}_i^\star \|_2}}_{\textrm{reconstruction data fidelity }}  \,\, +  \, \alpha \underbrace{\mathcal{R}\big( \mathbf{M}_{\textrm{b},i} \big)}_{\textrm{sparsity reg.\ }} + \,\,
    \lambda \underbrace{\varphi\big(\texttt{MNet}_\Gamma (\mathbf{z}_i) , \mathbf{M}_{\textrm{b},i} \big)}_{\textrm{consistency}} \bigg),
\end{aligned}\\
&\texttt{MNet}:\nonumber\\
&\begin{aligned}
\label{eqn::mnet_loss}
    &\min_{\Gamma} \sum_{i} \varphi\big(\texttt{MNet}_{\Gamma}(\mathbf{z}_i), \mathbf{M}_{\textrm{b},i} \big).
\end{aligned}
\end{align}
In loss function~\eqref{eqn::mb_loss} of \texttt{mask-backward}{}, the first term characterizes the fidelity between the reconstructed image and the ground truth image using normalized root mean square error (NRMSE), the second term is the regularization term to implicitly control the sparsity of the returned mask, and the third term imposes the consistency between the returned mask and the mask predicted by the current \texttt{MNet}{}. We include the consistency term to accelerate the convergence of this alternating training framework. We take $\mathcal{R}(\mathbf{M}_{b,i}) = \|\mathbf{M}_{b,i}\|_1$ to promote sparsity on the object-adaptive masks.
The exact binary constraint and desired sampling budget (as in~\eqref{prob::milp}) are enforced on $\mathbf{M}_{b,i}$ after the gradient step as shown in Subroutine~\ref{alg::mb}.
The function $\varphi$ denotes the binary cross entropy loss in both loss functions~\eqref{eqn::mb_loss} and~\eqref{eqn::mnet_loss}. The non-negative parameters $\alpha$ and $\lambda$
are weights for the sparsity regularization and consistency regularization terms, respectively.

In this training process, we separate the training of \texttt{MNet}{} and the image reconstructor \texttt{Recon} into two alternating steps to enable easier training. We first use the \texttt{mask-backward}{} method to find the ideal mask $\mathbf{M}_{\textrm{b},i}$ for each given image $\mathbf{x}_i$ along with updating the reconstructor.
We start our alternating training framework with a warmed-up reconstructor (e.g., \texttt{Unet}) $U_0$ that is pre-trained for de-aliasing zero-filled \texttt{IFFT} reconstructions from variable density random undersampling of $k$-space.

Generating mask labels through \texttt{mask-backward}{} method and updating the mask predicting network are executed alternatively.
The complete training algorithm is summarized in
Algorithm~\ref{alg::alternating_training}.

Since the \texttt{mask-backward}{} process involves nontrivial gradient update and post binarization, prior to updating \texttt{MNet}{},
we use the reconstruction data-fidelity as a quality function,  $\mathcal{Q}(\mathbf{M},U,\mathbf{x}^\star)$, to evaluate how good a mask $\mathbf{M}$ is with the reconstructor $U$ for the image $\mathbf{x}^\star$. We define $\mathcal{Q}(\mathbf{M},U,\mathbf{x}^\star) = -\frac{\|U(\mathcal{F}^{-1}\mathrm{diag}(\mathbf{M})\mathcal{F}\mathbf{x}^\star) - \mathbf{x}^\star\|_2}{\|\mathbf{x}^\star\|_2}$.
We need to ensure that the refined mask is improved in terms of contributing more for image reconstruction and that the refined mask is non-trivial. Therefore, the condition to accept a refined mask is that its quality should be better than that of the previous direct output from \texttt{MNet}{} (pre-refinement mask) and that of random masks. The quality of the random mask is always evaluated with the initial reconstructor $U_0$ pre-trained for random-masked observations. Meanwhile, the quality of adaptive masks and their refined counterparts is evaluated by the reconstructor $U_s$ that is updated during training.

After a refined mask passes the quality check, we use the ideal binary $\mathbf{M}_{\textrm{b},i}$ to update the parametrization of the \texttt{MNet}. The direct output of the deployed \texttt{MNet}{}, as shown in Figure \ref{fig::mnet_structure} represents the significance of each high frequency. Before we evaluate the loss function $\varphi$ in equation \eqref{eqn::mnet_loss}, we need to process the primitive output \texttt{MNet}{$_\Gamma(\mathbf{z}_i)$} to obtain a binary mask, and ensure that this output mask has the target sampling ratio.

Regarding \textit{sampling budget control}, we have used the normalization trick in~\cite{Loupe_20} to ensure the output mask from \texttt{MNet}{} as well as the refined mask returned by the \texttt{mask-backward}{} method has the exact target sampling ratio. Assume the sampling budget of lines is $b$. Thus, the sampling ratio is $\alpha=\frac{b}{m}$, or for a corresponding unbinarized mask $\mathbf{M}$, it is $\frac{\|\mathbf{M}\|_1}{m}=\frac{\|\sigma(\boldsymbol{\xi})\|_1}{m}$, where $\sigma(\cdot)$ is the sigmoid function and $\boldsymbol{\xi}$ is mask parametrization parameters (see Remark 2). Assume $\mathbf{P}$ is the mask characterization before normalization, thus $\|\mathbf{P}\|_1\ne b$. We define $\widetilde{p} = \frac{\|\mathbf{P}\|_1}{m}$, which is the pre-normalization sampling ratio, and thus $1-\widetilde{p}$ is the average value of $1-\mathbf{P}$. The normalization is done by
\begin{equation}
        N_\alpha(\mathbf{P}) =
        \begin{cases}
        \frac{\alpha}{\widetilde{p}}\mathbf{P}, \, \textrm{if } \widetilde{p}\ge \alpha\\
        1-\frac{1-\alpha}{1-\widetilde{p}}(1-\mathbf{P}), \, \textrm{otherwise}.
        \end{cases}
\end{equation}

The \textit{binarization} is necessary to make ultimate output sampling masks meaningful for practical use. The threshold we use for binarizing vectors is 0.5. During the training process, each entry in the mask characterization
$\mathbf{M}$ is in $(0,1)$. We should point out that the reconstructor in
Subroutine~\ref{alg::mb}
(e.g., the Unet or the MoDL~\cite{MoDL_18} model) sees the subsampled $k$-space information $\mathbf{y}_t$ filtered by the binarized mask, while the gradient with respect to unbinarized $\mathbf{M}$ in~\eqref{eqn::mb_loss} can still be computed with the definition of a (approximate) \textit{backward} function in e.g., the PyTorch implementation of binarization (see~\cite{Bengio_13_gradient,Loupe_extension_20,Bouman_samp_21}).

The sampling budget control and binarization are both applied before a refined mask is returned by the \texttt{mask-backward}{} method to update the \texttt{MNet}{} model, thus rendering the labels for training the \texttt{MNet}{} to have the pre-specified sampling ratio. They are also applied on the direct output of \texttt{MNet}{} at testing time.

\begin{algorithm}[!ht]
\caption{Alternating training framework for sampler network \texttt{MNet}{} and reconstructor}
\label{alg::alternating_training}
\begin{algorithmic}[1]
\Require A mask-predicting network \texttt{MNet}{}, a reconstructor $U_0$, ground truth images $\{\mathbf{x}^\star_s\}$, initial (low-frequency) measurement count $l$, sampling budget $b$, maximal iteration steps $T$ for \texttt{mask-backward}, sparsity control parameter $\alpha$, consistency control parameter $\lambda$, number of iterations $c$ for \texttt{MNet}{} updates.

\For{epoch $<$ maximal epoch}
    \For{$s < \textrm{maximal batch number}$}
        \State Compute $\mathbf{y}_s = \mathcal{F}[\mathbf{x}^\star_s]$ and extract $k$-space information $\mathbf{z}_s = \mathbf{y}^{\textrm{lf}}_s$ corresponding to lowest $l$ frequencies.
        \State Generate the adaptive masks $\mathbf{M}_s = \texttt{MNet}{}(\mathbf{z}_s)$ for image batch $\mathbf{x}^\star_s$.
        \State Apply sampling budget control with budget $b$ and binarization on $\mathbf{M}_s$.
        \State $\allowbreak [\mathbf{M}^{\textrm{refined}}_s, \widetilde{U}_s] = \texttt{mask-backward}(\mathbf{x}^\star_s,\mathbf{M}_{s}, U_s, \allowbreak T,\alpha,\lambda, b)$
        \If{$\mathcal{Q}(\mathbf{M}^{\textrm{refined}}_s,\widetilde{U}_s,\mathbf{x}^\star_s) > \max\big\{ \mathcal{Q}(\mathbf{M}_s,\widetilde{U}_s,\mathbf{x}^\star_s), \allowbreak \mathcal{Q}(\mathbf{M}_{\textrm{random}},U_0,\mathbf{x}^\star_s)\big\}$}

            \State Make $c$ steps of supervised training of \texttt{MNet}{} with labels as the refined mask $\mathbf{M}^{\textrm{refined}}_s$ with respect to objective \eqref{eqn::mnet_loss}.
            \State $U_{s+1} = \widetilde{U}_s$
        \Else
            \State $U_{s+1} = U_s$
        \EndIf
    \EndFor
\EndFor
\State \textbf{return} \texttt{MNet}, image reconstructor $U$

\end{algorithmic}
\end{algorithm}

Next, we list a set of remarks about our approach and some extensions.

\subsection{Remarks} \begin{enumerate}
    \item Algorithm~\ref{alg::alternating_training} follows a co-design approach to identify a sampler and a reconstructor simultaneously with respect to a training dataset. Finally, one can also take a trained \texttt{MNet}{} sampler from the output of Algorithm~\ref{alg::alternating_training} and train a
    reconstructor in a separate process. We have investigated the performance of both of these practices in Section~\ref{sec::experiments}.

    \item \textit{Computing gradient with respect to mask parametrization parameter} $\boldsymbol{\xi}$: When implementing~\eqref{eqn::mb_loss}, we let $\mathbf{M}_{b,i} = \sigma(\boldsymbol{\xi}_i)$ where $\sigma(\cdot)$ is the typical sigmoid function and $\boldsymbol{\xi}_i$ is the parameter vector in $\mathbb{R}^m$ characterizing the mask for image $\mathbf{x}_i$. Compared to parametrizing the adaptive mask corresponding to the image $\mathbf{x}_i$ using nonnegative real numbers between 0 and 1 (ideally binary), using sigmoid improves the numerical stability when updates are made to the free $\boldsymbol{\xi}_i$'s.

    \item For the mask initialization step in Subroutine~\ref{alg::mb}, the input mask $\mathbf{M}_\textrm{init}$ is binarized, and we set $\boldsymbol{\xi}_0[i] = 0.1$ for frequencies with mask value 1 and $\boldsymbol{\xi}_0[i] = -0.1$ for frequencies with mask value 0.

    \item One can replace a Unet with other parametric reconstructors such as MoDL and unrolling iterative blocks in the \texttt{mask-backward}{} method in our joint training framework, as long as the reconstructor is differentiable to let the gradient to propagate through.

    \item In Step 4 in Algorithm~\ref{alg::alternating_training}, if $\mathbf{M}_s$ are repetitive for more than half of the images in a batch, then we use random masks as the input for the \texttt{mask-backward}{} process to avoid any degeneracies.

    \item \textit{Multi-coil setting}: In this case, the $k$-space measurements of an image are obtained with $n$ coils. For each coil, the measurements can be expressed as multiplication of the image with coil sensitivity maps, followed by the Fourier transform and subsampling of $k$-space (same across coils).

    To apply the \texttt{mask-backward}{} procedure in the multi-coil setting, we only need to replace the \texttt{IFFT} operation in step 4 of subroutine~\ref{alg::mb} with the adjoint of the multi-coil measurement operator $\mathbf{A}$.

    The input to \texttt{MNet}{} can have multiple channels to accommodate data from different coils and a learnable convolution across coils/channels can be added as a first step in the \texttt{MNet}{} architecture to explore correlation among coils. After this first step, the architecture can stay the same as in the single-coil case.

\end{enumerate}

redpajama_set_name: RedPajamaArXiv


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B i o
Leigh Jones is a Portland, Oregon based musician.
After growing up steeped in musical theater in North Carolina, Leigh Jones earned a Drama degree from New York University's Tisch School of the Arts, then spent several years in the NYC music scene before moving to Portland in 2014. She has toured across the US and abroad both solo and with her former band, modern bluegrass outfit Crow and the Canyon.
In the fall of 2012 Leigh released her debut solo LP, Worth Wild, which showcases the intersection of her theater background and southern roots. Most recently, Leigh has turned her focus to a new project, harmony-driven folk trio Five Letter Word. With an unforgettable voice that has been compared to Dolly Parton and Alison Krauss, Leigh's musical performances within the band show remarkable versatility as she bounces between stringed and percussive instruments, occasionally leaning on her dance background to round out the show, and always lending her clear voice to the melodies and harmonies that form the backbone of the band's sound.
© 2013 by Leigh Jones Music. Built by WYTHE
R É S U M É | E P K

redpajama_set_name: RedPajamaCommonCrawl


text:
ACCEPTED

#### According to
The Catalogue of Life, 3rd January 2011

#### Published in
Bull. Soc. mycol. Fr. 118(3): 179 (2003)

#### Original name
Cortinarius tigrellus Soop, 2003

### Remarks
null

redpajama_set_name: RedPajamaGithub


We are a small publishing house located in Calgary, Alberta. Initially established in 2016 as a vehicle for B.G. Cousins to self-publish his first novel (The Locket), he hopes to use his self-publishing experience to assist other authors to self-publish their works under the Glen Kelty banner.
Glen Kelty Books is a member of the Alberta Romance Writers Association (ARWA).
It's been an unusually long and cold winter up here in Calgary this year. Nights dropping into the -20C range and only a few chinooks to break away from the cold. This was an ideal month to curl up with a good book or two (or three or four). For me, it has been the hard slog through research and story development for the third book in my Rainey Chronicles series, Dark Before Light. I'm about a quarter the way through the first draft and I'm hoping to have it published by June (not holding my breath, though). For March, I have another book signing on March 10 at the Indigo Signal Hill store and The Locket will again be part of an online 99 cent promotion March 23 to 27.


text:
Bassano Virtus 55 S.T. is een Italiaanse voetbalclub uit Bassano del Grappa.

Geschiedenis
De club werd opgericht in 1920 als US Bassano en fuseerde in 1968 met Virtus Bassano tot Bassano Virtus 55 ST. De clubkleuren zijn rood en wit.

Renzo Rosso, de oprichter en eigenaar van Diesel, is eigenaar van de club. De 55 in de naam van de club verwijst naar het geboortejaar van Rosso. Medio 2018 nam Bassano Virtus 55 ST de rechten van het in januari 2018 failliet gegane Vicenza Calcio over en verhuisde de club naar Vicenza waar het als LR Vicenza Virtus ging spelen. De jeugdwerking bleef wel in Bassano del Grappa.

Externe links
 Officiële site Bassano Virtus

Italiaanse voetbalclub
Sport in Veneto
Vicenza (provincie)

redpajama_set_name: RedPajamaWikipedia


Betsy Gotta of North Brunswick, New Jersey has been square dancing since 1952 and calling since 1962. As a full-time caller Betsy currently calls regularly for 5 square dance clubs, calls all programs from Contra through C-3B and is an accredited Callerlab Caller Coach. Averaging approximately 300 dances/classes per year, in addition to her regular home calling program, Betsy travels throughout the United States & Canada for guest calling engagements, weekends, festivals, and callers' schools. Betsy also is the featured caller for instructional videos produced by Square Dance Videos Inc. and sold by Gottadance Videos, and has recorded for County Line and Desert Gold Records. Betsy is a former Vice-Chairman of CALLERLAB. She is a founding member of the Callers' Council of NJ, a charter member of the Square Dance Council of NJ (and it's current Chairman), and with her husband Roy, a member of the Northern NJ Round Dance Leader's Council and Roundalab. In 2001, at the IAGSDC Convention in Vancouver, Betsy received the Golden Boot Award, for her efforts to bridge the gap between gay and straight square dance clubs.
In 2006, at the CALLERLAB Convention in Charlotte, Betsy received the Milestone Award, CALLERLAB's highest honor, for her numerous contributions to the square dance activity. At the 2012 National Square Dance Convention in Spokane, WA Betsy was honored by the National Executive Committee for her Support, Devotion and Calling at the National Square Dance Convention 1963-2012. Betsy has been an official Times Squares club caller since 2012.
Howard Richman began his training with classes in ballet and jazz at the Alvin Ailey, Steps and Broadway Dance studios. Howard then began calling Square Dances and teaching Western Dance in 1989. He attended numerous classes and seminars with CallerLab, Callers Council of New Jersey (CCNJ) and the Gay Callers Association (GCA) organizations. He has choreographed for numerous performance groups including The Times Squares, the New York Silver Spurs, and New Jersey's Red Rose. Howard Calls and teaches for over 30 square dance clubs and groups in the Tri-State area and has called in England, France, Switzerland, Holland, Norway, Belgium and toured Germany. For the last few years Howard has also been teaching Ballroom and Square Dancing on various cruise lines including Norwegian, Holland America and Royal Caribbean. Howard has been directing and acting with the Hudson Theatre Ensemble's children's theatre department "Silly on Sixth" in Hoboken New Jersey for 4 seasons. You can watch him calling for Times Squares on YouTube.
Spring break, 1981 I attended a square dance in PA with my parents. They said "Doesn't this look like fun?" I said "Sure, but I like that guy, the caller. He gets to tell everyone what to do!!" First I had to learn to dance. 3 days later, done.
Now back at college, I was befriended by a local square dance caller. The next 3 years, for 4-6 nights/week, we would ride together into Pittsburgh. We'd talk calling the whole way there, at the dance, he'd let me call a tip, then he'd critique me the whole way home. Honest immediate and direct criticism was very motivating.
Late 1984, I moved to the Washington DC area and decided to give up square dancing. Well, after attending one dance, I was hooked again. Within a year I was the club caller for 2 clubs. I'm proud to still be with Chesapeake Squares, more than 33 years later.
In the ensuing years, things continued to move steadily forward as I called dances, fly-ins, weekends, festivals, and conventions. I strive to never rest on my laurels and try to continue to improve at every dance.
Now, 36 years after that fateful spring break, I am loving my 5 clubs, the chance to entertain the best people in the world, and make wonderful, deep lasting friends.
Although he lives in Columbia, MD, Dayle is a frequent caller for Times Squares, especially at our very popular White Party in August.
Todd started dancing at age 7 and started calling at age 13 and in 1999 Todd received his certificate from Callerlab for 25 years of calling.. He took square dance lessons as part of an extra curriculum course in elementary school and graduated from Quinnipiac University with a Bachelors Degree in Accounting. Todd has called in 27 states, Sweden, Germany, and Canada. His home programs range from A-2 through C-3A. Todd also calls Fun Night and Basic level through C-4. He also travels on a regular basis for other clubs, weekends, and conventions. Todd's choreography is interesting, different, and extremely accurate. He's noted for his quick wit and sense of humor on and off stage, always wanting the dancers to succeed while at the same time keeping the level challenging. Todd's at ease calling for one square or 40. Dancers have commented on how much they enjoy his choice of music for the dance. Todd uses the sign of the Tiger as his logo. Todd is a frequent caller for Times Squares, eagerly awaited every year for our Holiday dance in December.


New Advent: Do you know the real history of Halloween?
Do you know the real history of Halloween?
Halloween is a uniquely American holiday, and Catholics understandably approach its traditions with mixed emotions. However, one may not realize that the origins of Halloween are actually very Catholic. Historian Fr. Augustine Thompson, O.P., gives us a fascinating insight into Halloween and shows how a thorough understanding of its history is essential to properly approaching the festivities with faith and joy in Christ.


text:
The Romania men's national basketball team () represents Romania in international basketball competition. The team is administered by the Romanian Basketball Federation (FRB).

Romania has qualified for the EuroBasket 18 times throughout their history. Their best tournament results occurred in 1957 and 1967 respectively, where they finished fifth. The national team has also appeared at the Olympic Games once, in 1952. However, Romania has yet to clinch qualification for their first trip to the FIBA World Cup.

History

EuroBasket 1935
At the first European Championship in Geneva, the Romania national team finished in last place in the then ten team tournament. Losing all three of their matches; 42-9 to Switzerland, 66-23 to France, and 24-17 to Hungary.

EuroBasket 1947
Romania returned to the European championship twelve years later in Prague, for the EuroBasket 1947. Romania finished with a record of 1-2 after the preliminary round, which eliminated them from championship contention but they earned their first ever EuroBasket victory against the Netherlands. The Romanians added two more in the classification semifinal, as they defeated Austria and Albania. This put them in a classification match against Italy for 9th and 10th place, which Romania lost 55-39. The national team thus placed tenth again, but this time they had outranked four teams to do so, due to the expansion of the tournament to 14.

EuroBasket 1951
The Romanian national team entered the EuroBasket 1951 in Paris, but withdrew before any games were played, so official records gave them three preliminary round losses, each with a score of 2-0.

1952 Summer Olympics
Romania managed to qualify for the 1952 Summer Olympics in Helsinki, where the national team finished 23rd. It was the first and only time they qualified for the Olympic Basketball Tournament.

EuroBasket 1953
The EuroBasket 1953 in Moscow was the next European tournament in which the Romanians actually played.  They finished in 3rd place in their preliminary round group with a 1-2 record and advanced to the classification rounds.  They won 3 of their 4 games there, finishing in a three-way tie for the top spot of the group.  They lost out on the tiebreaker though and were relegated to the 13th-16th place semifinals.  Winning that game and their next, the Romanians finished in 13th place of the 17 teams in the tournament.

EuroBasket 1955
Romania was much more successful two years later, at the EuroBasket 1955 in Budapest.  A preliminary round loss to the global power house Soviet Union mattered little, as the national team won their other three preliminary matches against Sweden, Switzerland and Luxembourg to take second place in the group behind the Soviets to advance to the final round.  There, they won games against Poland and Yugoslavia to finish 2-5 in the final round to take 7th place in the 18-team tournament.

EuroBasket 1957
Romania came within a single point of winning their preliminary pool at EuroBasket 1957 in Sofia, losing to Hungary in the narrow match.  They beat their other two opponents Belgium and Finland to finish in second place in the pool to advance into the final round.  There, they matched up again with Hungary, but the game wasn't as close as the Hungarians cruised to a 76-61 victory. The national team also lost to the other three pool winners, to finish with a 3-4 record to end the tournament in 5th place.

Later years
The team repeated its most successful campaign at the EuroBasket 1967 when it finished 5th again. The following years, the team had some successful performances but after 1987, it ceded completely from Europe's elite events. After 30 years of absence, Romania returned to the EuroBasket in 2017 when the country gained hosting rights alongside Finland, Israel and Turkey. However, the national team ended its participation with five losses in its five games played.

EuroBasket 2022 qualification
After defeating both Slovakia and Cyprus twice, the Tricolorii advanced to the final qualification round where they came short of their second EuroBasket qualification since 1987. The team was coached by Tudor Costescu.

Competitive record

FIBA World Cup

Olympic Games

EuroBasket

Results and fixtures

2021

2022

2023

Team

Current roster
Roster for the EuroBasket 2025 Pre-Qualifiers matches on 23 and 26 February 2023 against Bulgaria and Cyprus.

Depth chart

Head coach position

Notable players
Andrei Folbert – First Romanian Basketball "Maestru Emerit Al Sportului"
Gheorghe Mureșan – First Romanian to play in the NBA
Constantin Popa
Horia Demian
Ernie Grunfeld

Past rosters
1935 EuroBasket: finished 10th among 10 teams

3 Nae Alexandrescu, 4 Balteanu, 5 O.Belitoreanu, 6 M.Bercus, 7 E.Dumitrescu, 8 Nicu Grozavescu, 9 Samy Grunstein, 10 C.Riegler (Coach: C. O. Lecca)

1947 EuroBasket: finished 10th among 14 teams

3 Constantin Herold, 4 N.Badulescu, 5 H.Dlugos, 6 S.Ferencz, 7 H.Kevorkian, 8 P.Marossi, 9 Alexandru Popescu, 10 S.Sadeanu, 11 Gh.Teodorescu, 12 I.Vulescu, 13 Vasile Popescu, 14 A.Neagu, 21 C.Babaliescu (Coach: C.Virgolici)

1952 Olympic Games: finished 23rd among 23 teams

3 Cornel Călugăreanu, 5 Grigore Costescu, 6 Emanoil Răducanu, 7 Andrei Folbert, 8 Ladislau Mokos, 9 Liviu Naghy, 10 Mihai Nedef, 11 Cezar Niculescu, 12 Dan Niculescu, 13 Adrian Petroșanu, 15 Vasile Popescu, 16 Gheorghe Constantinide (Coach: Alexandru Popescu)

1953 EuroBasket: finished 13th among 17 teams

3 Mihai Nedef, 4 Cezar Niculescu, 5 Trajan Lita, 6 Andrei Folbert, 7 Emanoil Răducanu, 8 Dan Niculescu, 9 Adrian Petroșanu, 10 Alexandru Fodor, 11 Corneliu Calugareanu, 12 Mihai Erdogh, 13 Liviu Naghy, 14 Vasile Kadar (Coach: Alexandru Popescu)

1955 EuroBasket: finished 7th among 18 teams

3 E.Ganea, 4 R.Popovici, 5 Mihai Nedef, 6 Andrei Folbert, 7 Emanoil Răducanu, 8 Dan Niculescu, 9 Emil Niculescu, 10 Alexandru Fodor, 11 Mihai Erdogh, 12 Marian Spiridon, 13 Liviu Naghy, 14 Vasile Kadar, 15 Corneliu Calugareanu (Coach: Vasile Popescu)

1957 EuroBasket: finished 5th among 16 teams

3 Grigore Costescu, 4 A.Borbely, 5 Mihai Nedef, 6 Andrei Folbert, 7 Emanoil Răducanu, 8 Alexandru Berecky, 9 Emil Niculescu, 10 Alexandru Fodor, 11 Mihai Erdogh, 12 Armand Nováček, 13 Liviu Naghy, 14 A Koukouch (Coach: Alexandru Popescu)

1959 EuroBasket: finished 8th among 17 teams

3 Feodor Nedelea, 4 Armand Nováček, 5 Grigore Costescu, 6 Andrei Folbert, 7 Mihai Nedef, 8 Emeric Vizi, 9 Emil Niculescu, 10 Alexandru Fodor, 11 Dragos Nosievici, 12 Cristian Popescu, 13 Ludovic Toth, 14 Jzidor Milelman (Coach: Constantin Herold)

1961 EuroBasket: finished 7th among 19 teams

4 Aurel Popovici, 5 Emil Niculescu, 6 Horatiu Giurgiu, 7 Mihai Nedef, 8 Nicolae Viciu, 9 Mihai Albu, 10 Pavel Visner, 11 Dragos Nosievici, 12 Cristian Popescu, 13 Armand Nováček, 14 Horia Demian, 15 Gheorghe Valeriu (Coach: Constantin Herold)

1963 EuroBasket: finished 11th among 16 teams

4 Horia Demian, 5 Emil Niculescu, 6 Marian Spiridon, 7 Mihai Nedef, 8 Mihai Kiss, 9 Mihai Albu, 10 Horatiu Giurgiu, 11 Dragos Nosievici, 12 Cristian Popescu, 13 Pavel Visner, 14 Mircea Cimpeanu, 15 Gheorghe Valeriu (Coach: Vasile Popescu)

1965 EuroBasket: finished 13th among 16 teams

4 Horia Demian, 5 Gheorghe Novac, 6 Marian Spiridon, 7 Mihai Nedef, 8 Nicolae Ionescu, 9 Mihai Albu, 10 Alin Savu, 11 Dragos Nosievici, 12 Cristian Popescu, 13 Armand Nováček, 14 Ekehardt Jekely, 15 Aurel Popovici (Coach: Vasile Popescu)

1967 EuroBasket: finished 5th among 16 teams

4 Adolf Cernea, 5 Gheorghe Novac, 6 Dragos Nosievici, 7 Alin Savu, 8 Ekehardt Jekely, 9 Mihai Albu, 10 Gheorghe Barau, 11 Nicolae Birsan, 12 Cristian Popescu, 13 Titus Tarau, 14 Radu Diaconescu, 15 Horia Demian (Coach: Alexandru Popescu)

1969 EuroBasket: finished 9th among 12 teams

4 Francisc Dikay, 5 Matei Ruhring, 6 Dragos Nosievici, 7 Constantin Dragomirescu, 8 Ekehardt Jekely, 9 Mihai Albu, 10 Petru Czmor, 11 Radu Diaconescu, 12 Gheorghe Novac, 13 Titus Tarau, 14 Mihai Dimancea, 15 Adolf Cernea (Coach: Alexandru Popescu)

1971 EuroBasket: finished 8th among 12 teams

4 Dan Georgescu, 5 Mircea Chivulescu, 6 Nicolae Pirsu, 7 Alin Savu, 8 Dan Niculescu, 9 Mihai Albu, 10 Constantin Dragomirescu, 11 Radu Diaconescu, 12 Gheorghe Cimpeanu, 13 Titus Tarau, 14 Gheorghe Oczelak, 15 Vasile Popa (Coach: Alexandru Popescu)

1973 EuroBasket: finished 9th among 12 teams

4 Gheorghe Cimpeanu, 5 Mircea Chivulescu, 6 Nicolae Pirsu, 7 Nicolae Manaila, 8 Dan Niculescu, 9 Dan Georgescu, 10 Costel Cernat, 11 Radu Diaconescu, 12 Gheorghe Novac, 13 Titus Tarau, 14 Gheorghe Oczelak, 15 Vasile Popa (Coach: Alexandru Popescu)

1975 EuroBasket: finished 11th among 12 teams

4 Dan Georgescu, 5 Georghe Mihuta, 6 Vasile Zdrenghea, 7 Dan Moisescu, 8 Nicolae Pirsu, 9 Dan Niculescu, 10 Costel Cernat, 11 Radu Diaconescu, 12 Gheorghe Novac, 13 Titus Tarau, 14 Gheorghe Oczelak, 15 Vasile Popa (Coach: Mihai Nedef)

1985 EuroBasket: finished 10th among 12 teams

4 Florentin Ermurache, 5 Attila Szabó, 6 Viorel Constantin, 7 Ioan Ionescu, 8 Doru Rădulescu, 9 Dan Niculescu, 10 Costel Cernat, 11 Roman Opsitaru, 12 Anton Netolitzchi, 13 Victor Jacob, 14 Marian Marinache, 15 Alexandru Vinereanu (Coach: Gheorghe Novac)

1987 EuroBasket: finished 12th among 12 teams

4 Florentin Ermurache, 5 Sorin Ardeleanu, 6 Robert Reisenbuchler, 7 Ioan Ionescu, 8 Petre Brănișteanu, 9 Dan Niculescu, 10 Costel Cernat, 11 Andrei Constantin, 12 Anton Netolitzchi, 13 Constantin Popa, 14 Gabriel David, 15 Alexandru Vinereanu (Coach: Gheorghe Novac)

2017 EuroBasket: finished 23rd among 24 teams

4 Andrei Mandache, 6 Catalin Petrisor, 8 Radu Paliciuc, 9 Vlad Moldoveanu, 10 Bogdan Nicolescu, 11 Octavian Calota, 13 Alexandru Olah, 14 Titus Nicoara, 15 Emanuel Cățe, 25 Rolland Torok, 26 Catalin Baciu, 32 Nandor Kuti (Coach: Marcel Tenter)

Kit

Manufacturer
2014: Ancada
2015–2017: Peak
2019–present: Spalding

Sponsor
2020–present: Banca Transilvania

See also

Sport in Romania
Romania women's national basketball team
Romania men's national under-20 basketball team
Romania men's national under-18 basketball team
Romania men's national under-16 basketball team
Romania men's national 3x3 team

Notes

References

External links

Official website
Romania FIBA profile
Romanian National Team – Men at Eurobasket.com
Romania Basketball Records at FIBA Archive

Videos
Romania v Spain - Highlights - FIBA EuroBasket 2022 - Qualifiers

Men's national basketball teams
1932 establishments in Romania
Romania national basketball team


text:
Q: C# - Bitwise operations on large integers I have many flags that I want to store them in an integer, so that later I can perform bitwise AND operation on them.
The number of these flags may be up to 1000. However, there are no types in C# that can store 1000 bits.
For Example, I want to perform (&) operation onü
100000010000000011100000000000000000000000000000...........00001 --> (1000 flags)
000000000000000001111110000000000000000000010101.......... 11111 --> (1000 flags)

So, how can I perform this (&) operation on a two bit collections?

A:
I have many flags that I want to store them in an integer, so that later I can perform bitwise AND operation on them.

That's a non-sequitur. You can use BitArray with its And method to satisfy that requirement without artificially going via a numeric type.
If you haven't logically got a number (and "many flags" doesn't sound like a number to me) you shouldn't go out of your way to use a numeric representation.

A: I guess BigInteger is what you are looking for. It has BitwiseAnd operator

redpajama_set_name: RedPajamaStackExchange


To counter the view that women are systematically discriminated against, especially in the workplace, the authors assert that equality has been achieved in the key areas of education, work and the law. Their conclusions are illustrated in graphs and tables.
Diana Furchtgott-Roth is a resident Fellow and assistant to the president at the American Enterprise Institute in Washington, D.C. Christine Stolba is a historian in Washington, D.C., and a Ph.D. candidate in American History at Emory University.


L-A Community Little Theatre Presents "Lend Me A Tenor"
Community Little Theatre (CLT) is proud to bring back (after 25 years) Ken Ludwig's hilarious farce "Lend Me A Tenor," directed by Mitchell Clyde Thomas. home. The show will run January 11-20 at the Great Falls Performing Arts Center, at 30 Academy Street in Auburn.
In 1934, when opera star "Tito Merelli" comes to Cleveland, everyone is dying to meet him. But after a huge fight with his wife, Tito, in a fit of despair, takes too many pills and suddenly slips into unconsciousness only minutes before the curtain. Put on the spot by his future father-in-law and manager of the Cleveland Grand Opera Company, "Max", the lowly office assistant and opera singer wannabe, is ordered to take Tito's place and fool the opera aficionados of Cleveland. With countless doors and several mistaken identities, passion runs rampant as the situation resolves in a spectacular ending. This fast-paced comedy will leave sides aching from laughter.
The show will be performed January 11, 12, 17, 18, and 19 at 7:30 p.m., and January 13 and 20 at 2:00 p.m. Call 783-0958 or visit LACLT.com for tickets and information. Tickets are $18, $15 for seniors and $12 for students.


As if offering a pinch of incense to the gods, the article begins: "Blood transfusions have saved millions of lives." But doesn't the very next phrase almost douse the flame?
"yet stored red cells may be less effective than hoped for because they can quickly lose much of their ability to deliver oxygen."
They can? They do? If so, blood transfusions haven't saved as many lives as they are credited with, perhaps not any more than simple saline solution or any other fluid that aims only to replace lost volume. Jehovah's Witnesses have no objection to the latter. They do, however, for religious reasons, decline the former. They are well known for refusing blood transfusions.
If I had a nickel for every news article or web post lambasting us for our transfusion stand....some going so far as to call it tantamount to murder....I could buy myself a Maserati. And another, even fancier car for my wife, Mrs. Sheepandgoats. Blood substitutes are no good, they scream at us, because only the real thing, only real packed red cells, delivers the life-saving oxygen to the body. Yet according to this article, they don't!
The problem is that transfused blood needs nitric oxide to keep the blood vessels open, otherwise, the carried oxygen never reaches the tissues. But nitric oxide begins to break down within three hours of storage, and donated blood is presently stored up to 42 days. To be sure, researchers think they can remedy the problem. But that does nothing to improve the effectiveness of blood transfusions already given, each one of which was hailed as "life-saving," yet few of them actually qualifying as such, at least not any more so than saline solution, which offers no danger of rejection. We all know that the body spots foreign tissue in an instant, and tries hard to get rid of it.
All this reminds me of Bruce Spiess, addressing the Australian and New Zealand College of Anesthetists a few months ago. He declares blood transfusions have hurt more people than they've helped. Transfusions, he observes, are "almost a religion" because physicians practice them without solid evidence that they help. "Blood transfusion has evolved as a medical therapy and it's never been tested like a major drug," he said. "A drug is tested for safety and efficacy, blood transfusion has never been tested for either one."
Meanwhile, Jehovah's Witnesses steadfastly refuse blood transfusions (for religious reasons, not medical) and have created hundreds of Hospital Liaison Committees composed of members who interact with local hospitals and doctors. As a result, some in the medical field have pioneered bloodless techniques. By eliminating the risk of foreign tissue, human error, and blood-borne diseases, these new techniques offer a safety margin that conventional blood transfusions do not. The film Knocking states there are over 140 medical centers in North America that offer some form of bloodless surgical techniques. Might the day come, or is it even here already, when the number of lives saved through such medicine will outnumber those lost by a few members of a relatively tiny religious group that stuck to its principles amidst much opposition?
Jehovah's Witnesses refuse blood for religious reasons you say? Well I am Jewish and Jews see no religious conflict at all with blood transfusions. The command to not eat blood has nothing to do with blood transfusions. You see, all the blood transfusions in the world will not keep a starving man alive. Blood is food only if it is eaten, but transfused blood is only a medium to transport nutrients and oxygen- not food. The Bible says nothing about blood transfusions, but everything about the sacredness of life. JW's have been needlessly letting their loved ones die and as a result they must pay the price for both individual and Watchtower collective bloodguilt.
This was in new Scientist this week, along with the findings that sending oxygenated blood back into damaged tissue might make things worse, not better. Blood transfusions have saved lives, but they might also have ended a few prematurely. It's an accepted practice. It's accepted without question by doctors who say they can't accept homeopathy because it hasn't been studied in extreme detail. Go figure.
I'm fortunate never to have been in a position where I might need one so I can't say whether I'd accept or not. It's my understanding that the patient is rarely in a position to decide, often they're not even able to speak.
It's an organ transplant. It's therefore a matter for individual choice. If you'd rather be dead than have someone else's heart beating in you, why should anyone else argue with that?
If people say they don't like a particular group (in this case you -JW's) then why are they so adamant that you must be kept alive even when you don't want it?
Death holds no terrors for me. I've already seen them and they're not so bad. In an earlier post, you talked about the way our old people are kept alive as drooling vegetables using drugs, rather than being allowed a dignified exit. I'm with you on that one too. No adult diapers for me - I'd rather be worm-food than a profit margin for a drug company.
Only those who have no afterlife to look forward to are so filled with fear of death that they have to avoid it at any cost. My idea of what comes next differs from yours, but we can agree there IS a 'what comes next'.
There's another interesting article in that particular issue. It seems that if there was no ageing or disease, only accidental death or murder, the average lifespan of a human would be around 1400 years.
Even Adam only made it to 930. If he hadn't left Eden he might have had another 500 years in him.
Note that the reason has nothing to do with dietary or health considerations. The prohibition is extended in the New Testament, where Christians are told to "abstain" from blood. [Acts 15:20] "Abstaining" might be viewed as more than merely "eating," much as an alcoholic wouldn't really be abstaining from alcohol if he devised a method to inject it into his veins.
As mentioned in the post, doctors have developed bloodless medicine techniques, safer than transfusions, largely due to the tenacity (stubbornness?) of Jehovah's Witnesses. Made available to the general public, these methods should save considerable lives. Society will therefore owe a debt to JWs for medical advances, just as they now do in the field of constitutional law.
The pioneers take the arrows. It's always been that way.
Moshe Feldman: If this study is correct, transfused blood carries little to no oxygen to cells, which is the only reason to choose it over safer non-blood volume expanders. Are you saying the study is incorrect?
Romulus: As Leviticus shows, our abstaining from blood is neither a desire for martyrdom, nor exercise of a right to die. We view it simply as being obedient to a command of God. Historically, people have been willing to put their lives on the line, if need be, for many causes, most far less noble than obeying God.
Do you have a link for that 1400 lifespan article? I'd love to read it.
I do know about the Jehovah's Witness edict against blood trasfusions, but I don't honestly know if there is or isn't a direct benefit to having one or not. In times of surgery they do have ways of working with those who cannot tolerate blood transfusions for whatever reason.
However, as someone who cannot give blood in the field (but often becomes an unwitting recipient of unasked-for blood), we all give praise to oxygen. The one medicine that EMT-Bs are allowed to give is O2 - and that saves far more lives.
I would stand by that statement any time.
If blood subtitutes don't carry oxygen and transfused blood, for whatever reason, doesn't successfully make the transfer, can O2 administered through the airways compensate? I've never thought of (or heard of) that before. You know more than I.
I do know that the hematocrit value (% of red blood cells) that used to ring alarm bells and trigger efforts to give a blood transfusion was set completely arbitrarily. Much lower levels of oxygen in the bloodstream can be (temporarily) tolerated.
Thanks for your thoughts, Aislinge.
Interesting. It appears that the article is simply rewritten from the Associated Press article. Nice liberties taken. I like how the AP article comments how it is a necessary technology though it is detrimental in enough cases to concern medical professionals. I'm curious how they will restore the Nitrous Oxide at the time of transfusion. Will they suffuse it, like the air filters do for a fish tank?


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How Long Does It Take To Get Into Ketosis. It takes between one and ten days to get into ketosis if you stick to your ketogenic diet. The time depends on prior nutrition, the body's metabolism and other factors such as supplements and fasting.
What is the best time of day to take CBD oil? Where Can You Find High-Quality CBD Oil? How Long Does CBD Take to Work? CBD can generally take from a few minutes to a couple of hours to work properly, depending on the serving size, the method of consumption, and the symptom you want to alleviate.
Your Definition of "Improvement": Obviously, it will take a lot longer to improve your credit rating from "bad" to "good" than to increase your score by a few points. In that sense, perspective matters. Below, we'll take a closer look at how long it generally takes to rebound from some common sources of credit-score damage.


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MSC And Costa Plan to Restart Europe Cruising This Month
Publish Time:2021-01-11 18:09:46Source:Travel Weekly
【Introduction】:MSC Cruises and Costa Cruises plan to resume service at the end of this month, after a lengthy pause due to Italy s extensive travel restrictions related to the nation s Covid-19 surge.
MSC Cruises and Costa Cruises plan to resume service at the end of this month, after a lengthy pause due to Italy's extensive travel restrictions related to the nation's Covid-19 surge.
MSC plans to resume sailings on Jan. 24, with a departure from Genoa on the MSC Grandiosa. The ship will visit the Italian ports of Civitavecchia, Naples and Palermo, and Valetta in Malta, with embarkation available in each port in Italy. The MSC Magnifica will also resume its 11-day Genoa sailings, starting Feb 14, to Western and Eastern Mediterranean ports in Italy, Greece and Malta.
Costa canceled all departures until Jan. 31, when it plans to resume service with the Costa Deliziosa, which will take over the 3-, 4-, and 7-day sailings to Italian ports originally planned for the Costa Smeralda. That ship will remained paused while the line makes new plans.
MSC and Costa were the first two lines to resume big-ship cruising this summer, offering Mediterranean sailings starting August and September. They were among the few European lines sailings when a second wave of Covid-19 hit Europe in the fall, causing them to cancel sailings starting in November and through the holiday season.
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