text:
\section{Algorithm}\label{sec:algorithm}
The proposed SPULTRA algorithm is based on a pre-learned union of sparsifying transforms. The process of learning such a union of transforms from a dataset of image patches has been detailed in~\cite{pwls-ultra2018}. The learning problem in \cite{pwls-ultra2018} simultaneously groups the training patches into $K$ \textcolor{black}{classes }and learns a transform in each \textcolor{black}{class }along with the sparse coefficients (in the transform domain) of the patches. This learning is accomplished by an alternating algorithm (see~\cite{pwls-ultra2018}).
This section focuses on describing the algorithm in the reconstruction stage for SPULTRA, i.e., for \eqref{eq:P0}.

The data-fidelity term $\mathsf{L}(\x)$ in \eqref{eq:P0} is nonconvex when the electronic noise variance $\sigma^2$ is nonzero. It is challenging to directly optimize such a logarithmic nonconvex function. We propose to iteratively design quadratic surrogate functions for this data-fidelity term $\mathsf{L}(\x)$. In each iteration, we optimize the surrogate function that is a quadratic data-fidelity term together with the ULTRA regularizer using alternating minimization that alternates between an image update step and a sparse coding and clustering step that has closed-form solution~\cite{wen:14:sos}. We use the relaxed OS-LALM algorithm for the image update step~\cite{nien:16:rla}. We perform only one alternation between the two steps for each designed surrogate function, which saves runtime and works well in practice.

\vspace{-0.1in}
\subsection{Surrogate function design} \vspace{-0.05in}
We design a series of quadratic surrogate functions for $\mathsf{L}(\x)$ as follows:
\begin{equation}\label{eq:surrogate}
\begin{aligned}
\phi(\x;\x^n) &= \mathsf{L}(\x^n) + \bm{d}_h(l^n)\A(\x - \x^n)\\
&\qquad+\frac{1}{2}(\x - \x^n)^T\A^T\W^n\A(\x - \x^n),
\end{aligned}
\end{equation}
where $(\cdot)^n$ denotes values at the $n$th iteration and ${\bm{d}_h(l^n) \in \mathbb{R}^{N_d}}$ is a row vector capturing the gradient information and is defined as $\bm{d}_h(l^n) \triangleq [\dot{h_i}(l_i^n)]_{i = 1}^{N_d}$. The curvatures of the $n$th updated parabola (surrogate) are described by $\W^n\triangleq \text{diag}\{c_i(l^n_i)\}$. In this paper, we use the optimum curvatures \cite{TMI99} that are defined as follows:
\begin{equation} \label{eq:OptCuv}
c_i(l^n_i) =  \begin{cases}
\big[2\frac{h_i(0) - h_i(l_i^n) + (l_i^n)\dot{h_i}(l_i^n)}{(l_i^n)^2}\big]_+, &l_i^n > 0\\
\big[\ddot{h_i}(0)\big]_+, &l_i^n = 0,
\end{cases}
\end{equation}
where $\ddot{h}_i$ is the second-order derivative, and operator $[\cdot]_+$ sets the non-positive values to zero. In practice, we replace negative values with a small positive number so that the diagonal matrix $\W^n$ is invertible.
Due to numerical precision, \eqref{eq:OptCuv} might become extremely large when $l_i^n$ is nonzero but small. To avoid this problem, we use an upper bound of the maximum second derivative $\big[\ddot{h_i}(0)\big]_+$ for the curvature $c_i(l^n_i)$ when $l_i^n > 0$ \cite{TMI99}.

By ignoring the terms irrelevant to $\x$ in \eqref{eq:surrogate}, we get the following equivalent form of $\phi(\x;\x^n)$:
\begin{equation}\label{deduce2}
\phi(\x;\x^n) \equiv \frac{1}{2}||\tilde{\y}^n - \A\x||_{\W^n}^2,
\end{equation}
where ``$\equiv$'' means equal to within irrelevant constants of $\x$, and $\tilde{\y}^n \triangleq \A\x^n -\big(\W^n\big)^{-1}[\bm{d}_h(l^n)]^T$. The overall surrogate function at the $n$th iteration for the penalized-likelihood objective function \textcolor{black}{$G(\x)$} in \eqref{eq:P0} is then
\begin{equation}\label{eq:surr_all}
\Phi(\x;\x^n) = \frac{1}{2}||\tilde{\y}^n - \A\x||_{\W^n}^2 + \R(\x),\ \textcolor{black}{\text{s.t.}\ \x \in \mathcal{X}.}
\end{equation}

We \textcolor{black}{descend} the surrogate function $\Phi(\x;\x^n)$ in \eqref{eq:surr_all} by alternating \textcolor{black}{once} between an image update step, and a sparse coding and clustering step.

\vspace{-0.1in}
\subsection{Image Update Step}\vspace{-0.05in}
In the image update step, we update the image $\x$ with fixed sparse codes $\{\z_j\}$ and \textcolor{black}{class} assignments $\{C_k\}$. The relevant part of the majorizer for this step is
\begin{equation}\label{eq:Phi1}
 \begin{aligned}
 \Phi_1(\x;\x^n) = \phi(\x;\x^n)  + \beta \sum_{k=1}^{K} \sum_{j\in C_k} \tau_j \|\omg_k \P_j \x - \z_j\|^2_2
 \end{aligned}
 \end{equation}
\textcolor{black}{Although we have a box} constraint on $\x$\textcolor{black}{, i.e., $\x \in \mathcal{X}$, in practice, the upper bound $x_{\mathrm{max}}$ can be set high such that it will not be active. We applied the }relaxed OS-LALM algorithm~\cite{nien:16:rla} \textcolor{black}{to minimize }\eqref{eq:Phi1} \textcolor{black}{with the constraint. This algorithm is } shown in Algorithm \ref{alg: spultra-alg} (steps 7-10). The OS-LALM method uses majorizing matrices. In particular, the matrix $\A^T\W^n\A$ is majorized by \textcolor{black}{$\D_{\A} \triangleq \text{diag}\{\A^T\W^n \A \mathbf{1}\}$, where $\mathbf{1}$ denotes a vector of ones.} Denoting the regularization term in \eqref{eq:Phi1} as $\R_2(\x)$, its gradient is
\begin{equation}\label{eq:grad_r2}
\nabla \R_2(\x) = 2\beta\sum_{k=1}^{K} \sum_{j\in C_k} \tau_j \P_j^T \omg_{k}^T (\omg_k \P_j \x - \z_j ).
\vspace{-0.05in}
\end{equation}
The Hessian of $\R_2(\x)$ is majorized by the following diagonal matrix:
\begin{equation}\label{eq: D_R}
\D_{\R} \triangleq 2\beta \bigg\{\max_k \|\omg_{k}^T\omg_{k}\|_2\bigg\}\sum_{k=1}^{K} \sum_{j\in C_k}\tau_j \P^T_j \P_j .
\end{equation}
The (over-)relaxation parameter $\alpha \in [1,2)$ and the parameter $\rho_t >0$ decreases with iterations $t$ in OS-LALM according to the following equation \cite{nien:16:rla}:
\begin{equation}\label{eq:rho}
\rho_t(\alpha)= \begin{cases}
1,      &  t=0\\
\frac{\pi}{\alpha(t+1)}\sqrt{1-\big(\frac{\pi}{2\alpha(t+1)}\big)^2},     &  \text{otherwise.}
\end{cases}
\vspace{-0.1in}
\end{equation}
\vspace{-0.1in}
\subsection{Sparse Coding and Clustering Step}
Here, with $\x$ fixed, we jointly update the sparse codes and the \textcolor{black}{class }memberships of patches. The relevant part of the cost function for the sparse coding and clustering step is
\begin{equation}\label{eq:codes-clusters}
\min_{\{\z_j, C_k\}} \sum_{k=1}^{K}  \bigg\{  \sum_{j\in C_k} \tau_j \{\|\omg_k\P_j \x - \z_{j}\|^2_2 + \gamma_c^2\|\z_j\|_0 \}\bigg\} .
\end{equation}
\textcolor{black}{Problem \eqref{eq:codes-clusters} is separable in terms of the patches, so each patch is clustered and sparse coded independently in parallel. The optimal sparse code $\z_j$ in \eqref{eq:codes-clusters} is obtained by hard-thresholding, i.e., ${\z_j= \mathit{H}_{\gamma_c}(\omg_{k_j}\P_j\x)}$, where ${\mathit{H}_{\gamma_c}(\cdot)}$ represents a vector hard-thresholding operator that zeros out elements whose magnitudes are smaller than $\gamma_c$, and leaves other entries unchanged. Then the optimized class $\hat{k}_j$ for the $j$th patch is computed as follows~\cite{pwls-ultra2018}:}
\begin{equation}
\label{eq:clustering}
 \small{\hat{k}_j = \argmin_{1\leq k \leq K} || \omg_{k} \P_j \x - \mathit{H}_{\gamma_c}(\omg_{k}\P_j\x)||^2_2 + \gamma_c^2\|\mathit{H}_{\gamma_c}(\omg_{k}\P_j\x)\|_0. }
\end{equation}
\textcolor{black}{We compute the cost values on the right hand side of \eqref{eq:clustering} for each $k = 1, \cdots, K$, and determine the $\hat{k}_j \in \{1, \cdots, K\}$ that gives the minimal cost value, i.e., patch $\P_j \x$ is grouped with the transform that provides the smallest value of the cost in \eqref{eq:clustering}. Then, the corresponding optimal sparse code is ${\hat{\z}_j = \mathit{H}_{\gamma_c}(\omg_{\hat{k}_j}\P_j\x)}$.}
\begin{algorithm}[!htp]
	\caption{SPULTRA Algorithm}
	\label{alg: spultra-alg}
	\begin{algorithmic}[1]
		\Require~~\\
		initial image $\hat{\x}^{0}$; $\alpha = 1.999$ ; $\rho_0 = 1$; \\
		pre-computed $\D_\R $ according to \eqref{eq: D_R}\\
		number of outer iterations $N$, number of inner iterations $P$, and number of ordered-subsets $M$.
		\Ensure reconstructed image $\hat{\x}^N$.
		\For {$n=0,1,2,\cdots, N-1$}
		\State \textbf{(1) Image Update}: Fix $\hat{\z}_j^{n}$ and $\hat{C}_k^n$; \\
		\textit{Initializations:
		}\begin{enumerate}
			\item $\x^{(0)} =  \hat{\x}^{n}$,
			\item Determine $c_i(l^n_i)$ according to \eqref{eq:OptCuv},
			\item $\W^n = \diag\{c_i(l^n_i)\}$,
			\item $\D_{\A} \triangleq \diag\{\textcolor{black}{\A^T\W^n\A\mathbf{1}}\}$,
			\item \textcolor{black}{$\bm{d}_h(l^n) = [I_0e^{-f_i(l^n_i)}\dot{f_i}(l_i^n)(\frac{Y_i}{I_0e^{-l^n_i} + \sigma^2} - 1)]_{i = 1}^{N_d}$,
			}\item $\tilde{\y}^n = \A\x^{(0)} -{\W^n}^{-1}[\bm{d}_h(l^n)]^T$,
			\item $\ze^{(0)} = \g^{(0)} = M\A_M^T\W^n_M(\A_M\x^{(0)}-\tilde{\y}_M^n) $,
			\item $\bm{\eta}^{(0)} = \D_\A \x^{(0)} - \ze^{(0)}$,
			\item compute $\nabla \R_2(\x)$ according to \eqref{eq:grad_r2}.
		\end{enumerate}

		\For {$p=0,1,2,3,\cdots,P-1$}
		\For {$m=0,1,2,3,\cdots,M-1$}
		\begin{equation*}\label{eq:rlalm}
		\begin{aligned}
		&t ~= ~pM + ~m;\\
		&\hspace{-0.1in}\left\{
		\begin{aligned}
		\s^{(t+1)} &= \rho_t(\D_\A \x^{(t)} -\bm{\eta}^{(t)}) + (1-\rho_t)\g^{(t)} \\
		\x^{(t+1)} &= [\x^{(t)} - (\rho_t\D_\A+\D_\R)^{-1}(\s^{(t+1)} +\nabla \R_2(\x^{(t)}))]_\mathcal{C} \\
		\ze^{(t+1)}& \triangleq M\A_m^T\W^n_m(\A_m\x^{(t+1)}-\tilde{\y}_m^n)   \\
		\g^{(t+1)} &= \frac{\rho_t}{\rho_t+1}(\alpha \ze^{(t+1)} + (1-\alpha)\g^{(t)}) +  \frac{1}{\rho_t+1}\g^{(t)}\\
		\bm{\eta}^{(t+1)} &= \alpha(\D_{\A} \x^{(t+1)} -\ze^{(t+1)}) + (1-\alpha)\bm{\eta}^{(t)}
		\end{aligned}
		\right.\\
		&\text{Decrease $\rho_t$ according to \eqref{eq:rho}};
		\end{aligned}
		\end{equation*}
		\EndFor
		\EndFor
		\State  $\hat{\x}^{n+1} = \x^{(t+1)}$;
		\State \textbf{(2) Sparse Coding and Clustering}: Fix $\hat{\x}^{n+1}$, compute class assignments $\hat{k}_j^{n+1}$ using \eqref{eq:clustering}, and sparse codes  $\hat{\z}_{j}^{n+1} =  H_{\textcolor{black}{\gamma_c}}(\omg_{\hat{k}_j^{n+1}} \P_j \hat{\x}^{n+1}),\   \forall \ j$.
		\EndFor
	\end{algorithmic}
\end{algorithm}

\textbf{Algorithm} \ref{alg: spultra-alg} illustrates the proposed optimization algorithm for Problem \eqref{eq:P0}.
\subsection{Computational Cost} \label{sec:computation analysis}
The SPULTRA algorithm has a similar structure in each iteration as the recent PWLS-ULTRA \cite{pwls-ultra2018}, except for several initializations in the image update step. Since forward and backward projections are used to compute $\D_\A$ and $\tilde{\y}^n$ during initialization, the image update step of SPULTRA is slightly slower than PWLS-ULTRA. In our experiments, we observed that the initializations took around $20 \%$ of the runtime in each outer iteration. However, in practice, especially for low doses, SPULTRA reconstructs images better than PWLS-ULTRA for a given number of outer iterations. Or alternatively, SPULTRA takes much fewer outer iterations (and runtime) to achieve the same image reconstruction quality as PWLS-ULTRA. These results are detailed in Sec. \ref{sec:experiments}.

\section{Introduction}
\vspace{-0.03in}
X-ray computed tomography (CT) is a popular imaging modality in many clinical and industrial applications.
There has been particular interest in CT imaging with low X-ray dose levels that would reduce the potential risks to patients from radiation.
However, image reconstruction at low X-ray dose levels is challenging.
Conventional X-ray CT image reconstruction methods include analytical methods, and model-based iterative reconstruction (MBIR) methods.
A classical analytical method is the filtered back-projection (FBP) method \cite{feldkamp1984practical}.
That can be degraded excessively by noise and streak artifacts in low-dose situations~\cite{streak-artifact-mA09,nonlocalmean2014}.

MBIR methods incorporate the system physics, statistical model of measurements, and typically certain simple prior information of the unknown object~\cite{survey2013-ct}.
A typical method of this kind is the penalized weighted-least squares (PWLS) method, for which the cost function includes a weighted quadratic data-fidelity term that models the measurement statistics, and a penalty term called a regularizer that models the prior information \cite{pwls1994jf,thibault:07:atd,beister2012iterative}.
For PWLS, various optimization approaches and regularization designs have been exploited with efficiency and convergence guarantees.

Adopting appropriate prior knowledge of images for MBIR approaches is also important to improve CT reconstruction.
More recently, with the availability of data sets of \mbox{CT} images, methods based on big-data priors have gained interest, such as dictionary learning-based techniques~\cite{xu:12:ldx}.
The dictionary can be either pre-learned from training data, or adaptively learned with the reconstruction.
In particular, the synthesis dictionary learning approaches represent a signal or image patch as a sparse linear combination of the atoms or columns of a learned dictionary, and have obtained promising results in many applications \cite{ravishankar2010mr,mairal2007sparse,elad2006image}.
However, the dictionary learning based MBIR approaches are often computationally expensive due to expensive sparse coding (where typically NP-hard problems are optimized for estimating sparse coefficients). Different from synthesis dictionary learning, sparsifying transform (a generalized analysis dictionary model) learning techniques efficiently adapt an operator to approximately sparsify signals in transform domains, and the corresponding transform sparse coding problem can be solved exactly and cheaply by thresholding~\cite{STlearning13}.
Sparsifying transform learning techniques including using doubly-sparse transforms and unions of transforms have been applied to image reconstruction and obtained promising results~\cite{doublyST13,pwls-ultra2018,ye2018spultra}.

\vspace{-0.03in}
Very recently, there has been growing interest in \textcolor{black}{deep learning approaches for medical imaging problems \cite{yang2017dagan,yu2017deep,schlemper2018stochastic,jin:17:dcn,kang2017deep,WavResNet18}}. \textcolor{black}{In the LDCT image reconstruction field, typical deep learning methods} learn the reconstruction mapping from large datasets of pairs of (low-dose and regular-dose) scans.
\textcolor{black}{These methods} include image-domain learning, sensor-domain learning, and hybrid-domain learning.
For example, a particular image-domain learning approach is the FBPConvNet scheme~\cite{jin:17:dcn} that solves the normal-convolutional inverse problems by applying a (learned) CNN after the direct inversion that encapsulates the system physics.
The image-domain learning approaches can have many variations. For example, instead of directly working in the image domain, one can transform the images to a specific domain and learn in such domain the relationship between training pairs. Kang et al.~\cite{kang2017deep} designed a neural network that learns a mapping between contourlet transform coefficients of the low-dose input and its high-dose counterpart. This work was later extended to learn a wavelet domain residual network (WavResNet)~\cite{WavResNet18}.

\vspace{-0.03in}
In the sensor-domain deep learning category, W{\"u}rfl et al.~\cite{wurfl2016deep} proposed an end-to-end neural network for low-dose CT that maps the sinogram to the reconstructed image by mapping the filtered back-projection algorithm to a basic neural network. This allows one to take into account the artifacts in the sensor-domain, e.g., the scatter and beam-hardening artifacts, and compensate them in the learning process.
Another framework named the Automated Transform by Manifold Approximation (AUTOMAP) \cite{AUTOMAP2018} \mbox{learns} a direct mapping from the measurement domain to image domain. However, due to the high memory requirements for storing fully connected layers, it is challenging for AUTOMAP to handle large scale reconstruction tasks such as CT image reconstruction.
Hybrid-domain learning approaches exploit data-fidelity terms in the neural network architecture. The Learned ISTA (LISTA) \cite{LISTA2010} was one of the earliest work of this kind. LISTA unfolds the iterative soft-thresholding (ISTA) algorithm~\cite{fista2009}, and learns the weight matrices and the sparsifying soft-thresholding operator. Later, Yang et al. \cite{admm-net2016} proposed an ADMM-Net which unfolds the alternating direction method of multipliers algorithm for image reconstruction. Each step of the \mbox{algorithm} is mapped to a neural network module.
This idea was then extended to a learnable primal-dual approach based CNN~\cite{adler2018primaldual}.
These methods fall in the class of physics-driven deep learning methods~\cite{ravchfess17,ravacfess18,chfess18}.
Hybrid-domain approaches also include a type that applies a plug-and-play model. He et al.~\cite{3pADMM} applied the plug-and-play model to the ADMM algorithm and unfolded it into a deep reconstruction network, so that each network module is learnable and replaceable.

Most deep learning algorithms are learned in a supervised manner (using task-specific cost functions) and require large training sets. However, in CT imaging, it is often difficult to acquire large datasets of training image pairs. Even though in the AAPM X-ray CT Low-Dose Grand Challenge, both regular-dose and the matched quarter-dose images were provided, only the regular-dose images were reconstructed from real scans, while the matched quarter-dose images were synthesized by adding noise to the regular-dose sinogram data. Therefore, training with smaller number of paired data (and yet generalizing) or without reference data is highly conducive for CT image reconstruction.
Moreover, different machine learning (as well as conventional) approaches such as dictionary or transform learning and deep learning use different types of big-data priors and are advantageous in different ways. For example, transform learning approaches learn general image properties and features in an unsupervised or model-based manner, and can easily and effectively adapt to specific image instances.

\begin{figure*}[!t]
	\centering
	\includegraphics[scale=0.67, trim = 10 170 6 130, clip]{arch.pdf}

	\vspace{-0.09in}
	\caption{Overall structure of the proposed reconstruction framework.}
	\label{fig:flowchart}
	\vspace{0.02in}
\end{figure*}

In this work, we propose a new image reconstruction framework for LDCT dubbed Supervised-UnsuPERvised (SUPER) learning.
The algorithm architecture involves interconnected supervised (deep network) and unsupervised (iterative reconstruction) modules over many layers. The architecture enables effectively leveraging different kinds of big data learned priors
for CT reconstruction.
For example, we used FBPConvNet~\cite{jin:17:dcn} as the supervised (deep) learned module and PWLS-ULTRA~\cite{pwls-ultra2018} as the unsupervised module with a pre-learned union of transforms, which provided both high quality image reconstruction and image-adaptive clustering.
The proposed SUPER learning used relatively small training sets and dramatically outperformed both deep learning and transform/dictionary learning by effectively combining task-specific and image or instance-specific adaptivity.
The proposed framework is generalizable to include various constituent modules including non-learning based algorithms, as shown in our experiments.

\section{Proposed Model and Algorithms}
\vspace{-0.03in}
Here, we present the proposed reconstruction model, its motivations and interpretations, example architectures, and training method.

\vspace{-0.0in}
\subsection{Overview of the SUPER Model}
\vspace{-0.0in}
We propose a novel efficient physics-driven learning framework for CT reconstruction that effectively combines the benefits of supervised (deep) learning and unsupervised iterative reconstruction methods.
The proposed reconstruction architecture takes an initial image as input and processes it through multiple ``super" layers (Fig.~\ref{fig:flowchart}).
Each super layer consists of a network learned in a supervised manner (\emph{supervised module}) and an iterative reconstruction method (\emph{iterative module}) in sequence.
The supervised module is different in each super layer\textcolor{black}{, i.e., the weights in the supervised module are not shared among the super layers}.
Importantly, this module is learned in a supervised manner (e.g., to minimize reconstruction error) to remove artifacts and noise.
The iterative module on the other hand iteratively optimizes a regularized image reconstruction problem using image-adaptive priors or regularizers (e.g., the patches of the underlying image can be clustered and sparsified in a learned union of transforms or dictionaries~\cite{pwls-ultra2018}). The iterative algorithm is run for a fixed number of iterations in each super layer.

While the supervised module removes image noise and artifacts using a single learned network, the iterative module could adapt various image-specific features in an MBIR setup to further improve image quality and remove \mbox{artifacts}. Importantly, the iterative module is not learned in a supervised manner.
The SUPER model in Fig.~\ref{fig:flowchart} is flexible and could use various architectures for the supervised module (e.g., FBPConvNet~\cite{jin:17:dcn}, WavResNet~\cite{WavResNet18}, etc.) and a variety of iterative data-adaptive methods (e.g., PWLS-ULTRA~\cite{pwls-ultra2018}). The model could be potentially used in a variety of imaging as well as other applications.

\vspace{-0.0in}
\subsection{Interpretations and Generalization}
\vspace{-0.0in}
The SUPER model enables combining different kinds of machine learned models and priors in a common reconstruction framework. While the supervised module could be a deep convolutional network learned from a big dataset to optimize task-specific performance metrics, the iterative module could exploit models learned from images using criteria such as sparsity, manifold properties, etc.
For example, an operator could be pre-learned from CT images or patches to approximately sparsify them (a.k.a. transform learning~\cite{STlearning13}) and used to construct the regularizer for the iterative module.
Such image-based learned operators are not typically task-sensitive and often generalize readily to different settings and can help delineate or reconstruct various image features.

The proposed SUPER model can also help combine global adaptivity and image-specific adaptivity to obtain the best of both worlds. While each supervised module is learned from a dataset and fixed during reconstruction, the iterative module could optimize novel and specific features for each image being reconstructed (during training and testing) and thus capture the diversity of images and enable highly adaptive reconstructions.
For example, during training and testing, the iterative module could cluster image patches differently for each image~\cite{pwls-ultra2018}, or even learn novel models such as dictionaries for each image~\cite{ravishankar:16:tci}.

Another interpretation of the SUPER model arises from the perspective of iterative reconstruction. Many \mbox{recent} state-of-the-art MBIR schemes involve complex nonconvex optimization and priors, wherein the initialization of the algorithm is typically quite important, and better initializations can lead to better reconstructions.
In the SUPER model, the iterative module is ``initialized" with a different image (i.e., output of the corresponding supervised module) in each super layer. If the output of the supervised module improves in quality over layers, the iterative module will see increasingly better initializations and could thus provide better quality outputs over layers. Moreover, the \mbox{parameters} of the iterative module could also be varied over layers to provide optimal bias-noise trade-offs.
Thus, the SUPER model could be viewed as minimizing nonconvex costs in sequence with better initializations and parameters.

The proposed SUPER model for LDCT reconstruction can be generalized to incorporate a variety of iterative and MBIR techniques in the iterative modules.
For example, conventional techniques such as PWLS-EP (edge-preserving hyperbola regularizer)~\cite{cho:15:rdf} could be used in the iterative module.
PWLS-EP is a non-adaptive method that penalizes the differences between neighboring pixels in the reconstruction. We show later that combining PWLS-EP with supervised learning in the SUPER model boosts the performance of both methods. Note that we do not run the PWLS-EP modules to near convergence (as it involves a strictly convex problem and a unique minimum) but only for multiple iterations.

\vspace{-0.05in}
\subsection{Examples of SUPER Architectures}
\vspace{-0.03in}
We now discuss some example SUPER models and their properties.
To illustrate the proposed approach, in this work, we focused on the recent FBPConvNet~\cite{jin:17:dcn} for the supervised module. For the iterative module, we chose the conventional PWLS-EP approach that uses a hand-crafted prior (edge-preserving hyperbola regularizer) as well as the learning and clustering-based PWLS-ULTRA \cite{pwls-ultra2018}. Our experiments later show that combining such supervised and iterative methods improves image quality over the constituent methods. In the following, we further discuss the chosen architectures.

\vspace{-0.13in}
\subsubsection{Supervised Module}
We work with FBPConvNet, which is a CNN-based image-domain denoising architecture, originally designed for sparse-view CT.
The backbone of FBPConvNet is a U-net \cite{ronneberger:15:unc} like CNN that takes noisy images reconstructed by the FBP method (from low-dose scans) as input.
The neural network is trained so that its outputs closely match the reference high-quality (true) images, e.g., in an $\ell_2$ norm or root mean squared error (RMSE) sense.

The traditional U-net uses a multilevel decomposition, and employs a dyadic scale decomposition based on max pooling. Thus, the effective filter size in the middle layers is larger than that in the early and late layers. This scheme is important because the filters corresponding to the Hessian of the data fidelity term in \eqref{eq:recon} may have noncompact support. Similar to U-net, FBPConvNet employs multichannel filters, which is the standard approach in CNNs, to increase the capacity of the network. Compared with the traditional U-net, FBPConvNet adopts a residual learning strategy to learn the difference between the input and output.

\vspace{-0.15in}
\subsubsection{Iterative Module}
\vspace{-0.03in}

The iterative module optimizes an MBIR problem that estimates the linear attenuation coefficients $\x\in \mathbb{R}^{N_p}$ from the measurements $\y\in \mathbb{R}^{N_d}$. The typical PWLS approach involves a cost function of the following form:
\vspace{-0.06in}
\begin{equation}
	\x=\mathop{\arg\min}_{\x\ge \mathbf{0}} \|\y-\A\x\|^2_\W + \beta \R(\x),
	\label{eq:recon}
	\vspace{-0.06in}
\end{equation}
where $\A\in\mathbb{R}^{N_d\times N_p}$ is the CT system matrix, $\W$ is the weighting matrix related to the measurements (capturing measurement statistics), $\R(\x)$ is the regularizer, and $\beta$ is a positive scalar controlling the balance between the data-fidelity term and the regularizer.
In this paper, we used the unsupervised learning-based PWLS-ULTRA as well as the conventional PWLS-EP for \eqref{eq:recon}.

For PWLS-EP, the regularizer $\R(\x)$ can be written as ${\R(\x) = \sum_{j  =1}^{N_p} \sum_{k\in N_{j}}\kappa_{j} \kappa_{k} \varphi(x_j - x_k)}$, where $x_j$ is the $j$th pixel of $\x$, $N_j$ is the neighborhood of the $j$th pixel, and $\kappa_j$ and $\kappa_k$ are analytically determined weights that encourage resolution uniformity~\cite{cho:15:rdf}.
The potential function $\varphi (t)$ is often chosen as ${\varphi (t) = \delta^2(|t/\delta| - \log(1+|t/\delta|))}$, for $\delta>0$. PWLS-EP enforces approximate sparsity of gradients of the image, a non-adaptive prior.

PWLS-ULTRA pre-learns a union of sparsifying transforms from a dataset of image patches, and uses the learned model during reconstruction~\cite{pwls-ultra2018}. The formulation for learning the union of sparsifying transforms is as follows:
\vspace{-0.03in}
\begin{equation}
	\begin{aligned}
		\min_{\{\omg_k, \Z_i, C_k\}} &\sum_{k=1}^{K}\sum_{i\in C_k} \{ \|\omg_k\X_i - \Z_i\|_2^2 + \eta \|\Z_i\|_0 \} \\
		&+ \sum_{k=1}^{K}\lambda_k \Q(\omg_k), \text{ s.t. } C_k \in \bm{\mathcal{G}}.
	\end{aligned} \label{ultralearn}
	\vspace{-0.05in}
\end{equation}
Here, $\omg_k$, $C_k$, and $\Z_i$ represent the learned transform for the $k$th class, the set of indices of patches belonging to the $k$th class, and the sparse coefficient of the $i$th training signal or patch $\X_i$ ($N$ training signals assumed), respectively.
Each signal is grouped with a corresponding best matched sparsifying transform in \eqref{ultralearn}.
The set $\bm{\mathcal{G}}$ is the set of all possible partitions of $\left [ 1:N \right ]$ into $K$ disjoint subsets.
To avoid trivial solutions for $\omg_k$, the penalty terms $\Q(\omg_k)\triangleq\|\omg_k\|_F^2 - \log|\det \omg_k|$ for $1\leq k\leq K$ are used that also control the condition number of the transforms.
Parameters $\eta$ and $\lambda_k = \lambda_0 \sum_{i\in C_k} \left \| \X_i \right \|_{2}^{2}$ are positive weighting factors with $\lambda_0>0$~\cite{pwls-ultra2018}. Problem~\eqref{ultralearn} is solved efficiently using alternating optimization~\cite{pwls-ultra2018}.

With the pre-learned transforms $\{\omg_k\}$, the regularizer $\R(\x)$ for image reconstruction is as follows:
\vspace{-0.04in}
\begin{equation}
	\begin{split}
		\nonumber \R(\x)\triangleq \min_{\{\z_j, C_k\}} \sum_{k=1}^{K} \sum_{j\in{C_k}} \tau_j \left\{\|\omg_k\P_j\x-\z_j\|^2_2+\gamma^2\|\z_j\|_0\right\},

	\end{split}
	\vspace{-0.04in}
\end{equation}
where $\{\tau_j\}$ are patch-based weights to encourage uniform spatial resolution or uniform noise (see~\cite{pwls-ultra2018}), $\P_j\in\mathbb{R}^{l\times N_p}$ is a patch extraction operator that extracts the $j$th patch from $\x$. $\z_j$ is the sparse coefficient for the $j$th patch, and $\gamma>0$ is a parameter controlling sparsity.

The PWLS-ULTRA problem is efficiently solved by alternating between updating $\x$ (\textit{image update step}), and solving for $\{\z_j, C_k\}$ (\textit{sparse coding and clustering step}).
In the image update step, where $\{\z_j, C_k\}$ are fixed, the subproblem is quadratic with non-negativity constraints, and can be solved using fast iterative algorithms such as the relaxed linearized augmented Lagrangian  method with ordered-subsets (relaxed OS-LALM)~\cite{nien:16:rla,OGM2016}.
The sparse coding and clustering step with fixed $\x$ is solved exactly~\cite{ravishankar:16:tci}, with the optimal class assignment $\hat{k}_j$ for each patch given as
\begin{equation}
	\nonumber \mathop{\arg\min}_{1\le k \le K} \|\omg_k\P_j\x-H_\gamma(\omg_k\P_j\x)\|^2_2+\gamma^2\|H_\gamma(\omg_k\P_j\x)\|_0.
	\label{eq:cluster}
\end{equation}
The corresponding optimal ${\hat{\z}_j = \mathit{H}_{\gamma}(\omg_{\hat{k}_j}\P_j\x)}$, where $\mathit{H}_{\gamma}(\cdot)$ is the hard-thresholding operator that sets elements smaller than $\gamma$ to zero.
The hard-thresholding can be viewed as the non-smooth nonlinearity in PWLS-ULTRA. The clustering could vary from image to image and iteration to iteration in the PWLS-ULTRA algorithm.

\subsection{Training and Implementation}

We propose to train the SUPER model layer-by-layer (sequentially) from a dataset of pairs of low-dose and regular-dose CT measurements.
For example, the FBP method can be used to obtain reconstructions from the measurements. The initial low-dose reconstructed images are then used as inputs to the first supervised module, which is trained to minimize the reconstruction error (RMSE) at its output, with respect to the regular-dose reconstructions. The initial images are then passed through the trained network, following which the iterative algorithm in the first iterative module is run for each training image (in parallel) to produce iterative reconstructions.
The iterative reconstructions serve as inputs to the subsequent supervised model, which is trained to minimize reconstruction error. \textcolor{black}{The subsequent supervised modules are thus learned sequentially.}

Once trained, the SUPER model is readily implemented for test data by passing initial reconstructions sequentially through the supervised learned networks and iterative algorithms.

\vspace{-0.1in}
\section{Experiments}
\vspace{-0.04in}
Here, we first describe our experimental setup, training procedures, and evaluation metrics. Then we present results for the learned SUPER-ULTRA model, and compare these with those obtained by each individual module of SUPER, i.e., FBPConvNet and PWLS-ULTRA. We also tested the generalized SUPER model that replaces the unsupervised learning-based PWLS-ULTRA with the non-adaptive PWLS-EP. This scheme is dubbed FBPConvNet+EP.

\vspace{-0.03in}
\subsection{Experimental Setup}
\vspace{-0.04in}

We used regular-dose CT images of two patients from the Mayo Clinics dataset established for ``\textsl{the 2016 NIH-AAPM-Mayo Clinic Low Dose CT Grand Challenge}"~\cite{McC}, to evaluate the performance of the proposed SUPER learning.
The two patient datasets (L067 and L096) contain 224 and 330 real in-vivo slice images, respectively.
We simulated low-dose CT measurements $\y_{l}$ from the provided regular-dose images with GE 2D LightSpeed fan-beam CT geometry corresponding to a monoenergetic source. We projected the regular-dose images $\x^*$ to sinograms and added Poisson and additive Gaussian noise to them as follows:
\begin{equation}
\y_{l_i} = \text{Poisson} \{I_0 e^{-[\A\x^*]_i}\} + \mathcal{N}\{0, \sigma^2\}.
\end{equation}
We chose $I_0 = 1\times 10^5$ photons per ray \textcolor{black}{ and $\sigma=5$} in our experiments. \textcolor{black}{We approximated elements of the diagonal weighting matrix $\W$ of the data-fidelity in \eqref{eq:recon} by $\frac{\y_{l_i}^2}{\y_{l_i}+\sigma^2}$ \cite{pre-post-log}}.
The images are of size $512\times 512$ at a resolution of $0.9766 \text{ mm} \times 0.9766 \text{ mm}$, when reconstructed using the FBP method. These reconstructed low-dose FBP images were paired with their corresponding regular-dose CT images for training the SUPER model.

\vspace{-0.08in}
\subsection{Training Procedures}
\vspace{-0.04in}

In our experiments, the total number of training image pairs was 100,
of which 50 were chosen from patient L067 and 50 from patient L096. The image set was used to train the networks of FBPConvNet, FBPConvNet+EP, and SUPER-ULTRA.
Fig.~\ref{fig:training_data} shows some example (regular-dose or reference) images from the training and testing \mbox{datasets}.
Different body parts are included in the datasets.
In our experiments, we used the default network architecture in the FBPConvNet public implementation.
For PWLS-ULTRA, we pre-learned a union of five sparsifying transforms (corresponding to five classes) from twelve (regular-dose) \mbox{slices} that include three slices each from four patients (L109, L143, L192, L506).

\begin{figure}[htb]
	\centering
	\includegraphics[scale=0.19]{./fig1/L067Slice1_true.pdf}
	\includegraphics[scale=0.19]{./fig1/L067Slice161_true.pdf}
	\includegraphics[scale=0.19]{./fig1/L096Slice202_true.pdf} \\
	\includegraphics[scale=0.15]{./fig1/L067Slice50_true.pdf}
	\includegraphics[scale=0.15]{./fig1/L067Slice100_true.pdf}
	\includegraphics[scale=0.19]{./fig1/L096Slice291_true.pdf}
	\vspace{-0.05in}
	\caption{Example CT images in the training (top row) and testing (bottom row) datasets. The display window is [800 1200]~HU.}
	\label{fig:training_data}
	\vspace{-0.1in}
\end{figure}

We used a GTX Titan GPU  graphic processor for training and testing.
The union of transforms was learned efficiently using similar parameters as in~\cite{pwls-ultra2018}.
For SUPER-ULTRA and FBPConvNet+EP, the \textcolor{black}{training} time was about 46 hours and 13 hours, respectively, for 15 super layers.
Since the iterative reconstruction modules are relatively computationally expensive, we chose 15 super layers to balance reconstruction quality and computational time.
The training hyper-parameters for the CNN part of FBPConvNet were set as follows for the various models: the learning rate decreased logarithmically from 0.001 to 0.0001; the batchsize was 1; and the momentum was 0.99.
We initialized the filters in the various networks during training with i.i.d. random Gaussian entries with zero mean and variance 0.005, and initialized the bias with zero vectors.

\begin{figure*}[htb]
	\centering
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_fbp.pdf}
	\put(5,90){ \color{white}{\bf \large{PSNR:13.3 }}} 	\end{overpic}
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_fbpconvnet.pdf}
		\put(5,90){ \color{white}{\bf \large{PSNR:20.3 }}} 	\end{overpic}
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_ep.pdf}
		\put(5,90){ \color{white}{\bf \large{PSNR:23.4 }}} 	\end{overpic}
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_ultra.pdf}
		\put(5,90){ \color{white}{\bf \large{PSNR:27.1 }}} 	\end{overpic} \\
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_fbpconvnetep2.pdf}
		\put(5,90){ \color{white}{\bf \large{PSNR: \color{white}{29.0}}}} 	\end{overpic}
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_fbpconvnetultra.pdf}
		\put(5,90){ \color{white}{\bf \large{PSNR: 31.7}}} 	\end{overpic}
	\begin{overpic}[scale=0.305]{./fig3/L096Slice170_image_true.pdf}
		\put(5,90){ \color{white}{\bf \large{}}} 	\end{overpic}
		\vspace{-0.0in}
	\caption{Reconstructed testing image (Test \#4, from patient L096) obtained by FBP, FBPConvNet, PWLS-EP, PWLS-ULTRA, FBPConvNet + EP, SUPER-ULTRA, and regular-dose FBP. The display window is [800 1200]~HU.}
	\label{fig:L096Slice170_image}
\end{figure*}

For training FBPConvNet+EP, we ran 10 epochs of FBPConvNet training (using ADAM) in each super layer to ensure that the learned
networks capture layer-wise features.
For the constituent PWLS-EP modules,  we ran 4 iterations of the relaxed OS-LALM algorithm with 4 subsets, and set $\delta=20$~HU and the regularization parameter $\beta=2^{15}$.
For training SUPER-ULTRA, we ran 10 epochs of FBPConvNet training in each super layer along with 4 outer iterations of PWLS-ULTRA with 5 inner iterations of the image update step that used the relaxed OS-LALM algorithm with 4 subsets. We set $\beta=5\times 10^3$ and $\gamma=20$ for the PWLS-ULTRA module.

\begin{figure}[!t]
	\centering
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_fbp.pdf}
		\put(5,90){ \color{white}{\bf {PSNR:\color{white}{13.6}}} }	\end{overpic}
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_fbpconvnet.pdf}
		\put(5,90){ \color{white}{\bf {PSNR: \color{white}{30.3}}} }	\end{overpic}
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_ep.pdf}
		\put(5,90){ \color{white}{\bf {PSNR: \color{white}{23.0}}}} 	\end{overpic}\\
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_fbpconvnetultra.pdf}
		\put(5,90){ \color{white}{\bf {PSNR: \color{white}{31.6}}}} 	\end{overpic}
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_fbpconvnetep2.pdf}
		\put(5,90){ \color{white}{\bf {PSNR: \color{white}{31.6}}}} 	\end{overpic}
	\begin{overpic}[scale=0.19]{./fig2/L067Slice100_image_ultra.pdf}
		\put(5,90){ \color{white}{\bf {PSNR: \color{white}{29.6}}}} 	\end{overpic}
	\caption{Comparison of Test \#2 for FBP, FBPConvNet, PWLS-EP, PWLS-ULTRA, FBPConvNet + EP, and SUPER-ULTRA (clockwise top left). The display window is [800 1200]~HU.}
	\label{fig:L067Slice100_error}
	\vspace{-0.1in}
\end{figure}

We compared our learned models with the iterative schemes PWLS-EP and PWLS-ULTRA.
Since FBPConvNet+EP and SUPER-ULTRA already include the learned FBPConvNet modules, we used smaller regularization parameters for them compared to the usual or stand-alone PWLS-EP and PWLS-ULTRA, which worked well in our experiments.
For the stand-alone PWLS-EP iterative reconstruction algorithm that solves a convex problem, $\beta$ and $\delta$ were set as $2^{16}$ and $20$, respectively, and we ran 100 iterations of the OS-LALM algorithm to ensure convergence.
For the stand-alone PWLS-ULTRA, wherein the optimization problem is nonconvex, we set $\beta$ and $\gamma$ as $10^4$ and $25$, respectively, and ran $1000$ iterations of the alternating algorithm to ensure convergence. The above parameters provided optimal image quality in our experiments. Both PWLS-EP and PWLS-ULTRA were initialized with the simple FBP reconstructions.

\vspace{-0.01in}
\subsection{Evaluation Metrics}
\vspace{-0.01in}
To quantitatively evaluate the performances of the various reconstruction models, we chose three classic metrics, namely RMSE, peak signal to noise ratio (PSNR), and structural similarity index measure (SSIM)~\cite{wang:04:iqa}.
RMSE in Hounsfield units (HU) is defined as RMSE=$\sqrt{\sum_{i=1}^{N_p} \left(\hat{x}_i-x^*_i\right)^2/N_p}$, where $\x^*$ is the reference regular-dose CT image, $\hat{\x}$ is the reconstruction, and $N_p$ is the number of pixels.
We computed PSNR in decibels (dB).

\subsection{Results and Comparisons}
\label{sec:exper}

\vspace{-0.0in}
\subsubsection{Visual Results}
\vspace{-0.02in}
We applied the various methods
to six testing slices (three slices from L067 and three slices from L096) indexed as Test~\#1 to Test~\#6.
Figs.~\ref{fig:L096Slice170_image} and~\ref{fig:L067Slice100_error} show the \mbox{reconstructions} of
Test~\#4 (from patient L096) and Test~\#2 (from patient L067), respectively, using different methods.
Clearly, the traditional FBP reconstruction shows severe artifacts, while the results obtained by the other methods have much better image quality.
The stand-alone optimal PWLS-EP reconstruction still contains noise such as in the central soft-tissue area (Fig.~\ref{fig:L096Slice170_image}).
The (unsupervised) learning-based PWLS-ULTRA removes noise, but the edges of soft-tissues are more blurry in the results.
The supervised learning-based FBPConvNet achieves better trade-off between resolution and noise compared to PWLS-EP and PWLS-ULTRA, but many small structures were missed or distorted.
With the same training data as FBPConvNet, the learned FBPConvNet+EP and SUPER-ULTRA models provide much better reconstructions.
However, FBPConvNet+EP suffers from some streak artifacts in the central area (Fig.~\ref{fig:L096Slice170_image}) \textcolor{black}{as well as noise generally, and} SUPER-ULTRA mitigates these artifacts.
In both Figs.~\ref{fig:L096Slice170_image} and~\ref{fig:L067Slice100_error}, SUPER-ULTRA that combines supervised learned networks and transform learning-based iterative reconstructions achieved the best overall visual quality.
\footnote{\textcolor{black}{Additional comparisons between reconstructions for the other testing slices are included in the supplement.}}
\vspace{-0.15in}
\subsubsection{Quantitative Results}

Fig.~\ref{fig:L067Slice50_rmsessim_curve} shows the RMSE and SSIM evolution for the stand-alone PWLS-EP and PWLS-ULTRA along with those for FBPConvNet+EP and SUPER-ULTRA.
The latter two involve 15 super layers with 4 (outer) iterations in the iterative module per layer, and thus, the RMSE and SSIM evolution is plotted over these individual iterations.
For PWLS-EP and PWLS-ULTRA, we show the evolution of the metrics over $60$ iterations.
FBPConvNet+EP and SUPER-ULTRA clearly achieve much lower RMSE values and higher SSIM values over layers than PWLS-EP and PWLS-ULTRA. The plots also show \textcolor{black}{faster} convergence for the SUPER models.

\begin{table*}[htb]
	\centering
	\caption{PSNR, RMSE, and SSIM of reconstructed test images for different methods.}
	\vspace{-0.03in}
	\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
		\hline
		\multicolumn{3}{|c|}{} & FBP  & \tabincell{c}{FBPConvNet} & \tabincell{c}{PWLS-\\EP} & \tabincell{c}{PWLS-\\ULTRA} & \tabincell{c}{FBPConvNet+EP}  & \tabincell{c}{SUPER-ULTRA}\\
		\hline
		\multirow{9}{*}{L067} & \multirow{3}{*}{Test \#1} & PSNR & 11.0   & 29.8  & 23.5 &  29.3 & 30.5 & \textbf{30.9}\\
		\cline{3-9}
		& & RMSE & 245.5  & 26.2  & 54.1 & 28.3 & 24.5 &  \textbf{23.3}\\
		\cline{3-9}
		& & SSIM & 0.29  & 0.74  & 0.73 & 0.74 & 0.77 & \textbf{0.77}\\  \cline{2-9}
		& \multirow{3}{*}{Test \#2} & PSNR & 13.6   & 30.3  & 23.0 & 29.6 & 31.6 & \textbf{31.6}\\
		\cline{3-9}
		& & RMSE & 170.3  & 22.4  & 52.2 & 24.5 & 19.5 & \textbf{19.3}\\
		\cline{3-9}
		& & SSIM & 0.40  & 0.79  & 0.78 & 0.79 & 0.81 & \textbf{0.81}\\
		\cline{2-9}
		& \multirow{3}{*}{Test \#3} & PSNR & 9.5  & 19.9  & 24.4 & 28.6 & \textbf{32.2}  & 32.1\\
		\cline{3-9}
		& & RMSE & 299.5  & 81.7  & 47.9 & 29.6 & \textbf{19.6}  & 19.8\\
		\cline{3-9}
		& & SSIM & 0.24  & 0.56  & 0.69 & 0.69 & 0.72 & \textbf{0.72}\\ 	\hline

		\multirow{9}{*}{L096} & \multirow{3}{*}{Test \#4} & PSNR & 13.3   & 20.3  & 23.4 & 27.1 & 29.0 & \textbf{31.7}\\
		\cline{3-9}
		& & RMSE & 172.6  & 70.7  & 48.7 & 31.9 & 25.5 & \textbf{18.9}\\
		\cline{3-9}
		& & SSIM & 0.37 & 0.67 & 0.77 & 0.79 & 0.80 & \textbf{0.81}\\
		\cline{2-9}
		& \multirow{3}{*}{Test \#5} & PSNR & 9.3 & 29.7  & 23.3 & 25.7 & 30.6 & \textbf{30.8}\\
		\cline{3-9}
		& & RMSE & 304.4  & 25.8  & 53.8 & 40.6& 23.3 & \textbf{22.7}\\
		\cline{3-9}
		& & SSIM & 0.20  & 0.71  & 0.70 & 0.70 & 0.74 & \textbf{0.75}\\  \cline{2-9}
		& \multirow{3}{*}{Test \#6} & PSNR & 9.7   & 27.6  & 23.6 & 28.1 & 30.7 & \textbf{32.3}\\
		\cline{3-9}
		& & RMSE & 274.2  & 32.3  & 48.6 & 29.0 & 21.5 & \textbf{17.8}\\
		\cline{3-9}
		& & SSIM & 0.23  & 0.66  & 0.73 & 0.74 & 0.75 & \textbf{0.76}\\ \hline

	\end{tabular}
	\label{Tab:snrrmsessim_comp}
	\vspace{-0.1in}
\end{table*}
\begin{figure}[!t]
		\includegraphics[width = 0.235\textwidth]{./fig3/L096Slice170_rmse_ep_font18_v2.pdf}
		\includegraphics[width = 0.23\textwidth]{./fig3/L096Slice170_rmse_ultra_font18.pdf} \\
		\includegraphics[width = 0.235\textwidth]{./fig3/L096Slice170_ssim_ep_font18_v2.pdf}
		\includegraphics[width = 0.23\textwidth]{./fig3/L096Slice170_ssim_ultra_font18.pdf}
		\vspace{-0.05in}
	\caption{RMSE (first row) and SSIM (second row) values for Test \#4 over the iterations of PWLS-EP, \textcolor{black}{FBPConvNet+EP}, PWLS-ULTRA, and SUPER-ULTRA.}
	\label{fig:L067Slice50_rmsessim_curve}
	\vspace{-0.0in}
\end{figure}

Table.~\ref{Tab:snrrmsessim_comp} shows the RMSE, PSNR, and SSIM values with various methods for the testing slices.
The proposed SUPER-ULTRA \textcolor{black}{typically} achieves significant improvements in RMSE, SSIM, and PSNR over the other methods for all testing slices.
Importantly, FBPConvNet+EP and SUPER-ULTRA perform much better than PWLS-EP and PWLS-ULTRA, respectively, and also provide more promising results than FBPConvNet, demonstrating that the combination of the supervised module and iterative modules in the SUPER model works well and outperforms the individual modules.

\vspace{-0.0in}
\subsubsection{Visual Quality over SUPER layers}
\vspace{-0.01in}
Fig.~\ref{fig:L067Slice100_layer_res} shows the output of SUPER-ULTRA after different numbers of SUPER layers.
The initial FBP image and the final output (after $15$ layers) were shown in Fig.~\ref{fig:L067Slice100_error}.
The first several layers of SUPER-ULTRA mainly remove severe noise and artifacts, while the later layers recover some structural details.

\begin{figure}[!t]
	\centering
	\begin{overpic}[scale=0.295]{./fig2/L067Slice100_image_fbpconvnet_layer1_arrow-2.pdf}
	\put(20,90){ \color{white}{\bf \large{Super Layer 1}}}
	\end{overpic}
	\begin{overpic}[scale=0.295]{./fig2/L067Slice100_image_fbpconvnet_layer3_arrow-2.pdf}
		\put(20,90){ \color{white}{\bf \large{Super Layer 3}}}
	\end{overpic}  \\
	\begin{overpic}[scale=0.295]{./fig2/L067Slice100_image_fbpconvnet_layer7_arrow-2.pdf}
		\put(20,90){ \color{white}{\bf \large{Super Layer 7}}}
	\end{overpic}
	\begin{overpic}[scale=0.295]{./fig2/L067Slice100_image_fbpconvnet_layer13_arrow.pdf}
		\put(17,90){ \color{white}{\bf \large{Super Layer 13}}}
	\end{overpic}
	\caption{Outputs of 1st, 3rd, 7th, and 13th SUPER-ULTRA layers with display window [800 1200]~HU. The reconstruction is visually converging.  }
	\label{fig:L067Slice100_layer_res}
	\vspace{-0.0in}
\end{figure}

\vspace{-0.1in}
\subsubsection{Behavior of the ULTRA model in the SUPER architecture}
\vspace{-0.01in}
\begin{figure*}[!t]
	\centering
	\includegraphics[scale=0.35]{./fig2/cluster_mayoST_class1.pdf}
	\includegraphics[scale=0.35]{./fig2/cluster_mayoST_class2.pdf}
	\includegraphics[scale=0.35]{./fig2/cluster_mayoST_class3.pdf}
	\includegraphics[scale=0.35]{./fig2/cluster_mayoST_class4.pdf}
	\includegraphics[scale=0.35]{./fig2/cluster_mayoST_class5.pdf} \\

		\begin{overpic}[scale=0.235]{./fig2/cluster_L067Slice100_fbpconvnetultra_class1.pdf}\put(30,12){ \color{white}{\bf \large{Class 1}}} 	\end{overpic}
		\begin{overpic}[scale=0.235]{./fig2/cluster_L067Slice100_fbpconvnetultra_class2.pdf}
		\put(30,12){ \color{white}{\bf \large{Class 2}}} 	\end{overpic}
		\begin{overpic}[scale=0.235]{./fig2/cluster_L067Slice100_fbpconvnetultra_class3.pdf}
		\put(30,12){ \color{white}{\bf \large{Class 3}}} 	\end{overpic}
		\begin{overpic}[scale=0.235]{./fig2/cluster_L067Slice100_fbpconvnetultra_class4.pdf}
		\put(30,12){ \color{white}{\bf \large{Class 4}}} 	\end{overpic}
		\begin{overpic}[scale=0.235]{./fig2/cluster_L067Slice100_fbpconvnetultra_class5.pdf}
		\put(30,12){ \color{white}{\bf \large{Class 5}}} 	\end{overpic}
		\vspace{-0.02in}
	\caption{Pixel-level clustering results in Test \#2. The top row shows the transforms, with the transform rows shown as $8\times8$ patches. The bottom row shows the clustering results of SUPER-ULTRA with display window [800 1200]~HU.}
	\label{fig:clstering}
	\vspace{-0.2in}
\end{figure*}

Next, to better illustrate the image-adaptive learned clustering in the SUPER-ULTRA model, Fig.~\ref{fig:clstering} shows pixel-level clustering results from the last super layer for test slice \#1.
Since the ULTRA modules cluster image patches into specific classes, we cluster each pixel here using a majority vote among the patches overlapping the pixel.
Class 1 contains most of the more uniform soft-tissues; Classes 2, 3, and 5 contain many oriented edges (e.g., at 45-degree and 135-degree orientation); and class 4 contains most of the vertical edges and some horizontal edges as well as most of the bones.
The latter classes help provide sharper SUPER-ULTRA reconstructions.
The pre-learned transforms corresponding to each class are also shown in Fig.~\ref{fig:clstering}, and contain various directional and edge-like features.

\vspace{-0.1in}
\section{Conclusions}
\vspace{-0.03in}
This paper presented a new framework that combined supervised learned networks and unsupervised iterative algorithms for low-dose CT reconstruction.
The proposed SUPER framework effectively combines various kinds of priors and learning methods.
In particular, we studied SUPER-ULTRA that combines (supervised) deep learning (FBPConvNet) and the recent iterative (unsupervised) PWLS-ULTRA, as well as FBPConvNet+EP (or SUPER-EP).
Both methods showed better performance and faster convergence compared to their individual modules. FBPConvNet+EP substantially improved the performance of PWLS-EP, while SUPER-ULTRA \textcolor{black}{typically} performed the best by effectively leveraging deep learning and transform learning.
While SUPER model learning can exploit a variety of architectures and algorithms for the supervised and iterative modules, a more detailed study of various such architectures is left for future work.
We also plan to explore layer-dependent parameter selection for the iterative modules to further improve performance in future work.

{\small
\bibliographystyle{ieee}

\section{Additional Experimental Results}
\setcounter{figure}{7}

Here, we show the regular-dose FBP images for the other four testing slices (Test \#1, \#3, \#5, and \#6) in Fig.~\ref{fig:ref}, and their reconstructions in Figs.~\ref{fig:L067Slice50_image}, \ref{fig:L067Slice210_image}, \ref{fig:L096Slice291_image}, and \ref{fig:L096Slice330_image}.
\vspace{-0.07in}
\begin{figure}[htb]
	\centering
	\begin{overpic}[scale=0.23]{./fig4/L067Slice50_image_true.pdf}
		\put(25,90){ \color{white}{\bf \large{}}} 	\end{overpic}
	\begin{overpic}[scale=0.23]{./fig4/L067Slice210_image_true.pdf}
		\put(20,88){ \color{white}{\bf \large{}}} 	\end{overpic}
	\begin{overpic}[scale=0.23]{./fig4/L096Slice291_image_true.pdf}
		\put(25,90){ \color{white}{\bf \large{}}} 	\end{overpic}
	\begin{overpic}[scale=0.23]{./fig4/L096Slice330_image_true.pdf}
		\put(25,90){ \color{white}{\bf \large{}}} 	\end{overpic}
	\caption{Regular-dose FBP images of four testing slices: Test \#1 (top left), \#3 (top right), \#5 (bottom left), and \#6 (bottom right). The display window is [800 1200]~HU.}

	\label{fig:ref}
\end{figure}

\begin{figure}[htb]
	\centering
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_fbp.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: 11.0}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_fbpconvnet.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: 29.8}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_ep.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: 23.5}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_ultra.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: 29.3}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_fbpconvnetep2.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: \textcolor{white}{30.5}}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice50_image_fbpconvnetultra.pdf}
		\put(20,92){ \color{white}{\bf \normalsize{PSNR: 30.9}}} 	\end{overpic}

	\caption{Reconstructed testing image (Test \#1, from patient L067) obtained by FBP (top left), FBPConvNet (top right), PWLS-EP (middle left), PWLS-ULTRA (middle right), FBPConvNet + EP (bottom left), and SUPER-ULTRA (bottom right). The display window is [800 1200]~HU.}
	\label{fig:L067Slice50_image}
	\vspace{-0.2in}
\end{figure}
\begin{figure}[!t]
	\centering
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_fbp.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 9.5}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_fbpconvnet.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 19.9}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_ep.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 24.4}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_ultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: \textcolor{white}{28.6}}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_fbpconvnetep2.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: \textcolor{white}{32.2}}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L067Slice210_image_fbpconvnetultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 32.1}}} 	\end{overpic}

	\caption{Reconstructed testing image (Test \#3, from patient L067) obtained by FBP (top left), FBPConvNet (top right), PWLS-EP (middle left), PWLS-ULTRA (middle right), FBPConvNet + EP (bottom left), and SUPER-ULTRA (bottom right). The display window is [800 1200]~HU.}
	\label{fig:L067Slice210_image}
	\vspace{-0.2in}
\end{figure}

SUPER-ULTRA and FBPConvNet+EP obviously reduce artifacts and noise compared to FBP, PWLS-EP, and FBPConvNet.
Furthermore, SUPER-ULTRA also significantly improves the sharpness in the soft-tissue compared to PWLS-ULTRA.
Lastly, compared to FBPConvNet+EP, SUPER-ULTRA reduces noise while producing sharp reconstructions.

\begin{figure}[!t]
	\centering
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_fbp.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 9.3}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_fbpconvnet.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 29.7}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_ep.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 23.3}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_ultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 25.7}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_fbpconvnetep2.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: \textcolor{white}{30.6}}}} 	\end{overpic}
	\begin{overpic}[scale=0.22]{./fig4/L096Slice291_image_fbpconvnetultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 30.8}}} 	\end{overpic}

	\caption{Reconstructed testing image (Test \#5, from patient L096) obtained by FBP (top left), FBPConvNet (top right), PWLS-EP (middle left), PWLS-ULTRA (middle right), FBPConvNet + EP (bottom left), and SUPER-ULTRA (bottom right). The display window is [800 1200]~HU.}
	\label{fig:L096Slice291_image}
\end{figure}

\begin{figure*}[htb]
	\centering
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_fbp.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 9.7}}} 	\end{overpic}
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_fbpconvnet.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 27.6}}} 	\end{overpic}
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_ep.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 23.6}}} 	\end{overpic} \\
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_ultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 28.1}}} 	\end{overpic}
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_fbpconvnetep2.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: \textcolor{white}{30.7}}}} 	\end{overpic}
	\begin{overpic}[scale=0.26]{./fig4/L096Slice330_image_fbpconvnetultra.pdf}
		\put(20,88){ \color{white}{\bf \normalsize{PSNR: 32.3}}} 	\end{overpic}

	\caption{Reconstructed testing image (Test \#6, from patient L096) obtained by FBP (top left), FBPConvNet (top middle), PWLS-EP (top right), PWLS-ULTRA (bottom left), FBPConvNet + EP (bottom middle), and SUPER-ULTRA (bottom right). The display window is [800 1200]~HU.}
	\label{fig:L096Slice330_image}
\end{figure*}

\begin{figure*}[!b]
	\centering
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice50_fbpconvnetultra_class1.pdf}
		\put(30,90){ \color{white}{\bf \large{Class 1}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice50_fbpconvnetultra_class2.pdf}
		\put(30,90){ \color{white}{\bf \large{Class 2}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice50_fbpconvnetultra_class3.pdf}
		\put(30,90){ \color{white}{\bf \large{Class 3}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice50_fbpconvnetultra_class4.pdf}
		\put(30,90){ \color{white}{\bf \large{Class 4}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice50_fbpconvnetultra_class5.pdf}
		\put(30,90){ \color{white}{\bf \large{Class 5}}}
	\end{overpic}\\
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice210_fbpconvnetultra_class1.pdf}
		\put(30,87){ \color{white}{\bf \large{Class 1}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice210_fbpconvnetultra_class2.pdf}
		\put(30,85){ \color{white}{\bf \large{Class 2}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice210_fbpconvnetultra_class3.pdf}
		\put(30,85){ \color{white}{\bf \large{Class 3}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice210_fbpconvnetultra_class4.pdf}
		\put(30,85){ \color{white}{\bf \large{Class 4}}}
	\end{overpic}
	\begin{overpic}[scale=0.195]{./fig4/cluster_L067Slice210_fbpconvnetultra_class5.pdf}
		\put(30,85){ \color{white}{\bf \large{Class 5}}}
	\end{overpic}
	\caption{Top and bottom rows show the Pixel-level clustering results of SUPER-ULTRA for Test \#1 and Test \#3, respectively. The display window is [800 1200]~HU.}
	\label{fig:clstering}
\end{figure*}

Next, to better illustrate the learned clustering in the SUPER-ULTRA model, Fig.~\ref{fig:clstering} shows examples of pixel-level clustering results from the last super layer for test slices \#1 and \#3.

redpajama_set_name: RedPajamaArXiv


text:
Worthy Christian News > US News > Hoover Dam Explosion Investigated
Hoover Dam Explosion Investigated
Wednesday, July 20, 2022 | Tag Cloud Tags: Las Vegas, Nevada, Worthy News
By Stefan J. Bos, Chief International Correspondent Worthy News
WASHINGTON (Worthy News) – U.S. authorities were investigating an explosion Tuesday that rocked a transformer at the Hoover Dam, one of the nation's largest hydroelectric facilities, but caused no causalities.
Witnesses saw a thick cloud of black smoke and flames at the engineering marvel on the Colorado River that straddles the border of the U.S. states of Arizona and Nevada.
The blaze was extinguished within about 30 minutes before firefighters from the Boulder City fire department arrived on scene, the Nevada city reported.
"We arrived at the scene and turned away. Everything was all set," added Lisa LaPlante, Boulder City communications manager. "The fire was extinguished."
Electricity produced at Hoover Dam continued flowing to the 8 million people in Arizona, Nevada, and Southern California who rely on it, the Western Area Power Administration confirmed.
Yet authorities began investigating how an explosion could occur on the apron of the building housing turbines slightly downstream from the dam's base.
Officials are also working to determine the extent of damage to the transformer, one of 15 at the complex that control the voltages for power sent to customers
The Hoover Dam, about 25 miles (40 kilometers) southeast of Las Vegas, is one of the tallest concrete dams in the U.S., at 726 feet (221 meters). Each of its 17 generators can supply electricity to 100,000 households.
Hydropower from these and other dams has been threatened lately by the declining levels of Lake Mead and Lake Powell, the two largest human-made reservoirs in the U.S. that hold water from the Colorado River.
Federal officials say they have tried to prop up the lakes to preserve the dams' ability to generate power and keep water flowing to the Western states and Mexico that rely on it.
The lakes have sunk to their lowest levels in decades due to drought that activists have linked to climate change.
The cause of the fire was under investigation, and officials were working to determine the extent of damage to the transformer, one of 15 at the complex that controls the voltages for power sent to customers.
Police Investigated Over US School Attack
Russia Strikes Ukraine Dam As Losses Mount
Sierra Leone Mourns As Tank Explosion Kills Nearly 100
Palestinian Authority's Abbas warns Putin: West Bank poised for an 'explosion'
Explosion Hit Cuba Hotel; Several Killed
Hezbollah leader warns of possible 'major explosion' if Jerusalem Day Flag March passes through Old…
Massive Explosion Rocks Istanbul; Several Dead, Scores Injured
Tags: Las Vegas, Nevada, Worthy News
Copyright 1999-2022 Worthy News. All rights reserved.
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Tag Cloud Tags: Las Vegas, Nevada, Worthy News
Categories US News
Down Syndrome Child Among Dozens Killed In Russian Missile Strike
One Of Two Kidnapped Priests Found Dead

redpajama_set_name: RedPajamaCommonCrawl


text:
CASI INSTITUTE
About CaSi alloys
CaSi REACH Consortium
CaSi SIEF
Reach Compliant Companies
About CaSi alloys > Uses
Calcium-Silicon Alloys are used as deoxidizer and desulfurizer in the manufacturing of high grade steel. Indeed, Calcium and Silicon both have a strong chemical affinity for oxygen. Especially calcium, have a strong chemical affinity not only for oxygen, but also for sulphur and nitrogen. The steel industry accounts for around 90 % of global CaSi consumption.
The alloy is also used for the modification of non-metallic inclusions. Calcium Silicon is used to control the shape, size and distribution of oxide and sulfide inclusions improving fluidity, machinability, ductility, and impact properties of the steel product. The calcium liquefies the suspended solid inclusions of aluminium oxide), a principal source of fatigue failure in highly stressed alloy steels, that are always present in aluminium-killed steels(Al203 is transformed by Ca into CaO:Al2O3). Furthermore, the material properties of steel are improved: the calcium aluminates formed are largely undeformable in the solidified steel and allow high isotropy of the material properties to be maintained after working of the steel.
As a sulphide former, CaSi alloy promotes random distribution of sulfides, thereby minimizing chain-type inclusions.
Silicon-calcium alloy can replaces aluminium to accomplish the task of final deoxydation, and is used in the production of fine steel, special steel and special alloy. CaSi is indeed required in higher quality, higher strengh steels and alloy steels, which cannot tolerate the level of inclusions that remain following standard deoxidisation with ferrosilicon, silicomanganese or aluminium. For example, to turn out rail steel, mild steel, stainless steel, and special alloys such as nickel-based alloy and titanium-based alloy, which can be adopted Silicon-calcium alloy as deoxidant. Silicon-calcium alloy can also work as temperature-gaining agent in convertor steel making workshop , and as catalyst/inoculant in production of cast iron, as well as the additive in the production of nodular cast iron.
Calcium Silicon is also utilized in combination with Barium. The Barium increases the density of the alloy, thereby improving efficiency. In cast irons it is used mainly by Pipe Shops for the purpose of coating the inside of each mono cast machine.
Calcium Silicon can be obtained in lump, crushed or powder form. The most widely used method for intergrated steel applications is in the form of cored wire. This is mainly because calcium has a low density and high reactivity in the ladle making it difficult to introduce and retain in the melt. Cored wire eliminates these variables: CaSi powder is blown deep into the steel melt through immersion of steel tubes.
Fines of CaSi alloys can be used in the manufacturing of specialty pyrotechnic mixtures (component of firework or primer mix of munitions) and also as welding powder.
Prior to the 1980s, CaSi was dropped in lump form into the steel ladle. Most of the CaSi burned off into the slag on the top of the molten steel, with the CaSi only achieving a recovery of 2-5% into the molten steel.
In 1980, the European steelmaker Vallourec developed the process of adding CaSi into molten steel via the cored wire injection process
This involves enclosing CaSi powder within a hollow cored wire, and injecting it into the ladle of molten steel. The wire penetrates the steel slag to a much more successful extent than dropping in CaSi lumps. Recovery rates of 15-25% can be achieved by adding CaSi to steel by injecting cored wire
Certain steelmakers, notably TKS and Nippon Steel, developed an alternative method of adding CaSi into molten steel via gas injection. This achieves a recovery rate of around 10%, but has a high capital cost
Through the 1980s and 1990s, cored wire injection of CaSi became adopted by more and more steel producers. By the 2000s, this method has become the norm, accounting for 80% of the consumption of CaSi in steel
CaSi has been registered on its own for REACH in 2010
the position paper of the CaSi Institute can be found under Position Papers section
Site map Contact Copyright 2020, Casi Institute - Avenue de Broqueville, 12 1150 Brussels (Belgium) -
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text:
Противоречие

 Противоречие — утверждение, противоречащее предположению в доказательстве от противного.
 Противоречие — отношение между двумя суждениями в логике.
 Противоречие — в теории решения изобретательских задач ситуация, когда должны удовлетворяться противоположные требования.
 Диалектическое противоречие — реальное наличие противоположных свойств предмета в диалектике.

redpajama_set_name: RedPajamaWikipedia


The FCC is considering the implementation of new rules for wireless providers that will enhance competition leading to lower prices for consumers. The Wireless Competition Act passed in 2014 has also enhanced competition by allowing cell phone subscribers to "unlock" their numbers and go to any wireless carrier of choice.
Since the introduction of widespread cellular service over 20 years ago, the market for wireless services and products has expanded at incredible rates. Traditional landline products and services have recently begun to be dismantled and shutdown as wireless providers have consumed the telephone communications market nationwide.
The combination of numerous wireless service providers, along with advances in wireless technology speed and capacity have created a very competitive market for wireless services. A dramatic drop in landline usage has elevated service rates as traditional telephone line revenue has dropped because of fewer subscribers.


text:
// ----------------------------------------------------------------------------------
//
// Copyright Microsoft Corporation
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
// http://www.apache.org/licenses/LICENSE-2.0
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
// ----------------------------------------------------------------------------------

using Microsoft.Azure.Batch;
using Microsoft.Azure.Commands.Batch.Properties;
using System;
using System.Collections.Generic;

namespace Microsoft.Azure.Commands.Batch.Models
{
    public class JobScheduleOperationParameters : BatchClientParametersBase
{
        public JobScheduleOperationParameters(BatchAccountContext context, string jobScheduleId, PSCloudJobSchedule jobSchedule,
        IEnumerable<BatchClientBehavior> additionalBehaviors = null) : base(context, additionalBehaviors)
    {
        if (string.IsNullOrWhiteSpace(jobScheduleId) && jobSchedule == null)
        {
            throw new ArgumentNullException(Resources.NoJobSchedule);
        }

        this.JobScheduleId = jobScheduleId;
        this.JobSchedule = jobSchedule;
    }

    /// <summary>
    /// The id of the job schedule.
    /// </summary>
    public string JobScheduleId { get; private set; }

    /// <summary>
    /// The PSCloudJobSchedule object representing the target job schedule.
    /// </summary>
    public PSCloudJobSchedule JobSchedule { get; private set; }
}
}

redpajama_set_name: RedPajamaGithub


Our motel is located on scenic North Huron Avenue directly across from the Mackinaw Point Lighthouse and Lake Shore Park. This beautiful area of Mackinaw City boasts fantastic views of the Straits of Mackinac, Mackinac Island and the Mackinac Bridge. People come from far and wide to enjoy the beach, stroll the lakeside sidewalk or picnic in the park. In the evening watch the spectacular sunsets and the lighting of "Big Mac".
For the best value of all Mackinaw City Motels or Mackinaw City Hotels call us at 231-436-5304 or click below to make your reservations today!
Whether this is your first visit, or you have been a guest many times, we want your experience to be excellent. Our staff will help you with any questions or concerns you may have to ensure a successful stay each and every time.
We take cleanliness to a level second to none. Our staff will not place any room into inventory until there is absolutely no evidence that another guest has ever stayed in your room. We encourage you to come see for yourself.
Member of the Mackinaw City Chamber of Commerce - Copyright © 2017 Lighthouse View Motel - All Rights Reserved.


What does AMDR stand for?
AMDR stands for "Asset Management Data Register"
How to abbreviate "Asset Management Data Register"?
What is the meaning of AMDR abbreviation?
The meaning of AMDR abbreviation is "Asset Management Data Register"
AMDR as abbreviation means "Asset Management Data Register"


text:
<?php
/**
 * Created by PhpStorm.
 * User: Серега
 * Date: 01.02.2017
 * Time: 11:47
 */

namespace app\modules\admin\controllers;

use yii\filters\AccessControl;
use yii\web\Controller;

class AppAdminController extends Controller
{
    public function behaviors(){
        return [
            'acces' =>[
                'class' => AccessControl::className(),
                'rules' => [
                    [
                    'allow' => true,
                    'roles' => ['@']
                    ]
                ]
            ]
        ];
    }
}


text:
New features available in TimeApp
During the last couple of months we have been working on two new major features for TimeApp which we are proud to announce; Orders and Punch-clock. Punch-clock: This feature allows you to track the time worked by employees, mostly used for internal logging with support for multiple workplaces including geo-positioning – if enabled. Orders: Any…
Our new office
Take a look of our new office here
Soon to be released: TimeApp for iOS version 4.1.0
Shortly an updated version of our app (v 4.1.0) for iOS will finally be released on Apple's App Store. In this version we have, besides improving performance and the addition of some new features, also corrected a couple of bugs that caused headache for new users of TimeApp. The bug caused the app to crash…
Customer Testimonial: MetrTech Consultancy
John Powell retired from his civil service job in June 2014 and two months later he started his own firm; MetrTech Consultancy. They provide consulting services tailored to the government (US) sector, with a focus on the niche markets of investigative technology and digital multimedia evidence management. We asked John how he found TimeApp; –…
Inexpensive and easy to use
A law firm full of energy – that is how we can summarize Jennifer Loeb Cederwall and her law firm; Cederwalls International Law Firm. We spoke to Jennifer to ask her some questions about TimeApp and how they are using our app. She told us that they found it while researching which time keeping programs…
New article about TimeApp
Don't forget to read the latest article about TimeApp in Digital Journal, enjoy…
http://www.digitaljournal.com/technology/tarik-basic-talks-about-swedish-tech-company-timeapp/article/422242
A long time has passed since my latest blog, so it is time to sit down and write a couple of lines. In this post I will reflect on a company's credibility (also called ethos in the rhetoric world ). How do different companies work with their credibility? We know very well that there's an…
Christmas Contest
This Christmas we want to give our customers/users and Facebook followers a chance to become a popular Santa and give away a super nice Christmas present for someone you really like. It's not any old Christmas gift either; Two nights for two persons at the luxurious Kosta Boda Art Hotel in Sweden with breakfast, three-course dinner…
Customer Testimonial – Kenan Mandra, Farmis LTD Sarajevo
Customer Testimonial – Kenan Mandra, Farmis LTD Sarajevo Every now and then, we receive emails from customers who wish to highlight the advantages that they have experienced since implementing TimeApp. Kenan Mandra, who has been the head of marketing for Farmis LTD Sarajevo has now opened his own clinic, explains how TimeApp has facilitated and…
Article about TimeApp on Legal IT Insider: A big hit with Swedish lawyers
"Since its launch in 1995, Legal IT Insider has become the leading media, information and communications channel in legal technology. We cover and interpret legal IT news from UK & EMEA, the US and APAC regions in our famous free monthly digital newsletter – Legal IT Insider – and also here on our website via…
Article about TimeApp on SwedishStartupSpace.com
"Sweden is an emerging tech country – there's no doubt about that. Spotify, Klarna, Mojang and Wrapp are just a few examples of Swedish startups that have taken the world by storm. Reuters, Entrepreneur and PandoDaily have all praised Swedish entrepreneurs and their knack for building global businesses." So reads the introduction to SwedishStartupSpace.com –…
New version of TimeApp for Android
Recently we launched a brand new version of the Android app for TimeApp. It is reworked from the ground up to be even more stable so that you can use TimeApp even in cases of poor network coverage. The navigation is now placed at the left edge of the screen, and only appears when you…
Everything gets easier … step by step
We have now launched a new welcome guide on how easy it is to use our awesome time tracking software. In the wizard – which is presented on-screen when you first login – you can follow step-by-step how to use the program and get the maximum effect. We go over the various functions, with images and text, in a quick and informative matter.…
Now we change the name
The company behind TimeApp was founded in 1993 and was originally named Persson & Knapasjö Utvecklingskonsult AB. In September 2014 we changed the name to the more appropriate (and a lot shorter) P&K TimeApp AB. The change affects only the name, other details remains the same. P&K TimeApp AB Floragatan 3B SE-641 33 Katrineholm Sweden…
TimeApp expands with two new team members
On 1 October the TimeApp family grew with two additional skilled employees. Josefin Nässlander, who previously worked as an editorial secretary at the trade magazine Screen&Marknaden, assumed the role of Marketing Coordinator. Josefin will ensure that we expand our reach and will be working actively to strengthen the brand and spread awareness about TimeApp in…


IT'S time for Jack Wilshere to leave Arsenal.
The England playmaker deserves an enormous amount of credit in asking to leave The Emirates on loan over the summer and it's a decision that has paid off.
While he may not be scoring or setting up as many goals as he would like for the Cherries – at least he's playing.
And those returns will come in time once he's really back and in the flow. It's easy to forget just how much football he's missed in the last few years.
You can tell from watching him he's enjoying himself and after seeing his talent go to waste, the 24-year-old is starting to get back to where he should be.
While the south-coast experience is clearly benefitting Wilshere, he shouldn't consider his seaside holiday as a long-term solution.
Arsene Wenger has admitted he would like to tie Wilshere down to a new contract but we all know Arsenal is not the place for an injury-prone player.
It is no coincidence he's started playing regularly immediately after leaving.
Wilshere needs to do what is right for this career and his body and cut all ties with a club that has become cursed for him.
JACK WILSHERE completed 90 minutes for the first time since September 2014.
I think Jurgen Klopp would love to have him at Liverpool. He's exactly the type of player the German rates. Full of heart, committed, tough tacking, aggressive and attacking.
Liverpool's boss would get the best out him and, fitness depending, would have him back in the England team within months.
Tottenham's style would be a good fit for Wilshere, thought a move across north London is probably not on the cards.
Manchester City were linked with the Stevenage-born ace a few windows ago but would Pep Guardiola fancy him now? It's unlikely.
If Wilshere bites the bullet and forces an Emirates exit there will be no shortage of suitors.
No doubt Bournemouth would love to get him on a permanent basis but to stay there would be an emotional decision that could further stunt his development. Wilshere is Champions League standard player.
Wilshere should tell Arsenal he wants out in January, completely cut the ties and put his Emirates nightmare behind him.


The Enchanted Celebrations Team officially held down Neptune's Jumping Brook Country Club for the Mohr/Dorf Wedding. The day started at 2PM when our photographer arrived for the brides's preparation and ended at 1:30AM when the photographer and DJs left the after party in the "Club" room. The client used us for DJ, Photography, Videography, ZAP Photography, Photomontage as well as a wedding minister. The wedding went unbelievably smooth! That's what you get with Enchanted Celebrations one team with one goal, a picture perfect wedding.
For more information on Enchanted Celebrations, please visit our website at: www.enchantedcelebrations.com or call Alex directly at 609-618-7302.


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Q: Logistic regression + histogram with ggplot2 I have some binary data, and I want to plot both a logistic regression line and the histogram of relative frequencies of 0s and 1s in the same plot.
I ran into a very nice implementation using the package popbio here: shizuka lab's page
Here a MWE that runs with library(popbio) (courtesy shizuka lab)
bodysize=rnorm(20,30,2) # generates 20 values, with mean of 30 & s.d.=2
bodysize=sort(bodysize) # sorts these values in ascending order.
survive=c(0,0,0,0,0,1,0,1,0,0,1,1,0,1,1,1,0,1,1,1) # assign 'survival' to these 20 individuals non-randomly... most mortality occurs at smaller body size
dat=as.data.frame(cbind(bodysize,survive))

#and now the plot
library(popbio)
logi.hist.plot(bodysize,survive,boxp=FALSE,type="hist",col="gray")

which produces

Now, is it possible to do this with ggplot2?

A: Here are some idea's
ggplot(dat, aes(x = bodysize, y = survive)) +
  geom_dotplot(
    aes(fill = factor(survive)), method = "histodot", binpositions = "all",
    stackgroups = TRUE, stackdir = "centerwhole", binwidth = 1
  ) +
  geom_smooth(method = "glm", family = "binomial")

ggplot(dat, aes(x = bodysize, y = survive)) +
  geom_hex(bins = 10) +
  geom_smooth(method = "glm", family = "binomial")

ggplot(dat, aes(x = bodysize, y = survive)) +
  geom_bin2d(bins = 10) +
  geom_smooth(method = "glm", family = "binomial")

redpajama_set_name: RedPajamaStackExchange


I follow quite a lot of companies, so the usefulness of a watchlist to me cannot be overstated. Without my watchlist, I'd be unable to keep up on my favorite sectors and see what's really moving the market. Even worse, I'd be lost when the time came to choose which stock I'm buying or shorting next.
Today is Watchlist Wednesday, so I'm discussing three companies that have crossed my radar in the past week — and at what point I may consider taking action on these calls with my own money. Keep in mind that these aren't concrete buy or sell recommendations, nor do I guarantee I'll take action on the companies being discussed. What I can promise is that you can follow my real-life transactions through my profile and that I, like everyone else here at The Motley Fool, will continue to hold the integrity of our disclosure policy in the highest regard.
You can go ahead and give yourself all the reasons in the world why Apple should sell off after announcing its Street-topping second-quarter results, but I can give you 145 billion reasons why it remains an intriguing buy!
For the quarter, Apple Inc. (NASDAQ:AAPL) reported an 11% increase in revenue to $43.6 billion as it upped sales of the iPhone to 37.4 million from 35.1 million in the year-ago period, and iPad sales launched to 19.5 million from 11.8 million. Yes, PC sales remain a weak point with Mac sales dipping again, but overall this was another strong quarter of cash flow generation and consistent execution.
Not only did Apple Inc. (NASDAQ:AAPL) end the quarter with $145 billion in cash, but it also announced a long-awaited increase in its dividend and a big boost in its share repurchase program. Apple Inc. (NASDAQ:AAPL) upped its dividend by 15% to $3.05 per quarter and plans to return $100 billion in capital to shareholders through these dividends and share buybacks over a three-year period. As such, Apple Inc. (NASDAQ:AAPL)'s board authorized a 500% increase in its share repurchase program to $60 billion from $10 billion that is expected to be completed by the end of calendar 2015.
Again, what's wrong with Apple Inc. (NASDAQ:AAPL)? Other than a big group of emotional traders, nothing that I can tell!
Sticking with the theme of earnings-driven stock moves, fresh-Mex restaurant chain Chipotle Mexican Grill, Inc. (NYSE:CMG) rocketed higher last week after reporting better-than-expected first-quarter results.
For the quarter, Chipotle Mexican Grill, Inc. (NYSE:CMG) delivered a 13% increase in revenue to $726.8 million as adjusted EPS jumped 24%, but it achieved this mainly through the opening of new stores. Chipotle Mexican Grill, Inc. (NYSE:CMG)'s actual same-store sales growth improved just 1%. Looking ahead, Chipotle cautioned investors to expect same-store sales growth to be flat to up low-single-digits.


On Monday, President Obama took another step towards fighting climate change as his administration announced a new EPA Clean Power Plan to cut emissions from coal-fired power plants by 30% below 2005 levels by the year 2030.
We applaud the President's commitment to fighting climate change and creating a healthier and safer environment for our children. Capping carbon emissions is critical to the health of our communities, particularly children of color who are more likely to grow up breathing in the toxins from these very coal plants. The EPA's ruling provides flexibility for states to innovate ways in addressing air pollution from making plants efficient and producing more renewable energy to enacting broader policies such as carbon taxes or the very popular cap and trade.
While the move is deemed historic, we cannot stop here. Fighting climate change through carbon reduction alone will not solve the challenges climate change creates. As we discussed last month, inequity drives climate change, therefore, we need solutions that tackle both inequity and carbon.
How the EPA's rules are implemented will matter greatly and they must include the participation and voices of communities most impacted by coal. But if we rely on the implementation strategies such as "cap and trade," we will fail to address inequity. For example, the Center for Race Poverty and the Environment has found that "Cap and Trade has a disproportionate negative impact on communities of color because those communities do not receive the benefits of on site-reductions when major polluters buy pollution reductions from somewhere else".
Therefore, we need a cap and invest strategy to implement these rulings, where we reduce the carbon and invest in communities abilities to own, produce, and generate renewable energy that can spur a healthy economy and a cleaner environment. States are at the foundation of implementing how these rules take place.
They need to get it right.
Today, we offer a policy concept called Energy Investment Districts. Energy Investment Districts (EIDs) is a policy idea at the state, county or municipal level makes geographic areas that have suffered environmental and economic hardships eligible for funding and other financial and technical support for community-scale energy efficiency and renewable generation projects.
Together lets push for policies that will truly create opportunities for both.


Join the party led by Christina and have some fun while you get fit. The class is set to Latin and International rhythms and designed for everybody and every body. A total workout combing all elements of fitness - cardio, muscle conditioning, balance, and flexibility. Elevate yourself and keep moving!


text:
Q: Day vs 24h day? In Russian (my native language) there are two words - "день" and "сутки". First means day, just a day, probably lightest part of it, second - 24 hours from 00:00 to 00:00. There is a strict difference between them. As I tried all of the dictionaries - in English there is a single word "day" for both meanings. Is it true? Are there no two different words for a light part of day and for whole day 00:00-00:00?

A: It is true that "day" can have multiple shades of meaning, depending on context. If you need to disambiguate:

*

*A calendar day is a 24-hour period from midnight to midnight.

*Daytime or daylight hours refers to the time between sunrise and sunset.

A: As a translator and a learner of the English language, and Russian being my native language, I say the difference between the two terms does actually exist in English. Some may say that it doesn't though.
Let's take the word "day" first. It can refer to the daytime when it's bright outside or to the day itself as starting 0:00:00 at night and ending the next second after 11:59:59 the same night.
Now, we can state that a day is a timeline which equals 24 hours but has a defined start time and a defined end time. Besides that it is also a part within this defined timeline when the sun is up. Thus the word carries the same definition as the Russian word "день".
Now, if we take the Russian word "сутки" we can either represent it as a "day" or as a period of "24 hours" depending on the meaning we imply.
Assuming we are speaking about a period of 24 hours we can start them at any point of time within a day: 24 hours can start any time on one day but should strictly end at the same time the other day.
But besides that "сутки" can still refer to the day the way it starts and ends.
Here are a few examples that will help you feel the difference:

*

*Он занимается каждый день. - He studies every day.

*Весь день мы провели на улице. - We spent the whole day outside.

*Ночь была холодной, а вот день был тёплым. - The night was cold but the day was warm.

*Мы целые сутки потратили, чтобы это сделать. - We spent a whole day to finish it.

*У тебя есть сутки, чтобы добыть деньги. - You've got 24 hours to get the money.

*За сутки можно многое сделать. - Much can be done in a day.

*Компиляция занимает аж сутки. - The compilation takes a full day to complete.

As you can see, depending on the idea and the context we can choose an option that fits best.

A: Yes, the word "day" can mean the entire 24-hour period. But it can also mean just the part where the Sun is above the horizon. It can also mean the part of the day during which work and other "daytime" activities are accomplished.
Examples:
"A week is seven days long."
"Please respond within 30 days."
"The light he called day but the darkness he called night."
"I expect your response by the end of the day."
"I have been waiting for you all day!"
When it is necessary to specify a 24-hour period, this is done explicitly:
"He was under the rubble for 48 hours."
(Он провёл под завалом двое суток.)
The expressions "24 hours", "48 hours", and "72 hours" are used. After that the distinction is seldom important. When it is, you can say "full days".

A: The question succinctly could be expressed like this: How does one say сутки in English? That's really what the original poster is asking as this is one of the most common questions asked by Russian speakers who are new to English. He's just not being very clear about it (but that's fine).
Yes, it's true. The word сутки in Russian means an amount of time that equals 24 hours and is often used synonymously with the word день which in meaning is exactly equivalent to the English word day. By the same token, the word day in English is defined as exactly a 24-hour period when viewed as a length of time and, to the best of my knowledge, no other special term like сутки in Russian that specifically refers to a 24-hour time period exists in English. So, day does cover both meanings and, in general, you're really going to have to translate both день and cутки using a single word, day, when talking about a 24-hour period for which Russians have a special term.
However, all is not lost. In situations where you want to be more specific, you can qualify the word day by adding additional time-related information to it to make it more precise timewise. In other words, you would say something like this: entire day, 24-hour day (this one sounds a little bit tautological, though) or, if you want to be more technical, calendar day. Better yet, you could even say: twenty-four hour period or simply twenty-four hours (I think twenty-four hours is the exact English translation of сутки).
Examples:

It's been twenty-four hours and he still hasn't called. (Прошли сутки и он ещё не позвонил.)
This job will take me five calendar days to complete. (Эта работа займет у меня пять суток/календарных дней.)
I'm available twenty-four hours a day. (Я доступен каждые сутки.)

Here's one way to understand the problem. And this is just an example. Since the arrival of the Internet where there is really no difference between night and day, greetings like "Доброго всем времени суток!", which literally translates as "Good 24-hour day, everybody!" (not idiomatic English at all), have become very popular. And a lot of people learning English often ask how do you say the exact same thing in English? The answer is you can't. You simply have to say "Good day, everybody!" But the thing is that Russian has that expression too ("Добрый день всем!") which is distinct from the one used on the Internet. So, how to reconcile this situation is not obvious.

A: Yes, there is a word meaning "24 consecutive hours". It's nychthemeron. Not an everyday word, and few people would know it, but still it exists!

A: It sounds like день translates best to "daytime", while сутки translates to "day".  In many cases English speakers will use "day" for either, unless we want to distinguish the part when the sun is up from the nighttime.  In this case we say "during the day", which almost always means the part of the day when the sun is up.

She likes to paint during the day.

Otherwise use "daytime":

She likes to paint in the daytime.

These are standard expressions and you may want to substitute them for день as needed.

Can we see the moon during the day? (Можем ли мы видеть луну днем?)

Side note:  "Day", and "daytime" are rarely exact measurements in English. While, yes, a "day" is 24 hours, sometimes the speaker means, "From the time I wake up to the time I go to sleep," and so can vary with context.


KraftPal, with its proprietary product; manufacturing system and business-model for recyclable corrugated-cardboard pallets is excited by the appointment of James R. Keller to the team. James R. Keller will oversee both the KraftPal North American launch, and manage ensuing operations.
James R. Keller spent a 33 year career with Weyerhaeuser, including 9 years as Senior Vice-President of the Containerboard Packaging and Recycling Division. He was also an ICCA (International Corrugated Case Association) member for a number of years, including 4 years as President.
James R. Keller takes up his post as of January 15th, 2016; bringing enthusiasm, in-depth industry knowledge, plus a worldwide range of industry contacts to KraftPal.


text:
Lymantria apicebrunnea is een vlinder uit de familie van de spinneruilen (Erebidae), onderfamilie donsvlinders (Lymantriinae). De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1932 door Gaede.

Het mannetje heeft een voorvleugellengte van 25 tot 28 millimeter, het vrouwtje van 39 tot 41 millimeter. De vliegtijd is in juni.

De soort komt voor in China.

apicebrunnea
Endemisch dier uit China


text:
BLUE Kazimir
The Kazimir Watch measures 37mm (1.46″) in diameter and has an 18mm wide black silicone band. The watch measures 9 1/2″ from end to end with 6 ½″ to the closest hole and 8 1/2″ inches from the farthest hole.
BLUE Kazimir quantity
SKU: PJT-7296AS Category: Collections
YELLOW Kazimir $140.00
RED Kazimir $140.00
Diagram 17 STEEL $140.00
About Denis Guidone
Sped up Animation
Design: Denis Guidone (2015) 2019
We love this watch so much that it is our third color! First with the yellow in 2015 followed by the red in 2017 and now blue for 2019!
Blue Kazimir Suprematism, the invention of Russian artist Kazimir Malevich, was one of the earliest and most radical developments in abstract art. Its name derived from Malevich's belief that Suprematist art would be superior to all the art of the past, and that it would lead to the "supremacy of pure feeling or perception in the pictorial arts." Heavily influenced by avant-garde poets, and an emerging movement in literary criticism, Malevich derived his interest in flouting the rules of language, in defying reason. He believed that there were only delicate links between words or signs and the objects they denote, and from this he saw the possibilities for a totally abstract art. And just as the poets and literary critics were interested in deconstructing the language of literature, Malevich came to be intrigued by the search for art's barest essentials. It was a radical and experimental project that at times came close to a strange mysticism. Although the Communist authorities later attacked the movement, its influence was pervasive in Russia in the early 1920s. The movement was pivotal in shaping Constructivism, and continues to inspire abstract art to this day.
In homage to the art movement, the Blue Kazimir Suprematism watch frames bold shapes and colors which continue shift composition with each passing second. The hour is the gray hand, the minute is the black hand and the blue second hand disc rotates around the watch every 60 seconds.
Blue Kazimir is made of black IPB stainless steel with a black silicone band, and utilizes a high quality Japanese movement. It is considered the perfect unisex-size case measuring 37mm. Right in between our largest 40mm and smallest 33mm unisex sizes. The watch measures 9.5" from end to end; to 6.5″ at the closest hole and 8.5″ inches from the farthest hole.
Also, available are RED and YELLOW Kazimir Suprematisms with gray hour and black minute hands and complemented with black silicone band.
Each watch band has the accent loop color of the second hand…blue has a blue loop, red has a red loop and yellow has a yellow loop.
Japanese Quartz
9.65mm (.38")
Band (Colors)
Band (Material)
Band (Size)
Denis Guidone, as student of architecture, design and visual communication studied urban planning and architecture in Milan, Italy and Oporto, Portugal. His project process was born under the concept of subtraction, an ideology that follows the unification of opposites through a constant approach on the edge of architecture and visual communication, and at the same time mediated by an expressive intensity that addresses work on the basis of subtraction. Fascinated by the passing of time, Guidone carries out this intense didactic work for several universities and design schools, in Italy and abroad. His many awards and honorable mentions include: Ida international design award in Los Angeles, IDOT Best Italian Design Product, and The Bruno Munari international award.
February 2018 Denis Guidone has been appointed Ambassador of Italian Design and Culture from the Ministry of Foreign Affairs Farnesina and "la Triennale di Milano".
33mm Reveal Classic Watch
Tangency Watch
GREEN Reveal
Grand Tour Dual Time BLUE
STEEL Nadir
Mado Watch 2017 GOOD DESIGN Award
40mm Black Bodoni M&Co