text:
\section{INTRODUCTION}

For its practical applications in fields such as astronomy, medical, autonomous driving, etc., image denoising as an area of research has been around for a very long time. In spite of the advancements in technology which has made it possible to obtain crystal clear images in recent times, image capturing equipment have very little control over the environment in which such images are taken. For instance, medical images of human bones captured by X-ray machines have the various tissues and organs of the body as potential noise. This implies that noise in images cannot be totally removed by such equipment making image denoising a very active research area till today.

Image denoising is, therefore, the art of recovering a clean image $ \mathbf{X} $ from a noisy image observation $ \mathbf{Y} $. $ \mathbf{Y} $ is a combination of the clean image $ \mathbf{X} $ and a noise $ \mathbf{W} $ typically modeled as in \eqref{eq:noisyImage}.

\begin{equation}
	\mathbf{Y = X + W.}
	\label{eq:noisyImage}
\end{equation}

The entries of $ \mathbf{W} $ are assumed to be random and independently drawn from the same probability distribution (i.i.d) such as the Gaussian distribution. As such, $ \mathbf{W} $ is typically modeled as an additive, white and Gaussian noise (AWGN) with zero mean and a known standard deviation $\sigma$.

Over the years, several image denoising algorithms have been proposed. Some of them exploit various mathematical analysis and similarities in pixels and patches to perform denoising. Such algorithms are commonly referred to as classical denoising algorithms, such as \cite{buades2005, Behzad2017, Aharon2006a, Dabov2007a, mairal2009}. The non-local means (NLM) algorithm \cite{buades2005} uses the similarities in the pixels of the corrupted image to perform denoising. Its performance, however, has been surpassed by the patch-based algorithms. This is because a patch carries more information about an image as compared to a single pixel.

To perform patch-based denoising, the overlapping patches are usually transformed from the spatial domain to a transform domain. This transform domain is such that the noisy components are easily separable from the original image's components. Denoising then becomes a simple question of identifying which components are due to the noise and removing them. One of such domain which has been exploited by most patch-based denoising algorithms is the sparse domain. In this domain, the patches which are dense in the spatial domain are represented by a few nonzero components. This makes denoising easier and more effective. For such transformation, an overcomplete dictionary with the help of sparse recovery algorithms such as the orthogonal matching pursuit (OMP) \cite{Aharon2006a} and the support agnostic bayesian matching pursuit (SABMP) \cite{Masood2013a} algorithms are used. The overcomplete dictionary is obtained using analytical techniques such as the discrete cosine transform (DCT) and discrete wavelet transform (DWT) \cite{qayyum2016}, wavelet \cite{daubechies1990}, curvelet \cite{starck2002}, and contourlet \cite{do2005} dictionaries. Another approach to obtaining the dictionary is to learn it from the image patches themselves. This is done with the help of dictionary learning algorithms such as the K-SVD \cite{Aharon2006a}, the method of optimal directions (MOD) \cite{engan1999}, principal component analysis (PCA) \cite{vidal2003}, etc. Algorithms such as the Block-Matching and 3D transformation (BM3D) \cite{Dabov2007a}, Collaborative Support-Agnostic Recovery (CSAR) \cite{Behzad2017}, and K-SVD denoising \cite{Elad2006a} algorithms are some of a long list of algorithms which utilizes patches transformation into the sparse domain to perform denoising.

Owing to its effectiveness, the BM3D was used as the benchmark denoising algorithm until recently. The use of neural networks (NNs) in denoising has seen the performance of the state-of-the-art classical techniques surpassed. The NN algorithms mostly utilize several layers of convolutional NNs (CNNs) to perform denosing. Example of such algorithms are the deep CNN (DnCNN) \cite{Zhang2017} and the fast and flexible feed-forward NN (FFDNet) \cite{Zhang2018}.

Despite the progressive improvement in the performance of the denoising algorithms over the years, many of them fail in their inability to denoise images of lower resolutions. Noisy low resolution images are sometimes the output of medical imaging systems. As such, ability of the image denoising algorithms to denoise such images is of practical importance. Some of the algorithms also tend to smooth out the denoised image leading to the loss of some of the salient details of the image.

In this paper, we present a collaborative filtering-based (CoFiB) denoising algorithm which performs collaboration among similar patches in the sparse domain to perform denoising. Our contribution is the ability of our proposed algorithm to perform well across images of any given resolution. Most importantly, images of low resolution. While denoising, our algorithm is also smart enough to preserve the details of the image. This means that our algorithm is able to strike a balance between complete image denoising leading to smoothened effect on the image and details preservation.

The rest of the paper is organized as follows. We provide a detail description of our proposed CoFiB algorithm in Section \ref{sec:cofib}. Section \ref{sec:exp} provides the details of the expensive experiments conducted and the results. Finally, we conclude in Section \ref{sec:con}.

\section{THE PROPOSED COLLABORATIVE FILTERING-BASED DENOISING ALGORITHM}
\label{sec:cofib}

In this section, we present the proposed CoFiB denoising algorithm. Our algorithm is a patch-based classical image denoising algorithm. The patches are transformed to the sparse domain where collaborative denoising is performed. In the sparse domain, the algorithm takes advantage of the similar patches' sparse representations to obtain a better denoising performance. The denoised sparse  representations are transformed back to the spatial domain. All denoised patches are then brought together to generate the denoised image. All these are performed in 5 steps as shown in Fig. \ref{fig:cofibFlowchart}. These steps are described as follows.

\begin{figure*}
	\centering
	\includegraphics[height=9cm, width=\textwidth]{images/cofib2.jpg}
	\caption{Flowchart of the Proposed CoFiB Algorithm.}
	\label{fig:cofibFlowchart}
\end{figure*}

\subsection{Step 1: Patches Extraction and Transformation}
\label{sec:patchExtract}

The first step in the pipeline of the proposed CoFiB algorithm is to extract a set of overlapping patches from the noisy image. For ease of analysis, the patches are of squared sizes, i.e. having the same number of row and column pixels. The patches are formed in such a way that the number of patches is equal to the number of pixels in the noisy image. This is done by padding the image with $ \frac{\sqrt{R}-1}{2} $ pixels in all directions. Given that each patch is of size $ n \times n $, $ R = n^{2} $ is the size of each vectorized image patch. $ R $ is chosen in a way that $ n $ is always odd. The reason for choosing $ R $ in this manner is to ensure that $ \frac{\sqrt{R}-1}{2} $ will always produce an integer. This implies that all pixels in the image would have a corresponding patch, known as the pixel patch. The central pixel of a pixel patch corresponds to the pixel it is representing.

Some parts of both natural and man-made images have similar patterns that may or may not differ in intensity levels. For example, Fig. \ref{fig:simpatchesexample}(A) and Fig. \ref{fig:simpatchesexample}(B) have different intensity levels, but the underlying pattern/structure is similar. Similarly, a purely white image and a purely black image can be thought of as having the same pattern but, while the intensity value of a white image is 0, that of a black is 255 (or 0 for white and 1 for black for an image range of [0, 1]). In our algorithm, we want to ensure that all patches that have a similar structure are grouped together in an intensity invariant manner. This was achieved by dividing a patch by the maximum absolute pixel value. This ensure that patches are grouped together not by their intensity but by their underlying structural pattern.

\begin{figure}[htb]
	\centering
	\begin{center}
		\begin{tabular}{ c c c }
			\includegraphics[width= 3cm, height= 3cm]{images/20.jpg}
			&
			$ \qquad $
			&
			\includegraphics[width= 3cm, height= 3cm]{images/21.jpg}
			\\
			(A) & & (B)
		\end{tabular}
	\end{center}
	\caption{Patches with Similar Pattern}
	\label{fig:simpatchesexample}
\end{figure}

\subsection{Step 2: Patches Clustering}
\label{sec:patchCluster}

Our proposed algorithm performs collaboration in the sparse domain. To transform the intensity invariant patches from the spatial to the sparse domain, an overcomplete dictionary is needed. However, because the patches are coming from different parts of the image, they will be very different. Therefore, they will be scattered in the sparse domain. For collaboration, we need patches close enough to each other in the sparse domain. Similar patches are expected to have similar representations in the sparse domain. We take advantage of this fact and grouped similar intensity invariant patches together. These groupings are used to enhance the sparse domain collaborative denoising. The popular k-means clustering algorithm \cite{jain2010}  was used to group the intensity invariant patches ointo $ k $ clusters.

\subsection{Step 3: Dictionary Learning}
\label{sec:dictLearn}

The dictionary for each cluster was learned with the help of the K-SVD dictionary learning algorithm \cite{Aharon2006}. For each cluster, we learned a separate dictionary. While a little computationally expensive, this helped to ensure that each cluster is suitably represented and most of the sparse estimation components better project the patches of each cluster unto their corresponding sparse domain with negligible error compared to using a single dictionary for all the patches.

\subsection{Step 4: Collaborative Denoising}
\label{sec:collabDen}

The sparse representation of each patch in each cluster is then obtained using the Support Agnostic Bayesian Matching Pursuit (SABMP) sparse recovery algorithm \cite{Masood2013a}. Although there are several sparse recovery algorithms, our choice of the SABMP algorithm hinged on the following facts about the algorithm which helped in the strategy we employed for denoising the image.

\begin{enumerate}
	\item It is invariant to the distribution of the image intensity values.

	\item It does not impose any strict conditions on the cluster dictionary,

	\item and it gives, as an output, the probability of each element of the sparse representation of a patch to be active.
\end{enumerate}

The sparse representation of patches in a cluster should have a high number of common locations of nonzero elements provided that the dictionary used is perfect, that is, all the dictionary atoms are uncorrelated. This would mean that the uncommon nonzero elements in the sparse representation are due mainly to noise, to a high probability. By simply removing the uncommon nonzero location components will leave us with clean portions of the representation. However, because the dictionary of a cluster was trained from similar data, a perfect dictionary is not obtainable. In addition, the dictionary is overcomplete, as such, several sparse domain representations could be mapped to a patch. This gives rise to sparse domain representations that are not unique. This ultimately means that several atoms in the dictionary can combine in different ways to yield the same point in the sparse domain. Locations of nonzero elements could differ due to genuine differences in the patches themselves. This makes simple removal of uncommon elements, not a good move. The SABMP algorithm comes to our rescue in this case as it can take into consideration the apriori probability of an element in a patch to be active in order to estimate the sparse vector. To take advantage of this fact, patches in the same cluster are allowed to collaborate to update their active probabilities in a weighted average approach to yield the denoised sparse representation's nonzero locations' probability. This step is as given mathematically in equations (\ref{eq:alpha}) through (\ref{eq:lambdaupdate}) below.

\begin{subequations}
	\begin{align}
	\alpha_{j,p,k}=\dfrac{1}{\Delta(y_{j,k}, y_{p,k})} \quad j \neq p.
	\label{eq:alpha}
	\\[20pt]
	Normalized \qquad \alpha_{j,p,k}=0.5 \times \frac{\alpha_{j,p,k}}{\sum_{j=1}^{t-1}\alpha_{j,p,k}} \quad j \neq p.
	\label{eq:alphanormalized}
	\\[20pt]
	\lambda_{p,k}^{updated}=\sum_{j=1}^{t}{\alpha_{j,p,k}\lambda_{j,k}}, \quad \alpha_{t,p,k} = 0.5, \quad \lambda_{t,k} = \lambda_{p,k}.
	\label{eq:lambdaupdate}
	\end{align}
\end{subequations}

For each patch $ p $ in each cluster $ k $, $ t-1 $ similar patches are selected (where $ t $ is the number of patches for collaboration). The selection is based on how close the patches in the cluster $ k $ are to the patch $ p $. This can be obtained using any suitable distance measure. A common distance metric is the Euclidean distance between the patch $ p $ (given as $ y_{p,k} $) and other patches in the same cluster $ k $ (given as $ y_{j,k} $, for $ j = 1 $ to $ n_{cluster} $, where $ n_{cluster} $ is the number of patches in the cluster). This distance is given as $ \Delta(y_{j,k}, y_{p,k}) $. The $ t-1 $ similar patches obtained are therefore given as $ y_{j,k} $, for $ j = 1 $ to $ t-1 $.

The closer a patch is to the patch $ p $, the smaller the value of the distance between them, but the more similar they become. As such, a similar patch's denoising contribution to the patch $ p $ is inversely proportional to the distance between them. Hence, the denoising collaborative contribution ($ \alpha_{j,p,k} $) of a similar patch to the patch $ p $ in the same cluster $ k $ is obtained using equation(\ref{eq:alpha}). Observe the inverse proportionality in the equation. The distance of patch $ p $ to itself is $ 0 $, as such it should have the highest collaborative contribution to itself. This implies that patch $ p $ would have a 50\% normalized contribution value to itself, i.e. $ \alpha_{t,p,k} = 0.5 $. All the other similar patches would have to scale accordingly to contribute the other normalized 50\% value. This 50\%  normalized collaborative contribution value of the other similar patch to patch $ p $ is given by equation (\ref{eq:alphanormalized}). Observe how their contribution scale using $ 0.5 \times $ inverse of their summation as a scaling factor. This is the contribution of each obtained similar patch towards denoising the patch $ p $.

As earlier described, from the outputs of the SABMP algorithm, the location probabilities $ \lambda_{j,k} $ of the elements of the sparse representation of each of these similar patches including the patch $ p $ can be recovered. To obtain the denoised active probabilities of patch $ p $, its $ \lambda_{p,k} $ needs to be updated/denoised. The updated $ \lambda_{p,k}^{updated} $ is obtained using the normalized denoising collaborative contributions ($ \alpha_{j,p,k} $) of each of the patches with their respective $ \lambda_{j,k} $. This is as given in equation (\ref{eq:lambdaupdate}).

In Fig. \ref{fig:cofibcollab}, we show how the similar patches collaborate to update/denoise the location probability of the sparse representation of the currently considered noisy patch in each cluster. This updated/denoised probability is then fed back into the SABMP algorithm to re-estimate the sparse domain representation of the denoised patch.

\begin{figure*}
	\centering
	\begin{center}
		\begin{tabular}{ c }
			\includegraphics[width=0.8\textwidth, height=0.4\textheight]{images/24.jpg}
			\\
			It should be noted here that y represents the active location \\
			probabilities of the sparse representation, $\lambda$, of a patch \\
			and w is the same as the collaboration weight $\alpha$ in (\ref{eq:alphanormalized}).
		\end{tabular}
	\end{center}
	\caption{Collaborative Denoising of Patches in the Sparse Domain.}
	\label{fig:cofibcollab}
\end{figure*}

\subsection{Step 5: Image Reconstruction}
\label{sec:imageRecon}

The denoised sparse representation of the patches are then transformed back to the spatial domain. This is obtained by simply multiplying the denoised sparse representation of each patch with the dictionary of the cluster to which the patch belongs. This yields the denoised patches.

In order to reconstruct the denoised image, the denoised patches are simply replaced back to their original locations in the noisy image. Noting that these patches overlap, they are then aggregated to obtain the denoised image.

\section{EXPERIMENTAL RESULTS}
\label{sec:exp}

We performed several experiments on the proposed CoFiB algorithm. We also compared it to some of the classical denoising algorithms. For performance comparison, the widely used peak signal to noise ratio (PSNR) and the structural similarity index (SSIM) performance metrics have been used. These methods are termed objective measures. Despite these measures being widely used, visual inspection (subjective measure) of the denoised image is needed to truly ascertain the performance of an image denoising algorithm \cite{Al-Najjar2012}. We will show that in almost all cases, our proposed CoFiB algorithm performs subjectively (visually) better than most of the state-of-the-art denoising algorithms such as the K-SVD \cite{Elad2006a}, NLM \cite{buades2005}, and BM3D \cite{Dabov2007a}. However, using the objective metrics, some of the algorithms performed better in few cases. As such, an image denoising algorithm should be evaluated both subjectively and objectively to truly ascertain the performance of such algorithm.

To cluster the intensity invariant patches, we found $ k = 5 $ to be a good number to have yielded the best performance for our algorithm. This number can be explained by the fact that a natural image patch mostly consists of flat, horizontal, vertical, left diagonal, and right diagonal regions. We now proceed to present and describe the result of our experiments in the following sections.

\subsection{Performance Comparison of Different Algorithms on Different SNR}
\label{sec:cofibresultsdiffsnr}

Here, we used a $ 512\times512 $ boat image which is one of the widely used images in literature for testing image denoising algorithms. White Gaussian Noise (WGN) was added to the original boat image. The noise was assumed to be of zero mean and variance that corresponds to the different values of SNR that were selected for our experiments. The SNR have been given in decibel (dB). These values ranges from $ -5dB $ to $ 35dB $ with an interval of $ 5dB $. We use the K-SVD, NLM, and BM3D to compare our algorithm. Each algorithm was then used to denoise the noisy image. Table \ref{tab:cofibdiffsnr} shows a comparison of the performance of the different algorithms on the different SNR values. These results are also presented graphically as shown in Fig. \ref{fig:cofibdiffsnr}.

\begin{table*}
	\centering
	\caption{Performance Comparison of Different Algorithms w.r.t the SNR (Boat $ 512 \times 512 $)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t1.jpg}
	\end{tabular}
	\label{tab:cofibdiffsnr}
\end{table*}

\begin{figure*}
	\centering
	\includegraphics[width=\textwidth]{images/25.png}
	\caption{Performance Comparison of Different Algorithms w.r.t the SNR (Boat $ 512 \times 512 $)}
	\label{fig:cofibdiffsnr}
\end{figure*}

\begin{figure*}[htb]
	\centering
	\fbox{
		\begin{minipage}{13 cm}
			\begin{center}
				\begin{tabular}{ c c }
					\includegraphics[width= 0.4\textwidth]{images/26a.png} & \includegraphics[width= 0.4\textwidth]{images/26b.png}
					\\
					(a) & (b)
					\\
					\\
					\includegraphics[width= 0.4\textwidth]{images/26c.png} & \includegraphics[width= 0.4\textwidth]{images/26d.png}
					\\
					(c) & (d)
					\\
					\\
					\includegraphics[width= 0.4\textwidth]{images/26e.png} & \includegraphics[width= 0.4\textwidth]{images/26f.png}
					\\
					(e) & (f)
					\\
					\\
					\multicolumn{2}{c}{(a) Clean Image, (b) Noisy Image, (c) NLM} \\
					\multicolumn{2}{c}{(d) K-SVD, (e) BM3D, (f) CoFiB}
				\end{tabular}
			\end{center}
		\end{minipage}
	}
	\caption{Comparison of Results on Boat Image of SNR 20dB.}
	\label{fig:cofibdiffsnrimages}
\end{figure*}

It can be noticed from Table \ref{tab:cofibdiffsnr} and Fig. \ref{fig:cofibdiffsnr} that the CoFiB algorithm performed better than the results of some of the other state-of-the-art algorithms, except for severe noise level. Denoised images for SNR of 20dB are also presented in Fig. \ref{fig:cofibdiffsnrimages}. Emphasis should be placed on the flat regions and how edges are better preserved in CoFiB. It should also be observed in Fig. \ref{fig:cofibdiffsnrimages} that the visual performance of the CoFiB algorithm is closer to the original image compared to that of the other algorithms.

For the NLM result shown in Fig. \ref{fig:cofibdiffsnrimages}(c), the areas around the sand were smooth out, despite producing a PSNR of 30.73 and SSIM of 0.82. Also, K-SVD seems to produce a better result (Fig. \ref{fig:cofibdiffsnrimages}(d)) but its PSNR is lower than that of BM3D in Fig. \ref{fig:cofibdiffsnrimages}(e). The result of BM3D in Fig. \ref{fig:cofibdiffsnrimages}(e) was also smooth out compared to that of the proposed CoFiB algorithm. The CoFiB's result, Fig. \ref{fig:cofibdiffsnrimages}(f), can clearly be observed to have preserved all the details in the image, despite objective performance sometimes claiming otherwise. This detail preserving nature of our proposed algorithm would become more apparent when we compare the different algorithms on images of varying resolution in the next section.

\subsection{Performance Comparison of Different Algorithms on Different Image Resolutions}
\label{cofibresultsdiffresolution}

In this section, we compare the performance of the denoising algorithms for various image resolutions. We, again, selected the boat image for this purpose. We generated images of three different resolutions of $ 64\times64 $, $ 128\times128 $, and $ 256\times256 $ from the image. WGN was added to these images with SNR of 20dB and then denoised using the different algorithms. Table \ref{tab:cofibdiffresolution} shows the results of the various algorithms on the different boat image resolutions. It can be clearly observed that the proposed CoFiB algorithm performs objectively (in terms of PSNR and SSIM) better than all the other algorithms.

In addition, Table \ref{tab:cofibdiffresolutionresult} shows the denoised images of the various resolutions that were considered. From this table, it can clearly be seen that for lower resolution images of $ 64\times64 $ and $ 128\times128 $, the denoised output was completely distorted by all the other algorithms except our proposed CoFiB algorithm. CoFiB at the same time preserved the edges and flat regions of the image which the other algorithms tend to find a challenge at achieving. CoFiB gave clearer, sharper, and undistorted denoised images. This is a facr that we earlier highlighted that many algorithms find it challenging to denoise images of lower resolutions. Our proposed CoFiB algorithm can clearly be observed to have performed reasonably well for any given image resolution.

An ideal image denoising algorithm should work well for any given image. For this reason, in the next section, we present the comparison of our proposed algorithm with the other algorithms for different images.

\begin{table*}
	\centering
	\caption{Performance of Different Algorithms on Different Resolutions (Boat, SNR of 20dB)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t4.jpg}
	\end{tabular}
	\label{tab:cofibdiffresolution}
\end{table*}

\begin{table*}
	\centering
	\caption{Performance Comparison of Different Algorithms on Different Images (SNR = 20dB)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t2_2.jpg}
	\end{tabular}
	\label{tab:cofibdiffimages}
\end{table*}

\subsection{Performance Comparison of Different Algorithms on Different Images}
\label{cofibresultsdiffimages}

Here, we compare our proposed algorithm with the other algorithms on different images. The images used are some of the commonly used ones in literature to test denoising algorithms. In this paper, we have shown the results for 6 different images namely the boat, barbara, cameraman, mandrill, pepper and house images. For visual inspection of the performance, we have shown the results of the boat, cameraman, and the mandrill images.

Table \ref{tab:cofibdiffimages} shows the objective comparison of the performance of the different algorithms on the considered images. Similar to other experiments, WGN of fixed SNR of 20dB was added to the original images. The various algorithms are then used to denoise the images. From Table \ref{tab:cofibdiffimages}, it can easily be observed that our proposed CoFiB algorithm performed better than all the other algorithms both in terms of PSNR and SSIM except for the Cameraman image where the BM3D algorithm performed slightly better.

Furthermore, in Table \ref{tab:cofibdiffimagesresult}, we show three original and noisy images along with their denoised versions for all the selected algorithms for subjective performance comparison. It can be observed from this table that the proposed CoFiB result presented a more realistic denoised version of the original image. While results are close visually, differences can still be observed along the nose of the mandrill image.

\section{CONCLUSION}
\label{sec:con}

In this paper, a collaborative filtering-based method for image denoising was proposed where collaboration among the sparse representations of similar patches in the sparse domain was adopted to perform denoising. These sparse domain collaboration helped in improving the denoising capability of the algorithm. Unlike most of the other algorithms that tend to smooth out the denoised image resulting in loss of salient image details, our proposed CoFiB algorithm was able to preserve the details of the denoised image. This was achieved by the ability of the algorithm to strike a balance between noise removal and detail preservation. Additionally, most denoising algorithms tend to completely distort images at relatively lower resolutions. Our proposed algorithm was able to conveniently preserve the structure of the image for all the image resolutions considered. Extensive experimental results demonstrated that our proposed algorithm does not only performed objectively well (in terms of PSNR and SSIM) but also performed subjectively (i.e. visually) better. In future, we intend to translate our proposed algorithm into a deep learning denoising algorithm.

\section{ACKNOWLEDGEMENT}
\label{sec:ack}

We gratefully acknowledge the support of the KAUST Supercomputing Lab for providing us with the computing cluster (ibex) that were used to carry out this research.
\balance

\clearpage
\onecolumn
\begin{center}
	\begin{longtable}{c c c}
		\caption{Results of Different Image Resolutions.}
		\label{tab:cofibdiffresolutionresult} \\

		\hline
		\hline
		\multicolumn{1}{c}{\textbf{$ 64 \times 64 $}} & \multicolumn{1}{c}{\textbf{$ 128 \times 128 $}} & \multicolumn{1}{c}{\textbf{$ 256 \times 256 $}} \\ \hline
		\endfirsthead

		\multicolumn{3}{c}%
		{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
		\hline \multicolumn{1}{c}{\textbf{$ 64 \times 64 $}} &
		\multicolumn{1}{c}{\textbf{$ 128 \times 128 $}} &
		\multicolumn{1}{c}{\textbf{$ 256 \times 256 $}} \\
		\hline \\
		\endhead

		\multicolumn{3}{r}{{Continued on next page}} \\
		\endfoot

		\hline \hline
		\endlastfoot

		Original & Original & Original\\
		\includegraphics[width= 0.3\textwidth]{images/28a.png} &
		\includegraphics[width= 0.3\textwidth]{images/28b.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28c.png}
		\\
		\hline
		Noisy & Noisy & Noisy \\
		\includegraphics[width= 0.3\textwidth]{images/28d.png} &
		\includegraphics[width= 0.3\textwidth]{images/28e.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28f.png}
		\\
		\hline
		NLM & NLM & NLM\\
		\includegraphics[width= 0.3\textwidth]{images/28g.png} &
		\includegraphics[width= 0.3\textwidth]{images/28h.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28i.png}
		\\
		\hline
		K-SVD & K-SVD & K-SVD\\
		\includegraphics[width= 0.3\textwidth]{images/28j.png} &
		\includegraphics[width= 0.3\textwidth]{images/28k.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28l.png}
		\\
		\hline
		BM3D & BM3D & BM3D\\
		\includegraphics[width= 0.3\textwidth]{images/28m.png} &
		\includegraphics[width= 0.3\textwidth]{images/28n.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28o.png}
		\\
		\hline
		CoFiB & CoFiB & CoFiB\\
		\includegraphics[width= 0.3\textwidth]{images/28p.png} &
		\includegraphics[width= 0.3\textwidth]{images/28q.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28r.png}
		\\
	\end{longtable}
\end{center}

\newpage
\begin{center}
	\begin{longtable}{c c c}
		\caption{Results of Some Selected Images (SNR of 20dB)}
		\label{tab:cofibdiffimagesresult} \\

		\hline
		\hline
		\multicolumn{1}{c}{\textbf{Boat Image}} & \multicolumn{1}{c}{\textbf{Cameraman Image}} & \multicolumn{1}{c}{\textbf{Mandrill Image}} \\
		$ 512 \times 512 $ & $ 256 \times 256 $ & $ 256 \times 256 $ \\ \hline
		\endfirsthead

		\multicolumn{3}{c}%
		{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
		\hline \multicolumn{1}{c}{\textbf{Boat Image}} &
		\multicolumn{1}{c}{\textbf{Cameraman Image}} &
		\multicolumn{1}{c}{\textbf{Mandrill Image}} \\
		\hline
		\\
		\endhead

		\multicolumn{3}{r}{{Continued on next page}} \\
		\endfoot

		\hline \hline
		\endlastfoot

		Original & Original & Original \\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27a.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27b.png}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27c.jpg}
		\\
		\hline
		Noisy & Noisy & Noisy \\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27d.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27e.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27f.jpg}
		\\
		\hline
		NLM & NLM & NLM\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27g.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27h.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27i.jpg}
		\\
		\hline
		K-SVD & K-SVD & K-SVD\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27j.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27k.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27l.jpg}
		\\
		\hline
		BM3D & BM3D & BM3D\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27m.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27n.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27o.jpg}
		\\
		\hline
		CoFiB & CoFiB & CoFiB\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27p.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27q.png}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27r.png}
		\\
	\end{longtable}
\end{center}
\clearpage
\twocolumn

\bibliographystyle{IEEEtran}

\section{INTRODUCTION}

For its practical applications in fields such as astronomy, medical, autonomous driving, etc., image denoising as an area of research has been around for a very long time. In spite of the advancements in technology which has made it possible to obtain crystal clear images in recent times, image capturing equipment have very little control over the environment in which such images are taken. For instance, medical images of human bones captured by X-ray machines have the various tissues and organs of the body as potential noise. This implies that noise in images cannot be totally removed by such equipment making image denoising a very active research area till today.

Image denoising is, therefore, the art of recovering a clean image $ \mathbf{X} $ from a noisy image observation $ \mathbf{Y} $. $ \mathbf{Y} $ is a combination of the clean image $ \mathbf{X} $ and a noise $ \mathbf{W} $ typically modeled as in \eqref{eq:noisyImage}.

\begin{equation}
	\mathbf{Y = X + W.}
	\label{eq:noisyImage}
\end{equation}

The entries of $ \mathbf{W} $ are assumed to be random and independently drawn from the same probability distribution (i.i.d) such as the Gaussian distribution. As such, $ \mathbf{W} $ is typically modeled as an additive, white and Gaussian noise (AWGN) with zero mean and a known standard deviation $\sigma$.

Over the years, several image denoising algorithms have been proposed. Some of them exploit various mathematical analysis and similarities in pixels and patches to perform denoising. Such algorithms are commonly referred to as classical denoising algorithms, such as \cite{buades2005, Behzad2017, Aharon2006a, Dabov2007a, mairal2009}. The non-local means (NLM) algorithm \cite{buades2005} uses the similarities in the pixels of the corrupted image to perform denoising. Its performance, however, has been surpassed by the patch-based algorithms. This is because a patch carries more information about an image as compared to a single pixel.

To perform patch-based denoising, the overlapping patches are usually transformed from the spatial domain to a transform domain. This transform domain is such that the noisy components are easily separable from the original image's components. Denoising then becomes a simple question of identifying which components are due to the noise and removing them. One of such domain which has been exploited by most patch-based denoising algorithms is the sparse domain. In this domain, the patches which are dense in the spatial domain are represented by a few nonzero components. This makes denoising easier and more effective. For such transformation, an overcomplete dictionary with the help of sparse recovery algorithms such as the orthogonal matching pursuit (OMP) \cite{Aharon2006a} and the support agnostic bayesian matching pursuit (SABMP) \cite{Masood2013a} algorithms are used. The overcomplete dictionary is obtained using analytical techniques such as the discrete cosine transform (DCT) and discrete wavelet transform (DWT) \cite{qayyum2016}, wavelet \cite{daubechies1990}, curvelet \cite{starck2002}, and contourlet \cite{do2005} dictionaries. Another approach to obtaining the dictionary is to learn it from the image patches themselves. This is done with the help of dictionary learning algorithms such as the K-SVD \cite{Aharon2006a}, the method of optimal directions (MOD) \cite{engan1999}, principal component analysis (PCA) \cite{vidal2003}, etc. Algorithms such as the Block-Matching and 3D transformation (BM3D) \cite{Dabov2007a}, Collaborative Support-Agnostic Recovery (CSAR) \cite{Behzad2017}, and K-SVD denoising \cite{Elad2006a} algorithms are some of a long list of algorithms which utilizes patches transformation into the sparse domain to perform denoising.

Owing to its effectiveness, the BM3D was used as the benchmark denoising algorithm until recently. The use of neural networks (NNs) in denoising has seen the performance of the state-of-the-art classical techniques surpassed. The NN algorithms mostly utilize several layers of convolutional NNs (CNNs) to perform denosing. Example of such algorithms are the deep CNN (DnCNN) \cite{Zhang2017} and the fast and flexible feed-forward NN (FFDNet) \cite{Zhang2018}.

Despite the progressive improvement in the performance of the denoising algorithms over the years, many of them fail in their inability to denoise images of lower resolutions. Noisy low resolution images are sometimes the output of medical imaging systems. As such, ability of the image denoising algorithms to denoise such images is of practical importance. Some of the algorithms also tend to smooth out the denoised image leading to the loss of some of the salient details of the image.

In this paper, we present a collaborative filtering-based (CoFiB) denoising algorithm which performs collaboration among similar patches in the sparse domain to perform denoising. Our contribution is the ability of our proposed algorithm to perform well across images of any given resolution. Most importantly, images of low resolution. While denoising, our algorithm is also smart enough to preserve the details of the image. This means that our algorithm is able to strike a balance between complete image denoising leading to smoothened effect on the image and details preservation.

The rest of the paper is organized as follows. We provide a detail description of our proposed CoFiB algorithm in Section \ref{sec:cofib}. Section \ref{sec:exp} provides the details of the expensive experiments conducted and the results. Finally, we conclude in Section \ref{sec:con}.

\section{THE PROPOSED COLLABORATIVE FILTERING-BASED DENOISING ALGORITHM}
\label{sec:cofib}

In this section, we present the proposed CoFiB denoising algorithm. Our algorithm is a patch-based classical image denoising algorithm. The patches are transformed to the sparse domain where collaborative denoising is performed. In the sparse domain, the algorithm takes advantage of the similar patches' sparse representations to obtain a better denoising performance. The denoised sparse  representations are transformed back to the spatial domain. All denoised patches are then brought together to generate the denoised image. All these are performed in 5 steps as shown in Fig. \ref{fig:cofibFlowchart}. These steps are described as follows.

\begin{figure*}
	\centering
	\includegraphics[height=9cm, width=\textwidth]{images/cofib2.jpg}
	\caption{Flowchart of the Proposed CoFiB Algorithm.}
	\label{fig:cofibFlowchart}
\end{figure*}

\subsection{Step 1: Patches Extraction and Transformation}
\label{sec:patchExtract}

The first step in the pipeline of the proposed CoFiB algorithm is to extract a set of overlapping patches from the noisy image. For ease of analysis, the patches are of squared sizes, i.e. having the same number of row and column pixels. The patches are formed in such a way that the number of patches is equal to the number of pixels in the noisy image. This is done by padding the image with $ \frac{\sqrt{R}-1}{2} $ pixels in all directions. Given that each patch is of size $ n \times n $, $ R = n^{2} $ is the size of each vectorized image patch. $ R $ is chosen in a way that $ n $ is always odd. The reason for choosing $ R $ in this manner is to ensure that $ \frac{\sqrt{R}-1}{2} $ will always produce an integer. This implies that all pixels in the image would have a corresponding patch, known as the pixel patch. The central pixel of a pixel patch corresponds to the pixel it is representing.

Some parts of both natural and man-made images have similar patterns that may or may not differ in intensity levels. For example, Fig. \ref{fig:simpatchesexample}(A) and Fig. \ref{fig:simpatchesexample}(B) have different intensity levels, but the underlying pattern/structure is similar. Similarly, a purely white image and a purely black image can be thought of as having the same pattern but, while the intensity value of a white image is 0, that of a black is 255 (or 0 for white and 1 for black for an image range of [0, 1]). In our algorithm, we want to ensure that all patches that have a similar structure are grouped together in an intensity invariant manner. This was achieved by dividing a patch by the maximum absolute pixel value. This ensure that patches are grouped together not by their intensity but by their underlying structural pattern.

\begin{figure}[htb]
	\centering
	\begin{center}
		\begin{tabular}{ c c c }
			\includegraphics[width= 3cm, height= 3cm]{images/20.jpg}
			&
			$ \qquad $
			&
			\includegraphics[width= 3cm, height= 3cm]{images/21.jpg}
			\\
			(A) & & (B)
		\end{tabular}
	\end{center}
	\caption{Patches with Similar Pattern}
	\label{fig:simpatchesexample}
\end{figure}

\subsection{Step 2: Patches Clustering}
\label{sec:patchCluster}

Our proposed algorithm performs collaboration in the sparse domain. To transform the intensity invariant patches from the spatial to the sparse domain, an overcomplete dictionary is needed. However, because the patches are coming from different parts of the image, they will be very different. Therefore, they will be scattered in the sparse domain. For collaboration, we need patches close enough to each other in the sparse domain. Similar patches are expected to have similar representations in the sparse domain. We take advantage of this fact and grouped similar intensity invariant patches together. These groupings are used to enhance the sparse domain collaborative denoising. The popular k-means clustering algorithm \cite{jain2010}  was used to group the intensity invariant patches ointo $ k $ clusters.

\subsection{Step 3: Dictionary Learning}
\label{sec:dictLearn}

The dictionary for each cluster was learned with the help of the K-SVD dictionary learning algorithm \cite{Aharon2006}. For each cluster, we learned a separate dictionary. While a little computationally expensive, this helped to ensure that each cluster is suitably represented and most of the sparse estimation components better project the patches of each cluster unto their corresponding sparse domain with negligible error compared to using a single dictionary for all the patches.

\subsection{Step 4: Collaborative Denoising}
\label{sec:collabDen}

The sparse representation of each patch in each cluster is then obtained using the Support Agnostic Bayesian Matching Pursuit (SABMP) sparse recovery algorithm \cite{Masood2013a}. Although there are several sparse recovery algorithms, our choice of the SABMP algorithm hinged on the following facts about the algorithm which helped in the strategy we employed for denoising the image.

\begin{enumerate}
	\item It is invariant to the distribution of the image intensity values.

	\item It does not impose any strict conditions on the cluster dictionary,

	\item and it gives, as an output, the probability of each element of the sparse representation of a patch to be active.
\end{enumerate}

The sparse representation of patches in a cluster should have a high number of common locations of nonzero elements provided that the dictionary used is perfect, that is, all the dictionary atoms are uncorrelated. This would mean that the uncommon nonzero elements in the sparse representation are due mainly to noise, to a high probability. By simply removing the uncommon nonzero location components will leave us with clean portions of the representation. However, because the dictionary of a cluster was trained from similar data, a perfect dictionary is not obtainable. In addition, the dictionary is overcomplete, as such, several sparse domain representations could be mapped to a patch. This gives rise to sparse domain representations that are not unique. This ultimately means that several atoms in the dictionary can combine in different ways to yield the same point in the sparse domain. Locations of nonzero elements could differ due to genuine differences in the patches themselves. This makes simple removal of uncommon elements, not a good move. The SABMP algorithm comes to our rescue in this case as it can take into consideration the apriori probability of an element in a patch to be active in order to estimate the sparse vector. To take advantage of this fact, patches in the same cluster are allowed to collaborate to update their active probabilities in a weighted average approach to yield the denoised sparse representation's nonzero locations' probability. This step is as given mathematically in equations (\ref{eq:alpha}) through (\ref{eq:lambdaupdate}) below.

\begin{subequations}
	\begin{align}
	\alpha_{j,p,k}=\dfrac{1}{\Delta(y_{j,k}, y_{p,k})} \quad j \neq p.
	\label{eq:alpha}
	\\[20pt]
	Normalized \qquad \alpha_{j,p,k}=0.5 \times \frac{\alpha_{j,p,k}}{\sum_{j=1}^{t-1}\alpha_{j,p,k}} \quad j \neq p.
	\label{eq:alphanormalized}
	\\[20pt]
	\lambda_{p,k}^{updated}=\sum_{j=1}^{t}{\alpha_{j,p,k}\lambda_{j,k}}, \quad \alpha_{t,p,k} = 0.5, \quad \lambda_{t,k} = \lambda_{p,k}.
	\label{eq:lambdaupdate}
	\end{align}
\end{subequations}

For each patch $ p $ in each cluster $ k $, $ t-1 $ similar patches are selected (where $ t $ is the number of patches for collaboration). The selection is based on how close the patches in the cluster $ k $ are to the patch $ p $. This can be obtained using any suitable distance measure. A common distance metric is the Euclidean distance between the patch $ p $ (given as $ y_{p,k} $) and other patches in the same cluster $ k $ (given as $ y_{j,k} $, for $ j = 1 $ to $ n_{cluster} $, where $ n_{cluster} $ is the number of patches in the cluster). This distance is given as $ \Delta(y_{j,k}, y_{p,k}) $. The $ t-1 $ similar patches obtained are therefore given as $ y_{j,k} $, for $ j = 1 $ to $ t-1 $.

The closer a patch is to the patch $ p $, the smaller the value of the distance between them, but the more similar they become. As such, a similar patch's denoising contribution to the patch $ p $ is inversely proportional to the distance between them. Hence, the denoising collaborative contribution ($ \alpha_{j,p,k} $) of a similar patch to the patch $ p $ in the same cluster $ k $ is obtained using equation(\ref{eq:alpha}). Observe the inverse proportionality in the equation. The distance of patch $ p $ to itself is $ 0 $, as such it should have the highest collaborative contribution to itself. This implies that patch $ p $ would have a 50\% normalized contribution value to itself, i.e. $ \alpha_{t,p,k} = 0.5 $. All the other similar patches would have to scale accordingly to contribute the other normalized 50\% value. This 50\%  normalized collaborative contribution value of the other similar patch to patch $ p $ is given by equation (\ref{eq:alphanormalized}). Observe how their contribution scale using $ 0.5 \times $ inverse of their summation as a scaling factor. This is the contribution of each obtained similar patch towards denoising the patch $ p $.

As earlier described, from the outputs of the SABMP algorithm, the location probabilities $ \lambda_{j,k} $ of the elements of the sparse representation of each of these similar patches including the patch $ p $ can be recovered. To obtain the denoised active probabilities of patch $ p $, its $ \lambda_{p,k} $ needs to be updated/denoised. The updated $ \lambda_{p,k}^{updated} $ is obtained using the normalized denoising collaborative contributions ($ \alpha_{j,p,k} $) of each of the patches with their respective $ \lambda_{j,k} $. This is as given in equation (\ref{eq:lambdaupdate}).

In Fig. \ref{fig:cofibcollab}, we show how the similar patches collaborate to update/denoise the location probability of the sparse representation of the currently considered noisy patch in each cluster. This updated/denoised probability is then fed back into the SABMP algorithm to re-estimate the sparse domain representation of the denoised patch.

\begin{figure*}
	\centering
	\begin{center}
		\begin{tabular}{ c }
			\includegraphics[width=0.8\textwidth, height=0.4\textheight]{images/24.jpg}
			\\
			It should be noted here that y represents the active location \\
			probabilities of the sparse representation, $\lambda$, of a patch \\
			and w is the same as the collaboration weight $\alpha$ in (\ref{eq:alphanormalized}).
		\end{tabular}
	\end{center}
	\caption{Collaborative Denoising of Patches in the Sparse Domain.}
	\label{fig:cofibcollab}
\end{figure*}

\subsection{Step 5: Image Reconstruction}
\label{sec:imageRecon}

The denoised sparse representation of the patches are then transformed back to the spatial domain. This is obtained by simply multiplying the denoised sparse representation of each patch with the dictionary of the cluster to which the patch belongs. This yields the denoised patches.

In order to reconstruct the denoised image, the denoised patches are simply replaced back to their original locations in the noisy image. Noting that these patches overlap, they are then aggregated to obtain the denoised image.

\section{EXPERIMENTAL RESULTS}
\label{sec:exp}

We performed several experiments on the proposed CoFiB algorithm. We also compared it to some of the classical denoising algorithms. For performance comparison, the widely used peak signal to noise ratio (PSNR) and the structural similarity index (SSIM) performance metrics have been used. These methods are termed objective measures. Despite these measures being widely used, visual inspection (subjective measure) of the denoised image is needed to truly ascertain the performance of an image denoising algorithm \cite{Al-Najjar2012}. We will show that in almost all cases, our proposed CoFiB algorithm performs subjectively (visually) better than most of the state-of-the-art denoising algorithms such as the K-SVD \cite{Elad2006a}, NLM \cite{buades2005}, and BM3D \cite{Dabov2007a}. However, using the objective metrics, some of the algorithms performed better in few cases. As such, an image denoising algorithm should be evaluated both subjectively and objectively to truly ascertain the performance of such algorithm.

To cluster the intensity invariant patches, we found $ k = 5 $ to be a good number to have yielded the best performance for our algorithm. This number can be explained by the fact that a natural image patch mostly consists of flat, horizontal, vertical, left diagonal, and right diagonal regions. We now proceed to present and describe the result of our experiments in the following sections.

\subsection{Performance Comparison of Different Algorithms on Different SNR}
\label{sec:cofibresultsdiffsnr}

Here, we used a $ 512\times512 $ boat image which is one of the widely used images in literature for testing image denoising algorithms. White Gaussian Noise (WGN) was added to the original boat image. The noise was assumed to be of zero mean and variance that corresponds to the different values of SNR that were selected for our experiments. The SNR have been given in decibel (dB). These values ranges from $ -5dB $ to $ 35dB $ with an interval of $ 5dB $. We use the K-SVD, NLM, and BM3D to compare our algorithm. Each algorithm was then used to denoise the noisy image. Table \ref{tab:cofibdiffsnr} shows a comparison of the performance of the different algorithms on the different SNR values. These results are also presented graphically as shown in Fig. \ref{fig:cofibdiffsnr}.

\begin{table*}
	\centering
	\caption{Performance Comparison of Different Algorithms w.r.t the SNR (Boat $ 512 \times 512 $)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t1.jpg}
	\end{tabular}
	\label{tab:cofibdiffsnr}
\end{table*}

\begin{figure*}
	\centering
	\includegraphics[width=\textwidth]{images/25.png}
	\caption{Performance Comparison of Different Algorithms w.r.t the SNR (Boat $ 512 \times 512 $)}
	\label{fig:cofibdiffsnr}
\end{figure*}

\begin{figure*}[htb]
	\centering
	\fbox{
		\begin{minipage}{13 cm}
			\begin{center}
				\begin{tabular}{ c c }
					\includegraphics[width= 0.4\textwidth]{images/26a.png} & \includegraphics[width= 0.4\textwidth]{images/26b.png}
					\\
					(a) & (b)
					\\
					\\
					\includegraphics[width= 0.4\textwidth]{images/26c.png} & \includegraphics[width= 0.4\textwidth]{images/26d.png}
					\\
					(c) & (d)
					\\
					\\
					\includegraphics[width= 0.4\textwidth]{images/26e.png} & \includegraphics[width= 0.4\textwidth]{images/26f.png}
					\\
					(e) & (f)
					\\
					\\
					\multicolumn{2}{c}{(a) Clean Image, (b) Noisy Image, (c) NLM} \\
					\multicolumn{2}{c}{(d) K-SVD, (e) BM3D, (f) CoFiB}
				\end{tabular}
			\end{center}
		\end{minipage}
	}
	\caption{Comparison of Results on Boat Image of SNR 20dB.}
	\label{fig:cofibdiffsnrimages}
\end{figure*}

It can be noticed from Table \ref{tab:cofibdiffsnr} and Fig. \ref{fig:cofibdiffsnr} that the CoFiB algorithm performed better than the results of some of the other state-of-the-art algorithms, except for severe noise level. Denoised images for SNR of 20dB are also presented in Fig. \ref{fig:cofibdiffsnrimages}. Emphasis should be placed on the flat regions and how edges are better preserved in CoFiB. It should also be observed in Fig. \ref{fig:cofibdiffsnrimages} that the visual performance of the CoFiB algorithm is closer to the original image compared to that of the other algorithms.

For the NLM result shown in Fig. \ref{fig:cofibdiffsnrimages}(c), the areas around the sand were smooth out, despite producing a PSNR of 30.73 and SSIM of 0.82. Also, K-SVD seems to produce a better result (Fig. \ref{fig:cofibdiffsnrimages}(d)) but its PSNR is lower than that of BM3D in Fig. \ref{fig:cofibdiffsnrimages}(e). The result of BM3D in Fig. \ref{fig:cofibdiffsnrimages}(e) was also smooth out compared to that of the proposed CoFiB algorithm. The CoFiB's result, Fig. \ref{fig:cofibdiffsnrimages}(f), can clearly be observed to have preserved all the details in the image, despite objective performance sometimes claiming otherwise. This detail preserving nature of our proposed algorithm would become more apparent when we compare the different algorithms on images of varying resolution in the next section.

\subsection{Performance Comparison of Different Algorithms on Different Image Resolutions}
\label{cofibresultsdiffresolution}

In this section, we compare the performance of the denoising algorithms for various image resolutions. We, again, selected the boat image for this purpose. We generated images of three different resolutions of $ 64\times64 $, $ 128\times128 $, and $ 256\times256 $ from the image. WGN was added to these images with SNR of 20dB and then denoised using the different algorithms. Table \ref{tab:cofibdiffresolution} shows the results of the various algorithms on the different boat image resolutions. It can be clearly observed that the proposed CoFiB algorithm performs objectively (in terms of PSNR and SSIM) better than all the other algorithms.

In addition, Table \ref{tab:cofibdiffresolutionresult} shows the denoised images of the various resolutions that were considered. From this table, it can clearly be seen that for lower resolution images of $ 64\times64 $ and $ 128\times128 $, the denoised output was completely distorted by all the other algorithms except our proposed CoFiB algorithm. CoFiB at the same time preserved the edges and flat regions of the image which the other algorithms tend to find a challenge at achieving. CoFiB gave clearer, sharper, and undistorted denoised images. This is a facr that we earlier highlighted that many algorithms find it challenging to denoise images of lower resolutions. Our proposed CoFiB algorithm can clearly be observed to have performed reasonably well for any given image resolution.

An ideal image denoising algorithm should work well for any given image. For this reason, in the next section, we present the comparison of our proposed algorithm with the other algorithms for different images.

\begin{table*}
	\centering
	\caption{Performance of Different Algorithms on Different Resolutions (Boat, SNR of 20dB)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t4.jpg}
	\end{tabular}
	\label{tab:cofibdiffresolution}
\end{table*}

\begin{table*}
	\centering
	\caption{Performance Comparison of Different Algorithms on Different Images (SNR = 20dB)}
	\begin{tabular}{ c }
		\includegraphics[width= \textwidth]{images/t2_2.jpg}
	\end{tabular}
	\label{tab:cofibdiffimages}
\end{table*}

\subsection{Performance Comparison of Different Algorithms on Different Images}
\label{cofibresultsdiffimages}

Here, we compare our proposed algorithm with the other algorithms on different images. The images used are some of the commonly used ones in literature to test denoising algorithms. In this paper, we have shown the results for 6 different images namely the boat, barbara, cameraman, mandrill, pepper and house images. For visual inspection of the performance, we have shown the results of the boat, cameraman, and the mandrill images.

Table \ref{tab:cofibdiffimages} shows the objective comparison of the performance of the different algorithms on the considered images. Similar to other experiments, WGN of fixed SNR of 20dB was added to the original images. The various algorithms are then used to denoise the images. From Table \ref{tab:cofibdiffimages}, it can easily be observed that our proposed CoFiB algorithm performed better than all the other algorithms both in terms of PSNR and SSIM except for the Cameraman image where the BM3D algorithm performed slightly better.

Furthermore, in Table \ref{tab:cofibdiffimagesresult}, we show three original and noisy images along with their denoised versions for all the selected algorithms for subjective performance comparison. It can be observed from this table that the proposed CoFiB result presented a more realistic denoised version of the original image. While results are close visually, differences can still be observed along the nose of the mandrill image.

\section{CONCLUSION}
\label{sec:con}

In this paper, a collaborative filtering-based method for image denoising was proposed where collaboration among the sparse representations of similar patches in the sparse domain was adopted to perform denoising. These sparse domain collaboration helped in improving the denoising capability of the algorithm. Unlike most of the other algorithms that tend to smooth out the denoised image resulting in loss of salient image details, our proposed CoFiB algorithm was able to preserve the details of the denoised image. This was achieved by the ability of the algorithm to strike a balance between noise removal and detail preservation. Additionally, most denoising algorithms tend to completely distort images at relatively lower resolutions. Our proposed algorithm was able to conveniently preserve the structure of the image for all the image resolutions considered. Extensive experimental results demonstrated that our proposed algorithm does not only performed objectively well (in terms of PSNR and SSIM) but also performed subjectively (i.e. visually) better. In future, we intend to translate our proposed algorithm into a deep learning denoising algorithm.

\section{ACKNOWLEDGEMENT}
\label{sec:ack}

We gratefully acknowledge the support of the KAUST Supercomputing Lab for providing us with the computing cluster (ibex) that were used to carry out this research.
\balance

\clearpage
\onecolumn
\begin{center}
	\begin{longtable}{c c c}
		\caption{Results of Different Image Resolutions.}
		\label{tab:cofibdiffresolutionresult} \\

		\hline
		\hline
		\multicolumn{1}{c}{\textbf{$ 64 \times 64 $}} & \multicolumn{1}{c}{\textbf{$ 128 \times 128 $}} & \multicolumn{1}{c}{\textbf{$ 256 \times 256 $}} \\ \hline
		\endfirsthead

		\multicolumn{3}{c}%
		{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
		\hline \multicolumn{1}{c}{\textbf{$ 64 \times 64 $}} &
		\multicolumn{1}{c}{\textbf{$ 128 \times 128 $}} &
		\multicolumn{1}{c}{\textbf{$ 256 \times 256 $}} \\
		\hline \\
		\endhead

		\multicolumn{3}{r}{{Continued on next page}} \\
		\endfoot

		\hline \hline
		\endlastfoot

		Original & Original & Original\\
		\includegraphics[width= 0.3\textwidth]{images/28a.png} &
		\includegraphics[width= 0.3\textwidth]{images/28b.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28c.png}
		\\
		\hline
		Noisy & Noisy & Noisy \\
		\includegraphics[width= 0.3\textwidth]{images/28d.png} &
		\includegraphics[width= 0.3\textwidth]{images/28e.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28f.png}
		\\
		\hline
		NLM & NLM & NLM\\
		\includegraphics[width= 0.3\textwidth]{images/28g.png} &
		\includegraphics[width= 0.3\textwidth]{images/28h.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28i.png}
		\\
		\hline
		K-SVD & K-SVD & K-SVD\\
		\includegraphics[width= 0.3\textwidth]{images/28j.png} &
		\includegraphics[width= 0.3\textwidth]{images/28k.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28l.png}
		\\
		\hline
		BM3D & BM3D & BM3D\\
		\includegraphics[width= 0.3\textwidth]{images/28m.png} &
		\includegraphics[width= 0.3\textwidth]{images/28n.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28o.png}
		\\
		\hline
		CoFiB & CoFiB & CoFiB\\
		\includegraphics[width= 0.3\textwidth]{images/28p.png} &
		\includegraphics[width= 0.3\textwidth]{images/28q.png}
		&
		\includegraphics[width= 0.3\textwidth]{images/28r.png}
		\\
	\end{longtable}
\end{center}

\newpage
\begin{center}
	\begin{longtable}{c c c}
		\caption{Results of Some Selected Images (SNR of 20dB)}
		\label{tab:cofibdiffimagesresult} \\

		\hline
		\hline
		\multicolumn{1}{c}{\textbf{Boat Image}} & \multicolumn{1}{c}{\textbf{Cameraman Image}} & \multicolumn{1}{c}{\textbf{Mandrill Image}} \\
		$ 512 \times 512 $ & $ 256 \times 256 $ & $ 256 \times 256 $ \\ \hline
		\endfirsthead

		\multicolumn{3}{c}%
		{{\bfseries \tablename\ \thetable{} -- continued from previous page}} \\
		\hline \multicolumn{1}{c}{\textbf{Boat Image}} &
		\multicolumn{1}{c}{\textbf{Cameraman Image}} &
		\multicolumn{1}{c}{\textbf{Mandrill Image}} \\
		\hline
		\\
		\endhead

		\multicolumn{3}{r}{{Continued on next page}} \\
		\endfoot

		\hline \hline
		\endlastfoot

		Original & Original & Original \\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27a.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27b.png}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27c.jpg}
		\\
		\hline
		Noisy & Noisy & Noisy \\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27d.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27e.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27f.jpg}
		\\
		\hline
		NLM & NLM & NLM\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27g.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27h.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27i.jpg}
		\\
		\hline
		K-SVD & K-SVD & K-SVD\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27j.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27k.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27l.jpg}
		\\
		\hline
		BM3D & BM3D & BM3D\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27m.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27n.jpg}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27o.jpg}
		\\
		\hline
		CoFiB & CoFiB & CoFiB\\
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27p.jpg} &
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27q.png}
		&
		\includegraphics[width= 0.3\textwidth, height=0.3\textwidth]{images/27r.png}
		\\
	\end{longtable}
\end{center}
\clearpage
\twocolumn

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
"The emancipation of the working class will only be achieved by the working class itself." — Karl Marx
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French Workers Continue the Fight!
This article is from the January 30th issue of Lutte Ouvrière (Workers Struggle), the paper of the revolutionary workers group of that name active in France.
A fast-track bill in Parliament, a drafted law full of holes, a series of decrees … The French government wants its retirement reform adopted as quickly as possible. And the fact that it's a botch job, as the State Council has said, doesn't change a thing. The government's main preoccupation is to get past all the protests. But that's not going to work. Hundreds of thousands of workers have been mobilizing since December 5 and they're not ready to stop!
Even if the majority of SNCF union and the RATP union workers have had to return to work, they're still having strike days and holding interprofessional demonstrations. Workers are mobilized throughout the country—in ports, docks, schools, refineries, incinerators, and even in courts and at the Paris opera! They're all determined to keep up the pressure and be there on Wednesday, January 29. Friday, January 24 made it clear yet again that anger and determination are undiminished. Ministers and MPs keep trying to explain their reform but their unbearable hypocrisy, the lies they tell and the truths they leave out, just make people angrier.
"The reform won't change the legal retirement age," they say. But all the government's simulators use 65 as the equilibrium age. So it's out with the legal retirement age of 62 years, out with the pivotal age of 64 years and in with the equilibrium age of 65 years for all generations born after 1975!
The government is pushing retirement age to 65 years even though life expectancy in good health is around 62 years for men. And it's a lot lower for manual workers because working on a building site, on an assembly line or in shifts with staggered working hours is not a healthy program.
In the eyes of Labor Minister Pénicaud, working in painful positions, carrying heavy loads, exposure to chemical risks or mechanical vibrations are abstract criteria of "hardship at work." For workers, these criteria mean tendinitis, damaged spines, cancer and lifelong disabilities. When your health has been sacrificed on the altar of profits and capitalist greed, should you accept not having a retirement pension?
Looking us straight in the eye, every minister swears that this is a just and progressive reform. Well, they're lying, just like Cahuzac lied when he said that he could look everyone straight in the eye and say that he didn't have an overseas bank account!
They're claiming to be the defenders of the lowest paid. Really? Defenders of women, even though they have just trashed the unemployment rights of all women who have to juggle part-time jobs with periods of unemployment? Defenders of farmers, even though in March 2018 they vetoed reevaluating agricultural pensions up to a minimum of 85% of the minimum wage?
There will only be one winner in this reform: the ruling class. Just like all the major reforms made under Sarkozy, Hollande and Macron, this one is designed to divert billions from the pockets of workers to the bank accounts of the capitalists. This is because the bourgeoisie never has enough and, due to the pressure that the crisis is exerting, it's grabbing all it can from the workers.
The bourgeoisie wants to spend as little as possible on wages, jobs and working conditions. And it's even demanding that the money spent on retirement pensions, education and health be reduced to the strict minimum so that they can have more.
But something has changed in France: the government can no longer mask its intentions. Its policies that are exclusively in favor of a handful of predators are being challenged more and more. The inequality and injustice that lead to the major capitalists cashing in more per hour than millions earn in a month are fueling the discontent. And there is a growing awareness of the devastation caused by exploitation and the parasitism of the capitalist minority.
All this is increasing the exasperation of the working class. It made itself heard in the yellow-vest movement and in the opposition to the retirement pension reform which caused one of the biggest transport strikes ever. And this is only the beginning.
The same anger is also smoldering in private companies and it will eventually erupt because the workers have no choice but to hit back.
Trump's "Blue Collar Boom" Is Really a Great Depression
Pages 2-3
Hedge Funds and Drug Cartels Go Avocados
Only Ones Not Paying Are Boeing Big Cheeses
MOVE Members Freed
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Virginia: Which Politicians Help Working People?
Neither Party Has a Solution
Palestinians Shot Entering Israel to Work
Protests Catch Fire in Frozen Canada
Guadeloupe: Educators Strike
Syria: Endless War
Open Worries about Coronavirus in China—Hidden Dangers of the U.S. Health System
Michigan Prisoners Silenced with Solitary
U.S. Homeless Children
Cops Seize Cars
McKinley Park Asphalt Plant
Chicago: Ten Years to Build Highway Ramps?
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to get "sum" of true bools In this scenario I have 3x boolean vars (already set to either true or false)
The goal here is to identify if more than one of the set of bool vars are set to true
Right now I have the following written, which does work:
    boolval := 0
    if *firstbool {
        boolval++
    }
    if *secondbool {
        boolval++
    }
    if *thirdbool {
        boolval++
    }
    if boolval > 1 {
        // More than 1 of the bool vars are true
    }

I always flag my logic if I am writing successive if{} statements, so I figured I would ask you geniuses how you would accomplish this.

A:
identify if more than one of the set of bool vars are set to true

For example, write a set function:
package main

import "fmt"

func nTrue(b ...bool) int {
    n := 0
    for _, v := range b {
        if v {
            n++
        }
    }
    return n
}

func main() {
    v1, v2, v3 := true, false, true

    fmt.Println(nTrue(v1, v2, v3))

    if nTrue(v1, v2, v3) > 1 {
        // . .   .
    }
    // Or
    if n := nTrue(v1, v2, v3); n > 1 {
        // . .   .
    }
    // Or
    n := nTrue(v1, v2, v3)
    if n > 1 {
        // . .   .
    }
}

Playground: https://play.golang.org/p/g3WCN6BgGly
Output:
2

For example, range over the set,
package main

import "fmt"

func main() {
    v1, v2, v3 := true, false, true

    boolval := 0
    for _, v := range []bool{v1, v2, v3} {
        if v {
            boolval++
        }
    }
    if boolval > 1 {
        // . .   .
    }

    fmt.Println(boolval > 1, boolval)
}

Playground: https://play.golang.org/p/R6UGb8YYEFw
Output:
true 2

A: Depending on how you're getting the values of firstbool et. al, there might be a more idiomatic approach you should take here. Consider that channels and goroutines can handle doing the accounting in the background. If your booleans are the result of some heavier operations, it might make sense to do something like:
package main

import "fmt"

func firstFunc(ch chan bool) {
    // assume this works, true!
    ch <- true
}

func secondFunc(ch chan bool) {
    // assume this fails, false!
    ch <- false
}

func thirdFunc(ch chan bool) {
    // assume this works, true!
    ch <- true
}

func listener(ch chan bool, numCallers int) <-chan bool {
    outch := make(chan bool)
    THRESHOLD := 2
    go func(outch chan<- bool) {
        count := 0
        trues := 0

        for count < numCallers && trues < THRESHOLD {
            val := <-ch
            if val {
                trues++
            }
            count++
        }
        outch <- trues >= THRESHOLD
    }(outch)
    return outch
}

func main() {
    ch := make(chan bool)
    resultch := listener(ch, 3)

    go firstFunc(ch)
    go secondFunc(ch)
    go thirdFunc(ch)

    if <-resultch {
        fmt.Println("Two or more processes succeeded")
    }
}

However this is way over-engineered for a simple need, so consider this only if this pattern fits the larger design of your application.

redpajama_set_name: RedPajamaStackExchange


I noticed there are no pictures for it. Does who ever working on VR want to get some pictures of mine?
Might be after the new year but I will get some good ones. I will PM you the best one I got for now.
up once I figure out how.


text:
Q: Characteristic functions of infinite dimensional random elements I am trying to understand if it is possible to prove the convergence in distribution of a sequence of infinite dimensional random elements using characteristic functions.

Suppose that $\{X_n:n\ge1\}$ is a sequence of random elements with values in $L_2[0,1]$ (the Hilbert space of square integrable real functions defined on $[0,1]$). Is it possible to define a characteristic function of $X_n$ for each $n\ge1$ and to prove the convergence in distribution of $X_n$ by showing that the sequence of characteristic functions converges pointwise?

There is a somewhat related example on Wikipedia of a characteristic function of a stochastic process. If $X(s)$ is a stochastic process, then
$$
\varphi_X(t)
=\operatorname E\exp\biggl\{i\int_\mathbb R t(s)X(s)ds\biggr\}
$$
for all functions $t(s)$ such that the integral  $\int_\mathbb R t(s)X(s)\mathrm ds$ converges for almost all realizations of $X$. If the characteristic functions of a sequence of stochastic processes converge pointwise, does it mean that the finite-dimensional distributions of stochastic processes converge?
Any help is much appreciated!

A: It is possible to define characteristic functions for random variables with values in Hilbert/Banach spaces. If $\mathcal H$ is a separable Hilbert space and $X$ a random variable with values in $\mathcal H$, then we can define for $f\in \mathcal H$ the quantity
$$\varphi_X(f)=\mathbb E\left[\exp\left(i\langle f,X\rangle\right)\right].$$
The terminology "characteristic function" still makes sense since two random variables $X$ and $Y$ such that $\varphi_X(f)=\varphi_Y(f)$ for each $f\in\mathcal H$ have the same distribution.
However, this is not so useful to prove convergence in distribution. For example, if we take $(\xi_j)_{j\geqslant 1}$ a sequence of independent identically distributed standard Gaussian random variables and define $X_n:=\sum_{j=1}^n\xi_je_j$ (where $\mathcal H=\ell^2$ and $e_j=(0,\dots,0,1,0,\dots)$ and $1$ is at the $j$th position). We can show that for each $f\in \mathcal H$, the sequence $\left(\varphi_{X_n}(f)\right)_{n\geqslant 1}$ converges to $\exp\left(-\lVert f\rVert^2\right)$, hence we have the pointwise convergence. However, the sequence $\left(X_n\right)_{n\geqslant 1}$ is not tight in $\mathcal H$.


International philanthropy expert Russell Willis Taylor, President and CEO of National Arts Strategies (2001-14) and Executive Director of the English National Opera (1997-2001) will present two keynote addresses (development and governance). Antonia Ruffell, CEO of Australian Philanthropic Services, will also be featured. See the program schedule.
Russell Willis Taylor, President and CEO of US-based National Arts Strategies from January 2001 to January 2015, has extensive senior experience in strategic business planning, financial analysis and planning, and all areas of operational management. Educated in England and America*, she served as director of development for the Chicago Museum of Contemporary Art before returning to England in 1984 at the invitation of the English National Opera (ENO) to establish the Company's first fund-raising department. During this time, she also lectured extensively at graduate programs of arts and business management throughout Britain. From 1997 to 2001, she rejoined the ENO as executive director.
Russell has held a wide range of managerial and Board posts in the commercial and nonprofit sectors including the advertising agency DMBB; head of corporate relations at Stoll Moss; director of The Arts Foundation; special advisor to the Heritage Board, Singapore; chief executive of Year of Opera and Music Theatre (1997); judge for Creative Britons and lecturer on business issues and arts administration. She received the Garrett Award for an outstanding contribution to the arts in Britain, the only American to be recognised in this way, and has served on the boards of A&B (Arts and Business), Cambridge Arts Theatre, Arts Research Digest and the Society of London Theatre. She was part of the founding team with Diana, Princess of Wales, for the National Aids Trust in the UK.
Russell currently serves on the boards of the Salzburg Global Seminar, the Arts Management program at American University, the British Council's Arts & Creative Economy Advisory Group, Fractured Atlas and is a Fellow of the Royal Society of Arts. In 2013, Russell was honored with the International Citation of Merit by the International Society for the Performing Arts, presented in recognition of her lifetime achievement and her distinguished service to the performing arts. She has written a number of articles on arts management and policy issues in arts and culture and is currently taking a year away from work for research and relaxation.
* BA degree with honors, Randolph Macon Womens College; Undergraduate certificates in art history and literature from Kings College Cambridge University and New College Oxford respectively.
Antonia Ruffell is CEO of Australian Philanthropic Services, a not-for-profit organisation that establishes and administers private and public ancillary funds and provides grantmaking advice. She works with individuals and families, and the advisers who support them, to help them set up and run foundations, and be more effective in their giving.
Antonia's career spans the corporate and not-for-profit sectors. Prior to joining Australian Philanthropic Services, she held senior roles with Perpetual Trustees, the Royal Agricultural Society Foundation, Mission Australia, ING Australia, and The Prince's Trust in the UK. Antonia also sits on the Board of The Social Outfit, a charity that helps refugees with education and employment.


Medford is the fourth-oldest English settlement in the United States. Located along the Mystic River in Middlesex County, the city was founded back in 1630 and incorporated in 1892. It was originally settled by brick makers and ship builders, and its residents took an active role in the Revolutionary War. Today, the community is the home of Tufts University and is well-known as the birthplace of "Jingle Bells," a Christmas song written by a local resident, James Pierpont. If you need assistance with a legal matter in Medford or the nearby area, the real estate and workers' compensation attorneys at Pulgini & Norton can provide you with dedicated representation. With over four decades of combined experience, we have the knowledge needed to protect your interests.
You can qualify for workers' compensation benefits if you were hurt or became sick because of your job. The Massachusetts Workers' Compensation Law requires that all employers carry insurance to cover medical costs and lost wages for injured employees. The law does not consider fault, which means that an individual may be eligible for benefits irrespective of whether the worker, the employer, or a colleague was responsible for causing the harm.
Medical costs that are both reasonable and necessary to treat the injury or illness.
Weekly wage compensation, up to the state's average weekly wage cap. Wage benefits depend on the worker's average wage in the year prior to the injury, and whether the accident or illness results in a permanent disability, total temporary disability, or partial disability.
Payment for the loss of specific bodily functions or disfigurement.
Payment to a survivor or dependent if the worker died as a result of the job-related injury or illness.
To receive workers' compensation benefits, your employer's insurer must approve your claim. The insurer has the right to have you evaluated by a physician of its choice before you seek treatment from your preferred provider. The insurer will review your medical exams, proposed treatment, and the circumstances surrounding your condition to determine whether you qualify for benefits.
An insurer may deny a claim if it finds that an injury is not work-related or that the medical treatment is unnecessary. It may also approve a claim but then ask the Department of Industrial Accidents (DIA) for permission to lower your benefits. You can appeal both a denial of your claim and an insurer's request to modify your benefits. The DIA requires that you present medical evidence to support the gravity of your injury and need for the treatment you seek. We can review and procure the evidence you need to build a strong case and represent you throughout the DIA appeals process.
Medford homeowners and prospective buyers often come across issues that have a legal impact on their ability to sell, purchase, finance, or refinance a home. For instance, if you are buying or selling property, you will need to negotiate the terms of both the offer and purchase and sale agreement. If you are a buyer, you probably will want to conduct a home inspection, title search, and secure financing. If you are a seller, on the other hand, you will want to handle matters such as clearing title, disclosing any presence of lead, and drafting the deed.
If you own a home, you may encounter issues with your mortgage if you want to refinance or if you drop your homeowners' insurance. For instance, the cost to refinance may be higher if the terms of your current mortgage include a pre-payment penalty. Although you are not required by law to have homeowners' insurance, your lender may require that you carry it. For example, dropping the insurance may accelerate the payment, in full, of the balance on your mortgage.
The experienced work injury and property transactions lawyers at Pulgini & Norton represent Medford residents in real estate matters and workers' compensation claims. We offer counseling and guidance in the purchase and sale of homes, financing, and other related matters. Throughout our years of practice, we have also helped injured workers pursue the medical, wage, and other benefits they need after a job-related accident. For a consultation, call our office at 781-843-2200, or complete our contact form today.


KOHAR announces the release of the much-awaited Lebanon 2015 Live Concert recordings. The "All Time Armenian Favorites 5" DVD and Blu-ray formats are now available.
The interactive DVD and Blu-ray formats come with trilingual subtitles: Armenian, transliteration and English translation. You can also enjoy the bilingual 320-page songbook and a behind-the-scenes documentary.
This latest release comes to complement your All Time Armenian Favorites library.


text:
The multiplayer beta of Call of Duty: Black Ops – Cold War will not hit PC until October 15th, but will be released later this week – October 8th to be exact – on PlayStation 4. With the launch just a few In just a few days, Activision has announced details of what's coming, including how to get in and what to expect when the digital bullets fly.
Call of Duty : Cold War Beta for PC – Beta Trailer
https://youtu.be/8o-kI11cAx4
The data has already been confirmed: This week's round of testing is only available for PS4, while PC and Xbox players who pre-ordered the game will be able to start on October 15th. All others will be able to join in on October 17, when the beta is fully open. To participate in the promotion, a Battle.net account is required and can be accessed through the "Partner Games" list in the Battle.net client. Beta preloading on the PC will begin in October at 8:00 (PT) / 11:00 (ET) 13.
Activision said the beta will include "a mix of maps, modes and playlists" and a progress system that will allow unlocking weapons, equipment, slots to create a class and scorestreaks. There was also half warning, half teasing, that there will be frequent updates during the beta periods, and also "a few surprises". Players who reach level 10 in beta will receive a blueprint for the Mutual Animosity SMG when the entire game goes live.
Cold War Beta for PC – System Requirements
Operating system: Windows 7 64-bit (SP1) or Windows 10 64-bit (1803 or higher)
CPU: Intel Core i5 2500k or AMD equivalent
Video: Nvidia GeForce GTX 670 2 GB / GTX 1650 4 GB or AMD Radeon HD 7950.
Hard Disk: 45 GB HD storage space
Audio: DirectX compatible, requires DirectX 12 compatible system
Operating system: Windows 10 64 Bit (current service pack)
video: Nvidia GeForce GTX 970 4 GB / GTX 1660 Super 6 GB or AMD Radeon R9 390 / AMD RX 580
RAM: 16 GB RAM
There's not much to see yet, but Activision said more details on game modes and loadouts will be published in the multiplayer beta on callofduty.com. Last week it also revealed our first look at the all-new Cold War Zombies mode, and previously a new approach to character customization based on "psychological profiles".
Download Gmail Go APK For Android
Zayn Malik samples Mohammed Rafi's 'Chaudhvin Ka Chand' in his song 'Tightrope' from his latest album 'Nobody Is Listening' : Bollywood News
Is the Garden Rescue Star married? Why is she famous?


Computer programming jobs have previously struggled to attract women, but new technologies are providing substantial opportunities for a growing number of young girls seeking tech jobs, NZTech government relations director Andrea Hancox says.
Hundreds of Auckland schoolgirls will later this month have a hands on experience how some of New Zealand's most hi-tech companies such as Air New Zealand, Microsoft and Vodafone operate.
Job vacancies and recruitment within IT continues to rise, especially in retail and manufacturing. We are also seeing a steady demand for candidates with cloud experience. Meanwhile, IT salaries remain stable, some roles reporting a decline in remuneration whilst increases are reported in niche areas such as Network Designers, DevOps experts and Virtualisation Specialists.
More than 100 secondary school girls have jumped at the opportunity to spend a day shadowing IT experts in Wellington at some of the Capital city's well-known companies. For one day the girls will get to experience all aspects of ICT from programming to design and to see first-hand technology of all kinds inside a … Continue reading "Girls Go For ICT Experience in the Capital"
Young New Zealand female students have the opportunity to be mentored by high profile technology companies at Shadow Tech Day in Christchurch, Wellington and Auckland over the next three months. More than 230 students will take part – an increase from about 45 students in previous years.
The 10 job skills you will need by 2020!
In less than five years, over a third of skills that are considered important in the workplace today will have changed. We stand on the brink of a technological revolution known as the Fourth Industrial Revolution that will fundamentally transform the way we live and work. Also known as Industry 4.0, the revolution will bring … Continue reading "The 10 job skills you will need by 2020!"
Get your geek on: What skills and experience are IT employers looking for?
The information technology (IT) industry in New Zealand is booming and desperate for fresh talent. The number of jobs in the digital/IT field is expected to grow by up to 50% in the next 10 years. Tech is no longer the domain of the introverted maths genius. These days there are plenty of creative jobs … Continue reading "Get your geek on: What skills and experience are IT employers looking for?"
2016 is already well on its way and as promised we have some of the results from the recent Job Seeker Survey that was done before Christmas.


text:
Q: Hosting ApiControllers on windows service I have a REST API project with controllers and services behind them, and it works well on IIS.
Now I'm trying to find a way to host it on a non-IIS computer as a windows service. No luck this far, TopShelf seemed to be what I wanted but I have multiple controllers/services to host, and it seems to be able to handle only one at a time.
Is there a way to have a windows service project run another (referenced) project and host it? It's way easier to debug without having the service uninstall-install every time, so ideally the projects would be kept seperate.

A: You can host WCF service in windows service. To do that create a windows service project and refer the the WCF project.Also add System.ServiceModel and System.ServiceModel.Description dlls in in your windows service project then write a function like this
private ServiceHost host;

private void HostWcfService()
{
        //Create a URI to serve as the base address
         Uri httpUrl = newUri("http://localhost:1256/MyService/Service");

         //Create ServiceHost
         host = newServiceHost(typeof(MyService.IService), httpUrl);

         //Add a service endpoint
         host.AddServiceEndpoint(typeof(MyService.IService), newWSHttpBinding(), "");

         //Enable metadata exchange
         ServiceMetadataBehavior smb = new ServiceMetadataBehavior();
         smb.HttpGetEnabled = true;
         host.Description.Behaviors.Add(smb);

         //Start the Service
         host.Open();

}

and call The HostWcfService() in the On OnStart method like this
        protected override void OnStart(string[] args)
        {
            if (host != null)
            {
                host.Close();
            }
               HostWcfService();

        }

and Update OnStop  Method like this
protected override void OnStop()
{
    if (host != null)
    {
        host.Close();
        host = null;
    }
}


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St. George and the Dragon, 1506 by Raphael (Raffaello Sanzio Da Urbino) (1483-1520, Italy) | Museum Art Reproductions | ArtsDot.com
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St. George and the Dragon is a small cabinet painting by the Italian High Renaissance artist Raphael, painted between 1504 and 1506, and now in the National Gallery of Art in Washington, D.C. The saint wears the blue garter of the English Order of the Garter, reflecting the award of this decoration in 1504 to Raphael's patron Guidobaldo da Montefeltro, Duke of Urbino, by King Henry VII of England. The first word of the order's motto, "HONI" can be made out. The painting was presumably commissioned by the Duke, either to present to the English emissary who brought the regalia to Urbino, Sir Gilbert Talbot, or to Henry himself—recent scholarship suggests the latter. The honour paid to a minor Italian ruler reflected Henry's appreciation of the cultural prestige of Renaissance Italy as much as any diplomatic purpose.The traditional subject, Saint George and the Dragon, combining chivalry and Christianity, is appropriate for the occasion; like his father, Guidobaldo was a condottiero, or proprietor of a band of mercenary soldiers. In the early stages of his career Raphael painted a number of tiny cabinet paintings, including another St George in the Louvre, and the Vision of a Knight in the National Gallery in London.
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St. George and the Dragon
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St. George and the Dragon is a small cabinet painting by the Italian High Renaissance artist Raphael, painted between 1504 and 1506, and now in the Na
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Inventure Academy works to inculcate its Core Values and the school spirit in children, enabling them to achieve, and lead with wisdom, foresight and caring. We hope that these values and this spirit stays with them through their lives, manifesting in their every action, adding a confidence born of striving to always do the right thing in the right way. The Spirit of Inventure is invoked by individual and collaborative pursuits of charitable, literary, artistic and intellectual natures.
Every child is assigned to one of the four Houses for the duration of his or her school life. Houses also act as teams for social outreach initiatives and interactions with economically less advantaged and differently abled children from our neighborhood and the city. A discipline code is operative, through which students get positive or negative House points on behaviour related to uniform, hygiene, cleanliness, submission of homework, or for being a buddy to a new student. The honour system supports the philosophy of the school and the values of society as a whole. House points contribute towards the House Championship at the end of the year.
Belonging to Houses gives the children an opportunity to imbibe leadership qualities, sensitivity to others and develop team skills.
The Student Council is a student-led governing body comprising students elected by the entire school. It is led by the School Captain, Senior and Junior Vice-Captains, the Sports Captain and four House Captains and Beyond Academics Captain, who are elected office bearers. Being a part of this structure imposes discipline as well as gives a taste of democratic functioning and political responsibility to children, becoming a fertile training ground for the student voice to be heard.
New entrants into each class are welcomed, supported and assisted by "Buddies" specifically assigned by teachers. This system eases the transition for new students from their previous school to Inventure, and also inculcates sensitivity, empathy and helpfulness in our students.
Students are given opportunities to develop and present their skills and talents through active participation in weekly Assemblies. An additional assembly may be called for a specific event, lecture, or demonstration. Students also assemble for the celebration of national days, festivals, cultural presentations and recognition of achievement and awards, producing a feeling of togetherness, national and school pride among the student body. Attendance is compulsory for all.
Tournament of Minds
A very popular inter-house competition, which includes Music, Dance, Speech and Drama, Art and Craft. Students participate in quizzes, recitation, elocution, declamation, storytelling, word games, and debates, improving performances and setting challenging goals. The Tournament of Minds competitions are held during school year, and individual points gained by winners contribute to the overall House points and the self-esteem of participants.
Further, intra- and inter-school competitions hosted by the school and events such as Multiple Intelligences Xplored (M.I.X), Annual Day, Sports Day and Awards Day give children a sense of belonging and working in a team, secure in their individual abilities to contribute to it, and in an atmosphere of healthy competition with their peers.


The third and final session of the free community planned giving series put on by the Allegan General Hospital Foundation, Allegan County Community Foundation, Allegan County United Way, Perrigo Company Charitable Foundation and Wings of Hope Hospice is scheduled for Tuesday, November 15 from 11:30 a.m. until 1:30 p.m.
This session titled "Putting It All Together for Good" will discuss how to build a lasting legacy in your community. The featured guest speakers will be Attorney Danielle Streed from the firm of Redmond, Streed and Yokom and Steven Tibbitts of Tibbitts Financial Consulting. "Putting it All Together for Good" will be held in the Michigan Room at Allegan General Hospital (555 Linn Street, Allegan). Lunch will be provided.
Individuals who would like to attend are being asked to please R.S.V.P. in advance by contacting Brent Larson, Executive Director of the Allegan General Hospital Foundation, at (269) 686-4234 or via e-mail at brlar@aghosp.com.