\section{Comparison with several classic and recent methods
	\label{section:comparisons} }


In order to evaluate the real capacity of NL-Bayes, a fair and precise comparison
with other state-of-the-art methods had to be done. The other considered methods
were. For all of them, reliable software can be found. The four first methods have
public and commented implementations at Image Processing On Line \cite{bm3d_ipol},
\cite{ipol.2011.ys-dct}, \cite{KSVDipol}, \cite{ipol.2011.bcm_nlm}. Thus, the
experiments below can be verified on line for five of the compared algorithms.
\begin{itemize}
		\vspace{-3mm}\item BM3D;
	\vspace{-3mm}\item DCT denoising;
	\vspace{-3mm}\item NL-means;
	\vspace{-3mm}\item K-SVD; \vspace{-3mm}\item BM3D-SAPCA (only for grey
level images);
	\vspace{-3mm}\item BLS-GSM.
\end{itemize}
The following study has been led on the following noise-free color images
($\sigma_{real} << 1$). All algorithms have been processed on the same noisy images
obtained from noiseless images (saved in real values and not sampled on [0, 255])):
\ifpdf
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=0.25]{images/Alley.png} &  &
		\includegraphics[scale=0.25]{images/Computer.png}\\
		Alley &  & Computer\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=0.25]{images/dice.png} & &
		\includegraphics[scale=0.25]{images/flowers.png}\\
		Dice & & Flowers\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=0.25]{images/Girl.png} & &
		\includegraphics[scale=0.25]{images/traffic.png}\\
		Girl & & Traffic\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=0.25]{images/Trees.png} & &
		\includegraphics[scale=0.25]{images/Valldemossa.png}\\
		Trees & & Valldemossa\\
	\end{tabular}
\end{center}
\fi

\subsection{Comparative table}


\subsubsection{Color Images}


\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 2$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 45.39} & 44.95 & 44.54 & 41.22 & 42.79 & -\\
   \hline
   Computer & {\bf 45.88} & 45.21 & 44.55 & 44.80 & 44.07 & 44.69\\
   \hline
   Dice & {\bf 49.07} & 48.85 & 48.48 & 48.03 & 48.52 & 48.59\\
   \hline
   Flowers & {\bf 47.77} & 47.30 & 47.14 & 47.32 & 46.41 & 47.12\\
   \hline
   Girl & {\bf 47.64} & 47.38 & 46.94 & 47.24 & 46.95 & 47.14\\
   \hline
   Traffic & {\bf 45.26} & 44.58 & 44.22 & 43.52 & 43.58 & 44.15\\
   \hline
   Trees & {\bf 43.51} & 43.08 & 42.89 & 37.46 & 42.23 & -\\
   \hline
   Valldemossa & {\bf 45.14} & 44.71 & 44.42 & 38.96 & 43.34 & 44.41\\
   \hline
   \hline
   Mean & {\bf 46.21} & 45.76 & 45.40 & 43.57 & 44.74 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 5$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 39.35} & 38.96 & 38.29 & 38.42 & 37.28 & -\\
   \hline
   Computer & {\bf 40.73} & 39.98 & 38.98 & 39.58 & 38.99 & 39.30\\
   \hline
   Dice & {\bf 45.95} & 45.81 & 45.06 & 45.29 & 45.11 & 45.21\\
   \hline
   Flowers & {\bf 43.39} & 43.00 & 42.58 & 43.11 & 42.10 & 42.76\\
   \hline
   Girl & {\bf 44.23} & 44.05 & 43.36 & 43.57 & 43.48 & 43.70\\
   \hline
   Traffic & {\bf 39.59} & 38.67 & 38.09 & 38.78 & 37.61 & 38.10\\
   \hline
   Trees & {\bf 36.67} & 36.09 & 35.66 & 35.57 & 34.69 & -\\
   \hline
   Valldemossa & {\bf 38.77} & 38.35 & 37.89 & 37.91 & 35.97 & 38.02\\
   \hline
   \hline
   Mean & {\bf 41.08} & 40.61 & 39.99 & 40.28 & 39.40 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 10$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 35.12} & 34.83 & 33.92 & 34.31 & 33.59 & -\\
   \hline
   Computer & {\bf 36.98} & 36.26 & 35.00 & 35.76 & 35.54 & 35.47\\
   \hline
   Dice & {\bf 43.32} & 43.05 & 41.84 & 41.72 & 41.97 & 42.21\\
   \hline
   Flowers & {\bf 39.82} & 39.48 & 38.61 & 39.35 & 38.46 & 39.10\\
   \hline
   Girl & {\bf 41.72} & 41.45 & 40.35 & 40.31 & 40.48 & 41.14\\
   \hline
   Traffic & {\bf 35.47} & 34.56 & 33.76 & 34.72 & 33.98 & 33.92\\
   \hline
   Trees & {\bf 31.89} & 31.25 & 30.68 & 31.05 & 29.56 & -\\
   \hline
   Valldemossa & {\bf 34.14} & 33.79 & 33.17 & 33.31 & 32.13 & 33.41\\
   \hline
   \hline
   Mean & {\bf 37.31} & 36.83 & 35.92 & 36.32 & 35.71 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 20$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 31.37} & 31.21 & 30.13 & 30.55 & 29.94 & -\\
   \hline
   Computer & {\bf 33.27} & 32.69 & 31.08 & 31.94 & 31.61 & 31.89\\
   \hline
   Dice & {\bf 40.55} & 39.94 & 37.94 & 37.30 & 38.20 & 39.00\\
   \hline
   Flowers & {\bf 36.33} & 35.86 & 34.41 & 35.28 & 34.38 & 35.34\\
   \hline
   Girl & {\bf 39.10} & 38.72 & 36.74 & 36.42 & 36.92 & 38.49\\
   \hline
   Traffic & {\bf 31.49} & 30.83 & 29.79 & 30.70 & 30.11 & 30.14\\
   \hline
   Trees & {\bf 27.51} & 26.91 & 26.15 & 26.87 & 26.35 & -\\
   \hline
   Valldemossa & {\bf 29.86} & 29.59 & 28.69 & 29.08 & 28.44 & 29.17\\
   \hline
   \hline
   Mean & {\bf 33.68} & 33.22 & 31.87 & 32.27 & 31.99 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 30$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 29.37} & 29.34 & 28.20 & 28.60 & 27.82 & -\\
   \hline
   Computer & {\bf 31.12} & 30.70 & 28.75 & 29.88 & 29.24 & 29.90\\
   \hline
   Dice & {\bf 38.80} & 37.95 & 34.69 & 36.44 & 36.85 & 37.05\\
   \hline
   Flowers & {\bf 34.22} & 33.77 & 31.98 & 33.58 & 32.26 & 33.19\\
   \hline
   Girl & {\bf 37.32} & 36.98 & 33.73 & 35.40 & 35.55 & 36.91\\
   \hline
   Traffic & {\bf 29.35} & 28.85 & 27.64 & 28.59 & 27.71 & 28.20\\
   \hline
   Trees & {\bf 25.22} & 24.67 & 23.81 & 24.53 & 23.78 & -\\
   \hline
   Valldemossa & {\bf 27.53} & 27.30 & 26.34 & 26.79 & 25.87 & 26.97\\
   \hline
   \hline
   Mean & {\bf 31.62} & 31.19 & 29.39 & 30.48 & 29.89 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 40$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 28.05 & {\bf 28.07} & 26.89 & 27.27 & 26.47 & -\\
   \hline
   Computer & {\bf 29.50} & 29.12 & 26.92 & 28.22 & 27.52 & 28.52\\
   \hline
   Dice & {\bf 37.26} & 36.35 & 32.42 & 34.60 & 35.16 & 35.50\\
   \hline
   Flowers & {\bf 32.63} & 32.11 & 30.23 & 31.92 & 30.51 & 31.68\\
   \hline
   Girl & {\bf 36.05} & 35.70 & 31.26 & 33.81 & 34.13 & 35.61\\
   \hline
   Traffic & {\bf 27.86} & 27.46 & 26.14 & 27.14 & 26.20 & 26.93\\
   \hline
   Trees & {\bf 23.67} & 23.18 & 22.30 & 23.06 & 22.40 & -\\
   \hline
   Valldemossa & {\bf 26.02} & 25.80 & 24.88 & 25.32 & 24.44 & 25.50\\
   \hline
   \hline
   Mean & {\bf 30.13} & 29.72 & 27.63 & 28.92 & 28.35 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 60$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 26.37 & {\bf 26.40} & 25.49 & 25.66 & 24.66 & - \\
   \hline
   Computer & {\bf 27.52} & 27.04 & 25.57 & 26.28 & 25.26 & 26.68 \\
   \hline
   Dice & {\bf 35.71} & 34.09 & 32.08 & 32.94 & 33.26 & 33.61\\
   \hline
   Flowers & {\bf 30.80} & 29.94 & 28.81 & 30.04 & 28.27 & 29.70\\
   \hline
   Girl & {\bf 34.71} & 33.71 & 31.94 & 32.53 & 32.61 & 34.08\\
   \hline
   Traffic & {\bf 25.98} & 25.75 & 24.74 & 25.36 & 24.27 & 25.26\\
   \hline
   Trees & {\bf 21.78} & 21.19 & 20.68 & 21.27 & 20.45 & -\\
   \hline
   Valldemossa & {\bf 24.12} & 23.87 & 23.05 & 23.40 & 22.40 & 23.69\\
   \hline
   \hline
   Mean & {\bf 28.37} & 27.75 & 26.55 & 27.19 & 26.40 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 80$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 25.22 & {\bf 25.39} & 24.47 & 24.50 & 23.39 & -\\
   \hline
   Computer & {\bf 25.97} & 25.88 & 24.16 & 24.91 & 23.68 & 25.41\\
   \hline
   Dice & {\bf 34.57} & 32.37 & 30.04 & 31.04 & 31.38 & 32.16\\
   \hline
   Flowers & {\bf 29.41} & 28.53 & 27.27 & 28.47 & 26.58 & 28.29\\
   \hline
   Girl & {\bf 33.64} & 32.20 & 29.83 & 30.83 & 31.10 & 32.93\\
   \hline
   Traffic & {\bf 24.70} & 24.68 & 23.53 & 24.13 & 22.94 & 24.19\\
   \hline
   Trees & {\bf 20.58} & 20.43 & 19.70 & 20.17 & 19.30 & -\\
   \hline
   Valldemossa & 22.83 & {\bf 22.87} & 21.96 & 22.20 & 21.09 & 22.54\\
   \hline
   \hline
   Mean & {\bf 27.11} & 26.54 & 25.12 & 25.78 & 24.93 & -\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 100$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & {\bf 24.42} & 24.37 & 23.63 & 23.59 & 22.52 & -\\
   \hline
   Computer & {\bf 24.87} & 24.40 & 23.08 & 23.85 & 22.55 & 24.48\\
   \hline
   Dice & {\bf 33.42} & 30.42 & 28.53 & 29.52 & 29.91 & 31.06\\
   \hline
   Flowers & {\bf 28.20} & 27.04 & 26.09 & 27.29 & 25.43 & 27.23\\
   \hline
   Girl & {\bf 32.73} & 30.56 & 28.13 & 29.43 & 30.07 & 32.02\\
   \hline
   Traffic & {\bf 23.84} & 23.53 & 22.69 & 23.20 & 21.96 & 23.41\\
   \hline
   Trees & {\bf 19.82} & 19.48 & 19.03 & 19.38 & 18.56 & -\\
   \hline
   Valldemossa & {\bf 21.91} & 21.91 & 21.18 & 21.24 & 20.08 & 21.72\\
   \hline
   \hline
   Mean & {\bf 26.15} & 25.21 & 24.05 & 24.69 & 23.88 & -\\
   \hline
   \end{tabular}
\end{center}
According to the results, one can observe that the comparative performance of the
methods is quite independent of the noise value. The summary table below shows a
mean of the scores over all test images.
\begin{center}
	\begin{tabular}{||c||c|c|c|c|c|c||}
		\hline
		Methods & {\scriptsize NL-Bayes} & {\scriptsize BM3D} & {\scriptsize DCT denoising} & {\scriptsize K-SVD} & {\scriptsize NL-means} & {\scriptsize BLS-GSM}\\
		\hline
		Mean & {\bf 33.52} & 32.98 & 31.77 & 32.17 & 31.70 & - \\
		\hline
	\end{tabular}
\end{center}


\subsubsection{Grey level images}
For grey level images, the size of patches needs to be increased for both steps in
NL-Bayes. Then $k_{\hard}$ and $k_{\wien}$ are set to $5$ for small values of
noise, instead of $3$. The other parameters are identical to the color case.

\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 2$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means &
   BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & {\bf 43.49} & 43.27 & 42.90 & 43.08 & 42.58 & 43.01 & {\bf 43.49}\\
   \hline
   Computer & {\bf 45.02} & 44.65 & 44.24 & 44.45 & 44.01 & 44.06 & 44.98\\
   \hline
   Dice & 49.43 & 49.73 & 49.28 & 48.79 & 48.47 & 49.31 & {\bf 49.91}\\
   \hline
   Flowers & 47.79 & 48.15 & 48.18 & 47.69 & 46.04 & 48.10 & {\bf 48.38}\\
   \hline
   Girl & 47.81 & 47.95 & 47.44 & 47.19 & 46.95 & 47.59 & {\bf 48.11}\\
   \hline
   Traffic & {\bf 44.28} & 44.00 & 43.78 & 43.81 & 43.53 & 43.48 & {\bf 44.28}\\
   \hline
   Trees & {\bf 42.72} & 42.51 & 42.43 & 42.44 & 42.27 & 42.42 & 42.65\\
   \hline
   Valldemossa & {\bf 43.86} & 43.66 & 43.54 & 43.49 & 43.29 & 43.31 & 43.83\\
   \hline
   \hline
   Mean & 45.55 & 45.49 & 45.22 & 45.12 & 44.64 & 45.16 & {\bf 45.70}\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 5$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & 37.10 & 36.83 & 36.15 & 36.57 & 36.32 & 36.47 & {\bf 37.12}\\
   \hline
   Computer & 39.23 & 38.73 & 37.89 & 38.24 & 38.01 & 38.05 & {\bf 39.25}\\
   \hline
   Dice & 45.37 & 45.69 & 45.00 & 42.75 & 43.39 & 45.03 & {\bf 46.13}\\
   \hline
   Flowers & 42.27 & 42.83 & 42.56 & 41.64 & 40.33 & 42.76 & {\bf 43.03}\\
   \hline
   Girl & 43.46 & 43.54 & 42.87 & 41.63 & 42.07 & 43.23 & {\bf 43.78}\\
   \hline
   Traffic & 37.87 & 37.48 & 37.03 & 37.18 & 36.73 & 37.00 & {\bf 37.89}\\
   \hline
   Trees & {\bf 35.36} & 35.06 & 34.77 & 34.99 & 34.46 & 35.00 & 35.29\\
   \hline
   Valldemossa & {\bf 36.85} & 36.59 & 36.23 & 36.32 & 35.74 & 36.26 & 36.80\\
   \hline
   \hline
   Mean & 39.69 & 39.59 & 39.06 & 38.66 & 38.38 & 39.22 & {\bf 39.91}\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 10$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & 32.93 & 32.72 & 31.90 & 32.35 & 32.23 & 32.28 & {\bf 33.00}\\
   \hline
   Computer & 35.11 & 34.60 & 33.31 & 33.80 & 33.73 & 33.87 & {\bf 35.19}\\
   \hline
   Dice & 42.45 & 42.45 & 41.47 & 37.58 & 38.99 & 41.65 & {\bf 43.07}\\
   \hline
   Flowers & 38.42 & 38.85 & 38.25 & 36.74 & 35.95 & 38.68 & {\bf 38.97}\\
   \hline
   Girl & 40.65 & 40.61 & 39.64 & 37.09 & 38.01 & 40.21 & {\bf 40.87}\\
   \hline
   Traffic & 33.30 & 32.95 & 32.16 & 32.50 & 32.40 & 32.42 & {\bf 33.36}\\
   \hline
   Trees & {\bf 30.22} & 29.84 & 29.21 & 29.76 & 29.54 & 29.77 & 30.14\\
   \hline
   Valldemossa & {\bf 31.98} & 31.69 & 30.97 & 31.29 & 31.18 & 31.30 & 31.90\\
   \hline
   \hline
   Mean & 35.63 & 35.47 & 34.61 & 33.89 & 34.00 & 35.02 & {\bf 35.81}\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 20$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & 29.42 & 29.34 & 28.47 & 28.47 & 28.71 & 28.84 & {\bf 29.57}\\
   \hline
   Computer & 31.09 & 30.86 & 29.21 & 29.40 & 30.03 & 30.10 & {\bf 31.36}\\
   \hline
   Dice & {\bf 39.58} & 39.02 & 37.60 & 31.97 & 36.88 & 38.17 & {\bf 39.58}\\
   \hline
   Flowers & 34.65 & 34.91 & 33.83 & 31.62 & 32.84 & 34.71 & {\bf 34.94}\\
   \hline
   Girl & 37.80 & 37.72 & 36.19 & 31.88 & 35.74 & 37.36 & {\bf 37.99}\\
   \hline
   Traffic & 29.27 & 29.07 & 28.17 & 28.30 & 28.51 & 28.53 & {\bf 29.36}\\
   \hline
   Trees & {\bf 25.78} & 25.34 & 24.50 & 25.30 & 25.23 & 25.22 & 25.55\\
   \hline
   Valldemossa & {\bf 27.68} & 27.41 & 26.56 & 26.90 & 27.07 & 27.03 & 27.59\\
   \hline
   \hline
   Mean & 31.91 & 31.71 & 30.57 & 29.23 & 30.63 & 31.24 & {\bf 31.99}\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 30$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & 27.60 & 27.63 & 26.78 & 26.97 & 26.73 & 27.15 & {\bf 27.86}\\
   \hline
   Computer & 28.88 & 28.74 & 27.04 & 27.79 & 27.68 & 28.00 & {\bf 29.10}\\
   \hline
   Dice & {\bf 37.84} & 37.03 & 35.22 & 32.70 & 34.23 & 36.25 & 37.27\\
   \hline
   Flowers & 32.59 & 32.64 & 31.30 & 31.07 & 30.40 & 32.57 & {\bf 32.72}\\
   \hline
   Girl & {\bf 36.10} & 35.80 & 34.11 & 32.31 & 33.39 & 35.69 & 36.03\\
   \hline
   Traffic & 27.20 & 27.13 & 26.27 & 26.69 & 26.38 & 26.56 & {\bf 27.33}\\
   \hline
   Trees & {\bf 23.55} & 23.21 & 22.45 & 23.15 & 23.08 & 23.08 & 23.27\\
   \hline
   Valldemossa & {\bf 25.54} & 25.32 & 24.53 & 24.98 & 24.90 & 25.02 & 25.49\\
   \hline
   \hline
   Mean & {\bf 29.91} & 29.69 & 28.46 & 28.21 & 28.35 & 29.29 & 29.88\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{7}{c||}{$\sigma = 40$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM & BM3D-SAPCA\\
    \hline
   Alley & 26.36 & 26.55 & 25.69 & 25.65 & 25.51 & 26.02 & {\bf 26.65}\\
   \hline
   Computer & 27.38 & 27.27 & 25.60 & 26.20 & 26.10 & 26.64 & {\bf 27.70}\\
   \hline
   Dice & {\bf 36.26} & 35.41 & 33.26 & 30.42 & 33.70 & 34.74 & 35.46\\
   \hline
   Flowers & 31.01 & 31.03 & 29.59 & 29.17 & 28.83 & 30.94 & {\bf 31.10}\\
   \hline
   Girl & {\bf 34.81} & 34.47 & 32.38 & 30.19 & 32.67 & 34.40 & 34.28\\
   \hline
   Traffic & 25.82 & 25.86 & 25.06 & 25.31 & 24.89 & 25.34 & {\bf 26.03}\\
   \hline
   Trees & {\bf 22.20} & 21.99 & 21.30 & 21.89 & 21.66 & 21.84 & 21.99\\
   \hline
   Valldemossa & {\bf 24.21} & 24.08 & 23.35 & 23.67 & 23.53 & 23.72 & 24.14\\
   \hline
   \hline
   Mean & {\bf 28.51} & 28.33 & 27.03 & 26.56 & 27.11 & 27.95 & 28.42\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 60$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 24.91 & {\bf 25.07} & 24.32 & 24.35 & 23.73 & 24.63\\
   \hline
   Computer & {\bf 25.46} & 25.39 & 23.88 & 24.59 & 23.88 & 24.88\\
   \hline
   Dice & {\bf 34.47} & 33.16 & 31.97 & 29.73 & 31.88 & 32.67\\
   \hline
   Flowers & 28.92 & 28.83 & 28.14 & 27.83 & 26.71 & {\bf 29.03}\\
   \hline
   Girl & {\bf 33.30} & 32.45 & 31.72 & 29.66 & 31.21 & 32.79\\
   \hline
   Traffic & 24.19 & {\bf 24.31} & 23.46 & 23.82 & 23.11 & 23.83\\
   \hline
   Trees & {\bf 20.64} & 20.57 & 19.95 & 20.42 & 19.96 & 20.43\\
   \hline
   Valldemossa & {\bf 22.60} & 22.52 & 21.72 & 22.12 & 21.61 & 22.15\\
   \hline
   \hline
   Mean & {\bf 26.81} & 26.54 & 25.64 & 25.32 & 25.26 & 26.30\\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 80$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 23.74 & {\bf 24.02} & 23.39 & 23.14 & 22.66 & \\
   \hline
   Computer & 23.98 & {\bf 24.08} & 22.71 & 23.21 & 22.47 & 23.69\\
   \hline
   Dice & {\bf 33.09} & 31.21 & 30.16 & 27.35 & 29.92 & 31.23\\
   \hline
   Flowers & 27.40 & 27.31 & 26.62 & 26.30 & 25.26 & {\bf 27.63}\\
   \hline
   Girl & {\bf 32.18} & 30.99 & 30.27 & 27.70 & 30.02 & 31.64\\
   \hline
   Traffic & 23.08 & {\bf 23.25} & 22.53 & 22.66 & 21.92 & 22.84\\
   \hline
   Trees & 19.75 & {\bf 19.80} & 19.26 & 19.64 & 19.10 & 19.63\\
   \hline
   Valldemossa & 21.47 & {\bf 21.49} & 20.84 & 21.01 & 20.45 & 21.17\\
   \hline
   \hline
   Mean & {\bf 25.59} & 25.27 & 24.47 & 23.88 & 23.98 & \\
   \hline
   \end{tabular}
\end{center}
\begin{center}
   \begin{tabular}{||c||c||c||c||c||c||c||}
   \hline
    & \multicolumn{6}{c||}{$\sigma = 100$}\\
   \hline\hline
    & NL-Bayes & BM3D & DCT denoising & K-SVD & NL-Means & BLS-GSM\\
    \hline
   Alley & 22.97 & {\bf 23.06} & 22.71 & 22.14 & 21.84 & \\
   \hline
   Computer & {\bf 22.89} & 22.86 & 21.88 & 22.17 & 21.35 & 22.81\\
   \hline
   Dice & {\bf 31.57} & 29.14 & 28.84 & 25.53 & 28.44 & 30.16\\
   \hline
   Flowers & 26.31 & 26.09 & 25.59 & 25.02 & 24.17 & {\bf 26.56}\\
   \hline
   Girl & {\bf 31.07} & 29.24 & 28.95 & 26.16 & 28.73 & 30.71\\
   \hline
   Traffic & 22.25 & {\bf 22.36} & 21.79 & 21.70 & 20.99 & 22.11\\
   \hline
   Trees & 19.00 & {\bf 19.17} & 18.79 & 19.07 & 18.51 & 19.11\\
   \hline
   Valldemossa & 20.61 & {\bf 20.66} & 20.18 & 20.17 & 19.51 & 20.46\\
   \hline
   \hline
   Mean & {\bf 24.58} & 24.07 & 23.59 & 22.75 & 22.94 & \\
   \hline
   \end{tabular}
\end{center}
\end{changemargin}
According to the above results, one can observe again that the comparative
performance of the methods is quite independent of the noise value.  Here is a
summary table  showing a mean of the PSNR scores over all test images :
\begin{center}
	\begin{tabular}{||c||c|c|c|c|c|c|c||}
		\hline
		Methods & {\scriptsize NL-Bayes} & {\scriptsize BM3D} & {\scriptsize DCT denoising} & {\scriptsize K-SVD} & {\scriptsize NL-means} & {\scriptsize BLS-GSM} & {\scriptsize BM3D-SAPCA}\\
		\hline
		Mean & {\bf 32.02} & 31.79 & 30.96 & 30.40 & 30.59 &  ???& ???\\
		\hline
	\end{tabular}
\end{center}
One can conclude that NL-Bayes is not the best denoising algorithm for grey level
images. Indeed, for slight values of the noise, BM3D-SAPCA performs better in terms
of PSNR, and also visually. Having shape adaptive patches definitely improves  the
performance. Edges appear more ``straight'' with BM3D-SAPCA than with NL-Bayes, as
one can see on the next image. Nevertheless, for medium and large values of the
noise, due to its inherent artifacts, BM3D-SAPCA has slightly less convincing
results. Using the homogeneous area criterion in grey level images  with three
times less samples than in color images) blurs the result of NL-Bayes. This
criterion should therefore be refined for grey level images.  \ifpdf
\begin{changemargin}{-1.75cm}{0cm}
\begin{center}
	\begin{tabular}{c c}
		\includegraphics[scale=1.]{images/Comparisons/Traffic_green_part.png} &
		\includegraphics[scale=1.]{images/Comparisons/Traffic_green_noisy_30_part.png}\\
		Original image & Noisy image, $\sigma = 30$\\
		\\
	\end{tabular}
	\begin{tabular}{c c}
		\includegraphics[scale=1.]{images/Comparisons/Traffic_green_nl-bayes_30_part.png} &
		\includegraphics[scale=1.]{images/Comparisons/Traffic_green_sapca_30_part.png}\\
		NL-Bayes & BM3D-SAPCA\\
	\end{tabular}
\end{center}
\end{changemargin}
\fi The better visual results with BM3D-SAPCA seem to be due to the adaptive shapes
 and to the larger patch size ($8\times 8$ against $5\times
5$). Nonetheless, NL-Bayes is simpler than BM3D-SAPCA. Indeed BM3D-SAPCA employs
image patches (neighbourhoods) which can have data-adaptive shape. The PCA bases
are obtained by eigenvalue decomposition of empirical second-moment matrices that
are estimated from groups of similar adaptive-shape neighbourhoods. The anisotropic
shape-adaptive patches are obtained using the 8-directional LPA-ICI techniques. The
principal steps of this algorithm are:
\begin{enumerate}
	\vspace{-3mm}\item Obtain adaptive-shape neighbourhood centered at the
current pixel using the 8-directional LPA-ICI;
	\vspace{-3mm}\item Find patches similar to the reference patch using
block-matching, and extract an adaptive-shape neighbourhood from each of these
matched blocks using the shape obtained in Step 1;
	\vspace{-3mm}\item Determine the transform to be applied on the
adaptive-shape neighbourhoods (depending on the number of similar blocks),
which can be eigenvectors of a second-moment matrix, or a shape-adaptive DCT;
	\vspace{-3mm}\item Form a 3-D array by stacking together the adaptive-shape
neighbourhoods with highest similarity to the reference one;
	\vspace{-3mm}\item Apply the transform obtained in Step 3 on each of the
grouped adaptive-shape neighbourhoods. Subsequently, apply a 1-D orthogonal
transform (e.g., Haar wavelet decomposition) along the third dimension of the
3-D group;
	\vspace{-3mm}\item Perform shrinkage (hard-thresholding or empirical Wiener
filtering) on the 3-D spectrum;
	\vspace{-3mm}\item Invert the 3-D transform of Step 5 to obtain estimates
for all of the grouped adaptive-shape neighbourhoods;
	\vspace{-3mm}\item Return the obtained estimates to their original
locations using weighted averaging in case of overlapping.
\end{enumerate}
In terms of complexity, NL-Bayes is faster than BM3D-SAPCA. Comparative experiments
on a normal PC show that NL-Bayes takes between 4 and 32 seconds (depending on the
value of sigma) and BM3D-SAPCA needs about 15 minutes for the same image.


\subsection{Images}


In addition to the PSNR/RMSE results, it is really interesting to compare visually
all methods, especially to remark (and regret) the inherent and characteristic
artifacts of each one.


\subsubsection{$\sigma = 10$}


We shall start with a relatively low noise, in order to show each method to its
best :
\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_noisy_10.png}\\
		Original noise-free image & & Noisy image, $\sigma = 10$\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-bayes_10.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_bm3d_10.png}\\
		NL-Bayes & & BM3D\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_dct-denoising_10.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_ksvd_10.png}\\
		DCT denoising & & K-SVD\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-means_10.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_BLS-GSM_10.png}\\
		NL-means & & BLS-GSM\\
	\end{tabular}
\end{center}
\end{changemargin}


\subsubsection{$\sigma = 20$}


\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_noisy_20.png}\\
		Original noise-free image & & Noisy image, $\sigma = 20$\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-bayes_20.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_bm3d_20.png}\\
		NL-Bayes & & BM3D\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_dct-denoising_20.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_ksvd_20.png}\\
		DCT denoising & & K-SVD\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-means_20.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_BLS-GSM_20.png}\\
		NL-means & & BLS-GSM\\
	\end{tabular}
\end{center}
\end{changemargin}


\subsubsection{$\sigma = 30$}


\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_noisy_30.png}\\
		Original noise-free image & & Noisy image, $\sigma = 30$\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-bayes_30.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_bm3d_30.png}\\
		NL-Bayes & & BM3D\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_dct-denoising_30.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_ksvd_30.png}\\
		DCT denoising & & K-SVD\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-means_30.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_BLS-GSM_30.png}\\
		NL-means & & BLS-GSM\\
	\end{tabular}
\end{center}
\end{changemargin}


\subsubsection{$\sigma = 40$}


\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_noisy_40.png}\\
		Original noise-free image & & Noisy image, $\sigma = 40$\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-bayes_40.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_bm3d_40.png}\\
		NL-Bayes & & BM3D\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_dct-denoising_40.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_ksvd_40.png}\\
		DCT denoising & & K-SVD\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-means_40.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_BLS-GSM_40.png}\\
		NL-means & & BLS-GSM\\
	\end{tabular}
\end{center}
\end{changemargin}


The noise standard deviation limit beyond which artifacts appear with all methods
is $\sigma= 40$. This is the limit between moderate noise where we get visually
acceptable results, and high noise where no method gives visually acceptable
results.


\subsubsection{$\sigma = 80$}


Above $\sigma = 40$, inherent and characteristic artifacts become obvious. These
limitations are important to explore. They opens up the question of denoising high
noise.


\begin{changemargin}{-1.5cm}{0cm}
\begin{center}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_noisy_80.png}\\
		Original noise-free image & & Noisy image, $\sigma = 80$\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-bayes_80.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_bm3d_80.png}\\
		NL-Bayes & & BM3D\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_dct-denoising_80.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_ksvd_80.png}\\
		DCT denoising & & K-SVD\\
	\end{tabular}
	\begin{tabular}{c c c}
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_nl-means_80.png} & &
		\includegraphics[scale=2]{images/Comparisons/Traffic_part_BLS-GSM_80.png}\\
		NL-means & & BLS-GSM\\
	\end{tabular}
\end{center}
\end{changemargin}
