\section{Bibliography analysis
	\label{section:analysis_biblio} }
	

\subsection{Comments about ``Image denoising with patch based PCA: local versus global''
 \cite{Deledalle2011}}


The use of the PCA (\emph{Principal Component Analysis}) has been treated in
details in \cite{Deledalle2011}. This article proposes to use a PCA to learn an
orthonormal basis adapted locally or globally to the image. The interesting part in
this article is the confrontation of global versus local PCA. Indeed, a first
simple approach is to learn the PCA over the whole image, and then to denoise
patches by applying a Hard Thresholding on the coefficient of the patch on the new
learned orthonormal basis. This works quite well, but for patches which present a
rare structure, the denoising is not optimal, because this structure is
under-represented in the whole image. Thus, to adapt the method to every kind of
structure, even the rarest, the solution is to learn the PCA more locally, on small
windows around the current patch. Nevertheless, working on a small windows, there
is a risk of not finding enough patches to learn an unbiased PCA.


Another interesting point of this article is that it shows the first eigenvectors
and the last ones of each PCA. One can easily see that the first eigenvectors
describe the orientation of the gradient of the patches, whereas the last ones
present only noise. This illustrates experimentally how canceling the coefficients
of the last eigenvectors (those with the lowest corresponding eigenvalues) allows
to remove the noise in the image without affecting structure or details. Despite
this interesting study, one can finger-point some defaults of this article. In
fact, its
 PSNR results are worse than BM3D and the algorithm is slower. These defaults
 are  partly due to the  fact that this paper does not use all tricks that improve
 the denoising performance:
  only gray-level
images are treated, no second oracular step is applied. The size of the patches can
probably be reduced. Nevertheless, the algorithm applies a uniform aggregation. A
puzzling result  of their study is that applying an hard thresholding gives better
results than a Wiener filtering. But a second step would demonstrate that with the
right  oracle given by the first step, the Wiener filter is in fact better, being
optimal in theory. Regrettably only  small noise values $\sigma \in [5, 20]$ have
been tested.
\subsection{Analysis and comments about TSID}


This part considers the original method described in \cite{tsid}. As this algorithm
is really close to NL-PCA, it will be compared to it stepwise, and for this purpose
the same notation is adopted.


\subsubsection{Implementation}


The algorithm is  divided in two separate steps. Contrarily to NL-PCA, these steps
are identical, except for the used value of $\sigma$ which is upgraded after the
first step. Thus, only one step will be described. This algorithm was originally
developed for gray level images. In the following we therefore examine the
application to gray level images. The image is scanned pixel per pixel. Let us
denote by $\tilde P$ the current reference patch which size is $k_{\hard}\times
k_{\hard}$ (with $k_{\hard} = 5$) and $x_r$ the current central pixel of $\tilde
P$. The loop on the image is done on $x_r$. Like for NL-PCA and BM3D, each step is
divided into 3 parts:
\begin{enumerate}
\item \underline{Grouping:} Similar patches $\tilde Q$ are found by
    block-matching with the same distance as the one used for NL-Bayes. A patch
    is considered similar if its distance to $\tilde P$ is below a fixed
    threshold, depending on $\sigma$. Moreover, to stabilize the process of the
    PCA on the obtained 3D-block, a minimum number (denoted by $N_P$) of
    similar patches is requested. In this case the $N_P$ best patches obtained
    during block-matching are used.
\item \underline{Collaborative Filtering:} When the set of similar patches has
    been obtained, a matrix is built, containing all similar patches seen as
    vectors. Then, as for NL-PCA, the columns of the matrix are first centered
    around their common center of mass. After that, a singular value
    decomposition (SVD) of the centered matrix is obtained. Then, a LMMSE
    (Wiener filter) is applied to the new coefficients, and the patch is
    finally reconstructed with these new coefficients on the PCA.
\item \underline{Getting the estimate:} The principal difference with NL-PCA,
    is that only the estimate of the central pixel of the reference patch $x_r$
    is kept, which clearly decreases the PSNR, no aggregation step being
    possible.
\end{enumerate}


\subsubsection{Extending to color images}


In BM3D  the extension to color images is done by using a color space different
from the $RGB$ one (typically $Y_0U_0V_0$); the grouping part is done only on the
$Y_0$ channel and the rest of the algorithm is performed independently on the three
channels. For TSID instead this extension is done in the simplest way; the whole
algorithm is applied independently on the three channels $R$, $G$ and $B$. This is
another point that could have been very easily improved, but was not envisaged by
the authors.


\subsubsection{Comparisons between TSID and NL-PCA}


Here are some comparative results in terms of PSNR for TSID and  NL-PCA with color
images and grey level images.


\begin{center}
\begin{tabular}{c c}

	\begin{tabular}{||c||c|c||c|c||}
		\hline
		 & \multicolumn{2}{c||}{NL-PCA} & \multicolumn{2}{c||}{TSID}\\
		\hline
		$\sigma$ & {\scriptsize PSNR} & {\scriptsize RMSE} & {\scriptsize PSNR} & {\scriptsize RMSE}\\
		\hline
		\hline
		2 & 45.29 & 1.39 & 44.06 & 1.60\\
		\hline
		5 & 40.84 & 2.31 & 39.09 & 2.83\\
		\hline
		10 & 37.30 & 3.48 & 35.72 & 4.17\\
		\hline
		20 & 33.35 & 5.48 & 32.07 & 6.35\\
		\hline
		30 & 30.74 & 7.40 & 29.70 & 8.35\\
		\hline
		40 & 28.86 & 9.19 & 27.92 & 10.24\\
		\hline
		60 & 27.57 & 10.66 & 25.55 & 13.46\\
		\hline
		80 & 25.97 & 12.83 & 23.93 & 16.21\\
		\hline
		100 & 24.72 & 14.81 & 22.87 & 18.32\\
		\hline
	\end{tabular}
	&
	\begin{tabular}{||c||c|c||c|c||}
		\hline
		 & \multicolumn{2}{c||}{NL-PCA} & \multicolumn{2}{c||}{TSID}\\
		\hline
		$\sigma$ & {\scriptsize PSNR} & {\scriptsize RMSE} & {\scriptsize PSNR} & {\scriptsize RMSE}\\
		\hline
		\hline
		2 & 46.03 & 1.27 & 43.76 & 1.65\\
		\hline
		5 & 39.84 & 2.60 & 36.97 & 3.61\\
		\hline
		10 & 35.28 & 4.39 & 32.18 & 6.27\\
		\hline
		20 & 30.92 & 7.25 & 28.03 & 10.11\\
		\hline
		30 & 28.61 & 9.46 & 25.94 & 12.87\\
		\hline
		40 & 27.08 & 11.29 & 24.59 & 15.03\\
		\hline
		60 & 24.94 & 14.43 & 22.74 & 18.59\\
		\hline
		80 & 23.63 & 16.78 & 21.45 & 21.57\\
		\hline
		100 & 22.60 & 18.90 & 20.48 & 24.14\\
		\hline
	\end{tabular} \\
	Grey level images & Color images\\
	
\end{tabular}
\end{center}
