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\title{\huge A Multilayer Neural Network for Image Demosaicking}
\author{Yi-Qing Wang}
\institute[ENS Cachan]{CMLA, Ecole Normale Supérieure de Cachan, France}
\date[1er Septembre, 2014]{1er Septembre, 2014}

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            \begin{block}{Introduction}
              \begin{itemize}
              \item most digital cameras place a Bayer color filter array (CFA) in front of the sensor to record just one color (R, G or B) at each pixel site. 
              \item the process of restoring the missing color information in such a mosaiced image through cross-channel interpolation is termed \textit{demosaicking}.
	\item a body of work has been done in this field including two neural network inspired approaches. But recent successful applications of multilayer neural networks in image processing warrant a revisit of the subject. 
              \end{itemize}              
            \end{block}
            \vfill
            \begin{block}{Setting}
              \begin{itemize}
		\item a patch based demosaicking algorithm approximates the expectation of the missing pixels conditional on the visible ones under some patch distribution. 
		\item studies have shown that neural networks are universal approximators and that deep networks tend to represent signals more efficiently \cite{bengio2009learning}.
		\item key to training a deep, or multilayer, neural network seems to be an autoencoder-enabled unsupervised learning process.
              \end{itemize}
            \end{block}
            \vfill
            \begin{block}{Our neural network and its training}
		\begin{itemize}
		\item given a RGGB Bayer CFA filtered $\kappa$-by-$\kappa$ patch $v$, our network with two hidden and one linear output layers will make an educated guess of the missing $2\kappa^2$ values according to 
\begin{align}
\widehat{u} = W_3 [v^t, \tanh( W_2 \tanh(W_1 v+b_1) +b_2 )^t]^t + b_3. \label{all}
\end{align}
		\item an autoencoder's fan-in $x$ and fan-out $\widehat{x}$ are vectors of the same dimension, related by a non-linear hidden layer of potentially different size 
\begin{align*}
&\widehat{x} = Va + h \notag \\
&a = \tanh(Wx + b)  
\end{align*}
where $\tanh(\cdot)$ is applied element-wise. With $n$ inputs $(x_j)_{1\leq j \leq n} \in \mathbb{R}^{d\times n}$ which produce $n$ activations $\mathfrak{a}:=(a_j)_{1\leq j \leq n} \in \mathbb{R}^{\mathfrak{m}d \times n}$, the autoencoder's parameters are determined through minimizing the following objective
\begin{align*}
\frac{1}{nd}\sum_{j=1}^n\|\widehat{x}_j - x_j\|_2^2 + \frac{\alpha}{\mathfrak{m}d^2}(\|V\|_2^2 + \|W\|_2^2) + \beta \s (\mathfrak{a}, n, d, \rho) 
\end{align*}
with $\alpha, \beta, \rho$ three prefixed hyper-parameters and $\mathfrak{m}$ a redundancy control. The sparsity inducing penalty is a function of average activation
\begin{align*}
\s (a_1, \cdots, a_n, n, d, \rho) = \frac{1}{\mathfrak{m}d}\sum_{k=1}^{\mathfrak{m}d }\textsl{ KL}\big( \rho \big|\big| \frac{1}{n}\sum_{j=1}^n|a_{jk}| \big)
\end{align*} 
where $\textsl{KL}(\cdot || \cdot )$ is the standard Kullback-Leibler divergence between two Bernoulli probabilities. 
	\item a combination of supervised and unsupervised learning was used to set $(W_l, b_l)_{1\leq l \leq 3}$ in \eqref{all}. To $\kappa$-by-$\kappa$ random patches drawn from some high quality images, the RGGB Bayer CFA was applied to separate the visible pixels $v$ from the invisible ones $u$. An autoencoder was trained on $v$ and resulted in $(W_1, b_1)$. The activations from the first hidden layer were then computed, on which another autoencoder was trained to set up $(W_2,b_2)$. Finally, the output linear layer was initialized by a linear regression to fit the teaching signals $u$. 
          \item once fully configured, the network was fine-tuned with the backpropagation algorithm on fresh data in the hope of further improving its generalization error assessed on a distinct set of validation examples. Since we could not know in advance the hyper-parameters most conducive to good learning, an exhaustive exploration in the parameter space was carried out.
	\item evolution of the neural network's empirically assessed generalized error
	\vskip0.5em 
               \includegraphics[width=0.85\linewidth,height=0.45\linewidth]{ICIP/Figures/valerr}
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            \begin{block}{Inpainting}
	\begin{itemize}
	\item trained to restore color removed by the RGGB CFA on $2k$-by-$2k$ patches with $k \geq 1$, a network should be able to handle those masked by any of the other three CFAs as well because the patch distribution is rotation-invariant. Hence each missing color value can be estimated multiple times, thereby enhancing the algorithm's performance.\vskip0.5em 
		\begin{tabularx}{\linewidth}{ZZZZ}
	        RGGB 
		&
	        GRGB
		&
		BGGR
		&
		GBRG 
		\\
		\includegraphics[width=0.3\linewidth]{ICIP/Figures/1bayer}
		&
		\includegraphics[width=0.3\linewidth]{ICIP/Figures/2bayer}
		&
		\includegraphics[width=0.3\linewidth]{ICIP/Figures/3bayer}
		&
		\includegraphics[width=0.3\linewidth]{ICIP/Figures/4bayer}
		\end{tabularx}
	\end{itemize}
	\end{block}
            
            \begin{block}{Numerical Results}
\begin{itemize}
\item the RMSEs from our neural network trained on $4$-by-$4$ patches, KSVD \cite{mairal2009non}, CS \cite{getreuer2011contour}, SSD \cite{buades2009self}, HA \cite{hamilton1997adaptive}, Paliy \cite{paliy2007spatially} and ZW \cite{zhang2005color} on \textit{McMaster} and \textit{Kodak}. 

\begin{table}
\centering
\fontsize{10.0mm}{10.0mm}\selectfont

\begin{tabular}{l*{8}{c}r}
\\[3pt]\hline
\textit{McMaster} & \textbf{ NN} & \textbf{ KSVD} & \textbf{ CS} & \textbf{ SSD} & \textbf{ HA} & \textbf{ Paliy} & \textbf{ ZW}\\
\hline
\textbf{1} & \textbf{8.52} & 9.67 & 8.92 & 10.60 & 10.45 & 11.61 & 11.65\\
\textbf{2} & \textbf{4.66} & 4.89 & 5.03 & 5.03 & 5.08 & 5.36 & 5.44\\
\textbf{3} & 5.75 & \textbf{5.54} & 5.57 & 5.81 & 6.53 & 5.97 & 6.03\\
\textbf{4} & 3.70 & \textbf{3.49} & 4.37 & 3.90 & 4.53 & 4.48 & 5.09\\
\textbf{5} & \textbf{4.84} & 5.98 & 5.37 & 6.17 & 6.00 & 6.82 & 7.20\\
\textbf{6} & 3.61 & \textbf{3.36} & 3.82 & 4.33 & 4.41 & 5.14 & 5.65\\
\textbf{7} & 3.83 & 2.68 & \textbf{2.50} & 3.80 & 4.37 & 2.78 & 2.79\\
\textbf{8} & 3.16 & \textbf{2.96} & 3.26 & 3.46 & 3.76 & 3.23 & 3.49\\
\textbf{9} & \textbf{3.57} & 3.65 & 3.97 & 4.11 & 4.32 & 4.79 & 5.17\\
\textbf{10} & \textbf{3.05} & 3.13 & 3.25 & 3.30 & 3.50 & 3.88 & 4.02\\
\textbf{11} & 2.78 & \textbf{2.74} & 2.97 & 3.06 & 3.33 & 3.61 & 3.66\\
\textbf{12} & \textbf{2.96} & 3.20 & 3.32 & 3.33 & 3.54 & 3.82 & 3.89\\
\textbf{13} & \textbf{2.42} & 2.51 & 2.61 & 2.50 & 2.65 & 2.99 & 3.03\\
\textbf{14} & \textbf{2.96} & 3.09 & 3.28 & 3.15 & 3.27 & 3.76 & 3.60\\
\textbf{15} & \textbf{2.89} & 3.02 & 3.10 & 3.15 & 3.29 & 3.51 & 3.63\\
\textbf{16} & \textbf{4.82} & 6.32 & 6.15 & 7.15 & 6.94 & 8.92 & 8.11\\
\textbf{17} & \textbf{5.44} & 6.05 & 5.55 & 7.40 & 7.28 & 8.83 & 9.10\\
\textbf{18} & \textbf{4.38} & 4.71 & 5.03 & 5.16 & 5.22 & 5.34 & 5.34\\
\textsl{rmse avg.} & \textbf{4.07} & 4.27 & 4.33 & 4.74 & 4.91 & 5.26 & 5.38\\
\hline
\end{tabular}

\begin{tabular}{l*{8}{c}r}
\\[3pt]\hline
\textit{Kodak} & \textbf{ KSVD} & \textbf{ Paliy} & \textbf{ ZW} & \textbf{ NN} & \textbf{ SSD} & \textbf{ CS} & \textbf{ HA} \\
\hline
\textbf{01} & \textbf{2.23} & 2.44 & 2.73 & 4.62 & 4.70 & 4.09 & 5.35\\
\textbf{02} & \textbf{2.19} & 2.26 & 2.42 & 3.32 & 3.91 & 4.32 & 3.49\\
\textbf{03} & \textbf{1.54} & 1.69 & 1.86 & 2.57 & 3.05 & 3.88 & 3.04\\
\textbf{04} & \textbf{1.98} & 2.34 & 2.53 & 3.05 & 3.10 & 4.05 & 3.30\\
\textbf{05} & \textbf{2.64} & 3.33 & 3.25 & 4.19 & 4.36 & 4.72 & 4.88\\
\textbf{06} & \textbf{2.13} & 2.30 & 2.38 & 4.24 & 4.16 & 4.29 & 4.78\\
\textbf{07} & \textbf{1.64} & 1.82 & 2.06 & 2.45 & 2.94 & 3.82 & 2.84\\
\textbf{08} & \textbf{3.33} & 3.51 & 3.72 & 5.72 & 5.19 & 5.14 & 6.32\\
\textbf{09} & \textbf{1.62} & 1.67 & 1.74 & 2.37 & 2.64 & 3.69 & 2.75\\
\textbf{10} & \textbf{1.78} & 1.87 & 1.92 & 2.40 & 2.75 & 3.71 & 2.88\\
\textbf{11} & \textbf{2.16} & 2.36 & 2.43 & 3.80 & 3.71 & 4.11 & 4.09\\
\textbf{12} & \textbf{1.48} & 1.62 & 1.70 & 2.48 & 2.57 & 3.50 & 2.58\\
\textbf{13} & \textbf{3.74} & 3.95 & 4.19 & 7.76 & 6.48 & 5.10 & 8.09\\
\textbf{14} & \textbf{2.81} & 3.51 & 3.65 & 4.01 & 4.71 & 5.01 & 4.66\\
\textbf{15} & \textbf{2.25} & 2.62 & 2.73 & 3.17 & 3.94 & 4.59 & 3.97\\
\textbf{16} & \textbf{1.44} & 1.58 & 1.56 & 2.98 & 3.57 & 3.95 & 3.63\\
\textbf{17} & \textbf{1.95} & 2.02 & 2.02 & 2.95 & 2.74 & 2.20 & 2.97\\
\textbf{18} & \textbf{3.17} & 3.65 & 3.50 & 5.20 & 4.55 & 4.15 & 5.20\\
\textbf{19} & \textbf{1.95} & 2.20 & 2.19 & 3.52 & 3.02 & 2.54 & 3.40\\
\textsl{rmse avg.} & \textbf{2.21} & 2.46 & 2.55 & 3.72 & 3.79 & 4.04 & 4.11\\
\hline
\end{tabular}
\vskip0.5em
\end{table}
\item the experiments show that our tiny $16$-$80$-$80$-$32$ neural network compares favorably with these state-of-the-art algorithms. 
\item the training images' vibrant color explains in part that the resultant network does better on \textit{McMaster} than on \textit{Kodak}. 
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