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\title{Single Image Dehazing using a Multi-Layer Perceptron}

\author[a]{Sebasti\'an Salazar-Colores}
\author[b*]{ Ivan Cruz-Aceves}
\author[c]{ Juan-Manuel Ramos-Arreguin}
\affil[a]{Facultad de Inform\'atica, Universidad Aut\'onoma de Quer\'etaro, Av. de las Ciencias, s/n, Juriquilla, C.P. 76230 Santiago de Quer\'etaro, Qro., M\'exico.}
\affil[b]{CONACYT - Centro de Investigaci\'on en Matem\'aticas (CIMAT), A.C., Jalisco S/N, Col.Valenciana, C.P. 36000, Guanajuato, Gto, M\'exico.}
\affil[c]{Facultad de Ingenier\'ia, Universidad Aut\'onoma de Quer\'etaro, Cerro de Las Campanas, s/n, Las Campanas, Santiago de Quer\'etaro, Qro., M\'exico}


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\begin{abstract}
This paper presents a novel algorithm to improve images with hazing effects. Usually the dehazing methods based on the dark channel prior, make use of two different stages to compute the transmission map of the input image. That stages are the transmission map estimation stage and a transmission map refinement stage. However, the main disadvantage of these strategies is the trade-off between accurate restoration and computational time. The proposed method uses a Multilayer Perceptron to compute the transmission map directly from the minimum channel, and a contrast stretching technique to improve the dynamic range of the restored image the Multilayer Perceptron was trained in terms of Mean Squared Error (MSE) using a training set of 80 images. To evaluate the restoration quality, the metrics of Mean Absolute Error (MAE), and the Structural Similarity (SSIM) index were used. The experimental results have proven that the proposed method achieves superior performance in terms of restoration quality (MAE = $27.28$, SSIM index = $0.84$) compared with seven state-of-the-art dehazing methods. In addition, based on the average computational time achieved by the proposed method ($0.52$ seconds using a test set of 40 images), it can be highly suitable for real-time applications. 
\end{abstract}

% Include a list of up to six keywords after the abstract
\keywords{Artificial neural network, dark channel prior, defogging, image enhancement, multi-layer perceptron, single image dehazing}

% Include email contact information for corresponding author
{\noindent \footnotesize\textbf{*}Ivan Cruz-Aceves,  \linkable{ivan.cruz@cimat.mx} }

\begin{spacing}{2}   % use double spacing for rest of manuscript

\section{Introduction}
\label{sect:intro}  % \label{} allows reference to this section
Image enhancement is a very active research area since the image quality has a direct influence on computer vision systems. In general, outdoor images are susceptible to different atmospheric conditions, especially haze, fog, and smog. The main effects caused by these atmospheric conditions are images with low contrast, distorted colors, and reduced visibility. Consequently, the performance of systems, such as surveillance\cite{Cheng2016}, target tracking\cite{Baruwa2013}, remote sensing\cite{Long2014}, traffic control\cite{7103720}, and autonomous driving\cite{Yan2014} could be highly affected.

In the literature, the problem of eliminating or reducing the effects of atmospheric scattering has been widely studied. The most representative strategies can be classified according to the input information such as: multiple images \cite{schechner2001instant,Fang:14}, infrared radiation \cite{Fang2013}, 3D data \cite{kopf2008deep} and the use of a single image \cite{7162616}. However, the most studied approach is based on a single image, because additional data is not commonly available\cite{Vinkey}.

In the single image dehazing problem, several representative works have been proposed. Tarel et al. \cite{6442455}, proposed a method based on two different filters; a median filter and a filter for preserving edges and corners. The method estimate the scattered airlight for recovering a haze-free image. Fattal\cite{fattal2008single} assumes that there is a non-local relation between the shading and the signal. Since, the method try to solve a non-linear inverse problem by computing a transmission map, this approach generates suitable results in regions where the shading and transmission are different. However, it does not perform well in regions where they are similar. He et al. \cite{5567108} introduced the dark channel prior (DCP) based on a statistic analysis of haze-free images. They use the DCP to compute the transmission and restore the image utilizing the atmospheric scattering model. The DCP algorithm obtains accurate results in most cases. The main drawback of DCP method is that it requires an intensive computational processing to refine the generated coarse transmission making it inadequate for many applications where the short response time is required. In order to improve the computational time of the DCP along with the qualitative results of the DCP different works have been introduced. Pang et al. \cite {pang2011improved} proposed to refine the transmission an edge-preserving smoothing operator named guided filter using a patch radius of $30$ pixels. Nishino et al. \cite{raey1} used a probabilistic Bayesian method that combines the estimation of the albedo and depth of a scene. Gibson et al. \cite{Gibson} proposed a variant of DCP replacing the minimum operator with the median operator. Zhu et al. \cite{7128396} proposed a linear color attenuation prior based on the difference of the saturation and the brightness of the pixels to recover the image. Berman et al. \cite{Berman_2016_CVPR} use a pixel-oriented algorithm, where pixels are clustering in the RGB space to estimate a haze-line used to recovery directly the image. Ren et al. \cite{Ren2016} used a deep multiscale neural network, which was trained using raw images as input data and transmission maps generated with depth images as target variables.


In particular, Artificial Neural Networks (ANN) have begun to attract more attention in computer vision systems. ANN are computing systems inspired by biological neural networks trying to emulate the microstructure of the brain. One of the most popular ANN classes is the Multilayer Perceptron (MLP) because of its robustness and ease of implementation \cite{Gardner1998}. It is an ANN with at least three layers of neurons and uses a supervised learning strategy called backpropagation \cite{Kriesel2005}. Recently, it has been used to solve different problems in image processing, such as human face recognition \cite{Bhattacharjee2010}, skin segmentation \cite{Phung2005}, and denoising \cite{Burger2012}.

Given the suitable performance of the MLP in different image processing problems, in this work, a novel method for single image dehazing using an MLP to estimate the transmission map is proposed. In the training step of the MLP 1,144,000 samples were used, they were obtained from a subset of 80 real-world images. In order to define the optimal architecture, several MLP topologies were tested, and the best topology was selected based on the PSNR metric. To evaluate the performance of the proposed method an analysis was carried out using the Mean Absolute Error (MAE), and the Similarity Structural Index Measure (SSIM)\cite{Dosselmann2011}. The tests were performed in both real-world and synthetic images.

The content of the present paper is organized as follows: Section 2 describes the fundamentals of image dehazing along with the fundamentals of the MLP. Section 3 explains the proposed method. The details of the computational experiments are presented in Section 4, and conclusions are given in Section 5.

\section{Background}
\label{sect:back}  % \label{} allows reference to this section

\subsection{The atmospheric scattering model}
%\cite{Lamport94,Alred03,Goossens97}
The atmospheric scattering is a physical phenomenon where the light passing through the particles in the atmosphere is deviated from its straight path. The formation of an image can be explained using the atmospheric scattering model proposed by McCartney et al. \cite{McCartney1977}, and it is defined as follows:
\begin{equation}
I(x,y) = J(x,y)t(x,y)+A(1-t(x,y)),
\label{eq:scat}
\end{equation}

where \(I(x,y)\) corresponds to the captured image, \(J(x,y)\) is the image of the original scene without affectations, $A$ is the color of atmospheric light, and $t(x,y)$ is called transmission, which can be defined in a homogeneous atmosphere as:

\begin{equation}
t(x,y)=e^{{-\beta}d(x,y)},
\end{equation}

where \(\beta\) is the scattering coefficient of the atmosphere and \(d(x,y)\) is the scene depth.

In order to get a haze-free image, the Equation \ref{eq:scat} can be expressed as follows:

\begin{equation}
J(x,y)=\frac{I(x,y)-A}{t(x,y)}+A
\label{est}
\end{equation}

It is important to point out that in Equation \ref{est}, additionally to the input image $I(x,y)$, the transmission map $t(x,y)$ and atmospheric light $A$ must to be determined . The DCP method makes possible to obtain an accurate estimation of the variables $t(x,y)$ and $A$.

\subsection{The dark channel prior}

The dark channel prior (DCP) proposed by He et al. \cite{5567108}, is based on empirical observations about the statistical features in the intensity of dark channel for haze-free images. 

The dark channel $ {{I} ^ {dark}} (x, y) $ for a digital image $ I (x, y) $ is defined as follows:

\begin{equation}
{{I}^{dark}}(x,y)=\underset{C\in R,G,B}{\mathop{\min }}\,(\underset{z\in \Omega (x,y)}{\mathop{\min }}\,{{I}^{C}}(z)),               
\end{equation}

where $ \Omega (x, y) $ is a patch of $ m \times m $ (usually $ m = 15$)\cite{5567108} centered on $ (x, y) $, $ I ^ C $ is the color channel of $ I $: $ R $, $ G $ or $ B $, and $ z $ represents the index for a pixel in the domain of $ \Omega (x, y) $.

The transmission $ \hat{t} (x, y) $ is estimated as:

\begin{equation}
\hat{t}(x,y)=1-\omega \left(\frac{{I}^{dark}(x,y)}{A}\right),
\label{trans}
\end{equation}

where $ \omega $ is a value that defines the desired level of restoration, being $1$ the highest possible value.

The atmospheric light $ A $ is defined as:

\begin{equation}
A=\max \sum\limits_{C=1}^{3}{{{I}^{C}}\left[\underset{(x,y)\in (0.1\%*h*w)}{\mathop{\arg\max}}\,\left({I}^{dark}\left(x,y\right)\right)\right]},       
\label{ec_luz}
\end{equation}

where $ h $ and $ w $ are the height and width of $ I ^ {dark} (x, y) $ and $ C $ indicates each of the $ RGB $ channels  of the input image $ I (x, y) $. To estimate the image without affectations $ J (x, y) $, the Equation (\ref{est}) is applied.

The DCP algorithm gets accurate values of $ A $ and $ t (x, y) $. However, as it is shown in Fig. \ref{fig_sim}  the use of windows of $ m \times m $ when $ \hat{t} (x, y) $ is estimated, generates artifacts on the edges of the restored image $ J (x, y) $. The artifacts are generated because the morphology of the windows used in the DCP does not adapt to the complex geometric boundaries needing an aditional refinement step to remove or reduce this effect \cite{Vinkey}.

\begin{figure}[h]
\centering
\includegraphics[width=8.5cm]{images/res_dcp.eps}
\caption{Result of the DCP. (a) Input image $ I (x, y) $, (b) Dark channel map with $ \Omega = 15$, (c) Estimation of $ J (x, y) $ using $ \Omega = 15$, (d) Dark channel map with $ \Omega = 35 $, (e) Estimation of $ J (x, y) $ using $ \Omega = 35 $.}
\label{fig_sim}
\end{figure}


\subsection{Multi-layer perceptron}
Artificial Neural Networks (ANN) are learning machines inspired in a very simplified mathematical model from biological neural systems \cite{Fausett2014}. In these models, each Processing Element (PE) is called a neuron, and the connection between PE's form a network. One of the first PE developed was the Perceptron, that consist in multiple inputs $X_{i}$ multiplied by an scalar value known as weight $W_{ij}$, and a bias $b_{j}$ is added. The obtained result is processed by a predefined activation function \(f\), which can be described as follows:

\begin{equation}
Y_{j} = f\left(\sum_{i}(W_{ij}*X_{i}+b_{j})\right)
\end{equation}

The most common activation function \(f\) used in Perceptron is the hyperbolic tangent function \(tanh\), that is expressed by

\begin{equation}
tanh \left( x \right) = 2 \cdot \sigma \left( 2 x \right) - 1,
\end{equation}

where \(\sigma(x)\) is,

\begin{equation}
\sigma(x) = \frac{e^x}{1 + e^x}
\end{equation}

The training of the ANN adjusts the weight and bias values, in order to obtain the desired output according to the inputs combinations. For solving non-linear separation problems many Perceptrons in one or more hidden layer topologies are connected, this network is called Multilayer Perceptron (MLP). Backpropagation is an efficient training algorithm for MLP, and it is the most used from the mid-1980's to the present. Backpropagation is a generalization of the least squares method where the weights are updated according to the mean square error, which is determined with a comparison between the desired and the computed output \cite{Fausett2014}. The idea is to find the minimum error function in relation to the connection weights.
The error function is computed as

\begin{equation}
E = \frac{1}{2} \sum_{j \in M} \left(\hat{y}_m - y_m \right)^2,
\end{equation}

with $\hat{y}_m$ being the desired output of $m$-th $y_m$.\\

In the present work, an MLP is used to achieve accurate transmission maps in order to obtain high-quality free-haze images. This method is described in the following section.

\section{Proposed method for image dehazing}
\label{sect:Proposed}  % \label{} allows reference to this section
\subsection{The proposed method}
The dehazing algorithms based on the dark channel prior, make use of two different stages to compute the transmission map. In the first stage, a coarse transmission is estimated using square-patches of a predefined size. In the second stage, the coarse transmission map is improved using different strategies such as the soft matting\cite{5567108}, bilateral filter\cite{yeh:13}, median filter\cite{Gibson}, mean-shift\cite{6974568}, guided filter \cite{pang2011improved}, Wiener filter\cite{6738147}, and graph-cuts\cite{Zhu2017}. The main disadvantage of these strategies is the trade-off between accurate restoration and computational time. 

The proposed method is based on the transmission map $t_{min}$ defined as

\begin{equation}
t_{min}(x,y)=1-\omega{I}^{min}(x,y),
\label{eq3}
\end{equation}


where the minimum channel ${I}^{min}(x,y)$ is defined as follows\cite{Jeong2013a}
 
\begin{equation}
{I}^{min}(x,y)=\underset{c\in (R,G,B)}{\mathop{\min }}\,\frac{{{I}^{c}}(x,y)}{A^c}
\label{eq1}
\end{equation}

The minimum channel $t_{min}$ is equivalent to the the dark channel when the patch size $\Omega=1$. 

$t_{min}(x,y)$ has an excellent spatial resolution, however, its depth resolution computed is low because it omits the information about of neighboring pixels. From the computed transmission map $t_{min}(x,y)$, the application of an artificial neural network MLP to estimate a transmission map $t'(x,y)$ is introduced as follows:


\begin{equation}
t'(x,y)=MLP(t_{min}(x,y))
\label{eq6}
\end{equation}

The main idea behind the proposed method, is the strategy to estimate the transmission map and the introduction of an additional step for improving the contrast image. In order to improve the contrast of the recovered image, the luminance was modified by introducing the spacecolor L*a*b and a contrast stretching (CS)\cite{Yang2006} strategy. In Fig. \ref{proposed1} the procedure carried out by the proposed method is introduced.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=6cm]{images/diagrama.eps}
\end{tabular}
\end{center}
\caption{Flowchart of the proposed method} 
\label{proposed1}
\end{figure} 

\subsection{Synthetic ground-truth generation}
\label{ground}
Since the MLP is a supervised technique, a ground-truth is required to perform the training step. In the present work, the ground-truth data is computed using the soft matting algorithm proposed by Levin et al.\cite{Levin2008}, which has proven to be suitable for the dehazing problem\cite{5567108}, nevertheless, its computational processing time and memory requirements are both high.

The image soft matting procedure can be expressed as follows

\begin{equation}
I=F\alpha+B(1-\alpha),
\end{equation}

where $B$ and $F$ are the background and foreground color information, respectively, and $\alpha$ is the opacity of foreground. In the work of He et al.\cite{5567108} was proved the equivalence between  the $\alpha$ map and the transmission map $\bar{t}(x,y)$. Then the optimal $\bar{t}(x,y)$ can be computed by solving the sparse linear system expressed in the following equation

\begin{equation}
(L+\lambda U)\bar t=\lambda \hat{t},
\label{sparse}
\end{equation}
where the used parameters were $\lambda = 10^{-4}$, $\bar{t}$ is the optimal transmission, $\hat{t}$ is the transmission computed from the dark channel, $L$ is called matting Laplacian matrix, and $U$ is and identity matrix of the same size of $L$.

\subsection{The training process}
\label{trainingp}
In order to train the Multi-Layer Perceptron, a data set of 80 real-world images different size and features was used. From every image, two images of the same size were generated. The first one contains the estimated transmission map $t_{min}(x,y)$ using the Equation \ref{eq3}, and the second one is the transmission result $\bar{t}(x,y)$ which is obtained by applying the soft-matting method to $t_{min}(x,y)$.

The training step of a MLP requires two vectors of samples: the input data and the target vectors. In this work, each sample was acquired from a square window of size $l$ centered in positions $(x,y)$ where the length for each sample is $l\times l$. The input vector was obtained from $t_{min}(x,y)$ and the target vector from $\bar{t}(x,y)$. The setup to perform the training stage is illustrated in Fig. \ref{training}.


\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[height=9cm]{images/training_process.eps}
\end{tabular}
\end{center}
\caption 
{Training process for the MLP based on transmission maps.} 
\label{training}
\end{figure} 

\subsection{Application of MLP}
\label{application}
As it is shown in Fig. \ref{fig7}, to compute the transmission map $t'(x,y)$ using the trained MLP, an input vector is generated from square-windows of $t_{min}(x,y)$. Each two-dimensional square-window of size $s$ is converted to a one-dimensional signal of size $s\times s$. The interval of each sampling in the image is expressed by the variable $\delta$. If $\delta=s$ the sampling has not intersection in the pixels positions, and if $\delta=1$ the sampling is realized in every pixel. The value of each pixel in the transmission map $t'(x,y)$ is the minimum value of the superposition of windows.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{images/superposition}
\end{tabular}
\end{center}
\caption{Process to estimate the transmission map by using the MLP.}
\label{fig7} 
\end{figure} 

In Fig. \ref{fig111} an example of the proposed algorithm in four real-world images and the results of applying the proposed method are presented, along with the input $t_{min}(x,y)$ and output $t'(x,y)$ transmission maps of the MLP are shown. The parameters values are $\delta=16$ and $s=8$. 

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{images/transmission_map} 
\\
(a) \hspace{3.5cm} (b) \hspace{3.5cm} (c)\hspace{3.5cm} (d)
\end{tabular}
\end{center}
\caption 
{ \label{fig111}
Examples of the processing the proposed method: (a) input images $I(x,y)$, (b) initial transmission $t_{min}(x,y)$, (c) final transmission $t'(x,y)$, (d) recovered images $J(x,y)$.} 
\end{figure} 


\subsection{Evaluation metrics}
In order to measure the performance of the proposed method, two different metrics have been adopted: Mean Absolute Error (MAE), and Structural Similarity (SSIM) index\cite{Dosselmann2011}. 

The Mean Absolute Error (MAE) frequently used in statistics provides a measure of disparity between the restored image $ {{I} ^ {o} (x, y)} $ and the target image $ J (x, y) $ using the width ($w$) and height ($h$) of the input image, it is defined as:

\begin{equation}
\text{MAE}=\frac{1}{(w*h)}\sum\limits_{x=1}^{w}{\sum\limits_{y=1}^{h}{|J(x,y)-{{I}^{o}}(x,y)|}},
\end{equation}



The Structural similarity index (SSIM) is a normalized metric based on a perception-based model, and it is defined as:

\begin{equation}
S(x,y)=f(l(x,y),c(x,y),s(x,y)),
\end{equation}

where $l(x,y)$ is the luminance comparison, $c(x,y)$is the contrast comparison, and $s(x,y)$ is the structure comparison.


\section{Results and discussion}
The evaluation performance of the proposed method was assessed using a database of 120 rgb images. The database was divided into two subsets: a training set of 80 images and a test set of 40 images. The test set of images consists of 30 real-world images and 10 synthetic images obtained from Fattal et al.\cite{Fattal2014}. Furthermore, the computational experiments were performed using the $Matlab$ software, version $R2016a$ on  a computer with 3.10 GHz Intel Core i5-2400 and 12 GB RAM memory.


\subsection{Parameters tuning}

To obtain the best possible performance of the proposed method, several tests with different MLP configurations using the training set of 80 images were realized. In Fig \ref{fig15}, the results of the most significant configurations in terms of the MSE metric are shown. The best MLP performance is presented by the configuration $256-1024$, which corresponds to a window size $s=16$.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{images/arqui}
\end{tabular}
\end{center}
\caption 
{ \label{fig15}
Relation between MLP architecture and performance using the training set of images.} 
\end{figure} 

Once the value $s$ was determined, in order to obtain the optimal value of $\delta$ (window sliding value), the method was tested over the training set varying the $\delta$ value in the range $[1,..,16]$. Three aspects were considered to choose the $\delta$ value: a quantitative analysis using the index SSIM (Fig. \ref{fig31} (a)), a time processing analysis (Fig. \ref{fig31} (b)), and a qualitative analysis (Fig. \ref{fig21}). Based on the obtained results, the best value was $\delta=8$, representing the highest SSIM and lowest processing time.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16.5cm]{images/delta}  % fig2 includes two images 
\\
(a) \hspace{7cm} (b) 
\end{tabular}
\end{center}
\caption 
{ \label{fig31}
Relation between $\delta$ sliding windows used in the application of the MLP in the image, (a) SSIM index, (b) time processing.} 
\end{figure} 



\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=16cm]{images/tam_sliding} 
\\
(a) \hspace{2cm} (b) \hspace{2cm} (c)\hspace{2cm} (d)\hspace{2cm} (e) \hspace{2cm} (f)
\end{tabular}
\end{center}
\caption 
{ \label{fig21}
Transmissions and results generated using different $\delta$ sliding windows: (a) $\delta=1$, (b) $\delta=2$, (c) $\delta=4$, (d) $\delta=8$, (e) $\delta=12$, (f) $\delta=16$.} 
\end{figure} 


\subsection{Performance analysis}
The proposed method was compared with seven state-of-art dehazing methods: Tarel et al. \cite{6442455},He et al. method \cite{5567108}, Pang et al. method \cite{pang2011improved}, Gibson et al. \cite{Gibson}, Zhu et al. \cite{7128396}, Berman et al. and Ren et al. \cite{Ren2016}. The parameters used in each method are the proposed by every author. For the proposed method the parameters are: $\delta =8$ and window size $s=16$.
 
To perform a visual analysis, in Fig. \ref{fig3}, the dehazing results of the proposed and comparative methods over a subset of real-world images of the test set are presented. In this subset, a higher contrast along with more natural intensities are obtained by the proposed method. In addition, Fig. \ref{fig2} shows the results using synthetic images.

To evaluate the restoration level, Table \ref{tab:mae}, and Table \ref{tab:ssim}, present the MAE and SSIM index values, respectively. The proposed method achieves
the lowest average value of MAE, and the highest average in terms of the SSIM index. Finally, Fig. \ref{fig:time} presents the time processing comparison, where the proposed method obtains the best performance.

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=17cm]{images/natural_images}  % fig2 includes two images 
\end{tabular}
\end{center}
\caption 
{ \label{fig3}
Comparison of the recovered results for real-world images using the eight methods: (a) input images, (b) Tarel et al.\cite{6442455}, (c) He et al.\cite{5567108}, (d) Pang et al.\cite{pang2011improved}, (e) Gibson et al.\cite{Gibson}, (f) Zhu et al. \cite{7128396}, (g) Berman et al.\cite{Berman_2016_CVPR}, (h) Ren et al.\cite{Ren2016}, (i) proposed.} 
\end{figure} 

\begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=17cm]{images/synthetic_images}  % fig2 includes two images 
\\
(a) \hspace{1cm} (b) \hspace{1cm} (c)\hspace{1cm} (d)\hspace{1cm} (e)\hspace{1cm} (f) \hspace{1cm} (g) \hspace{1cm} (h) \hspace{1cm} (i) \hspace{1cm} (j)
\end{tabular}
\end{center}
\caption 
{ \label{fig2}
Comparison of the recovered results for synthetic images by the eight methods: (a) original images, (b) images with haze simulated, (c) Tarel et al.\cite{6442455}, (d) He et al. \cite{5567108}, (e) Pang et al.\cite{pang2011improved}, (f) Gibson et al.\cite{Gibson}, (g) Zhu et al.\cite{7128396}, (h) Berman et al.\cite{Berman_2016_CVPR}, (i) Ren et al.\cite{Ren2016}, (j) proposed.} 
\end{figure} 

   \begin{table}[htbp]
     \centering
     \caption{Comparative analysis using the Mean Absolute Error (MAE).}
		\scriptsize
       \begin{tabular}{lcccccccc}
       \hline
       \rule[-1ex]{0pt}{3.5ex}Images &  Tarel et al.\cite{6442455} & He et al.\cite{5567108}& Pang et al. \cite{pang2011improved} & Gibson et al\cite{Gibson}. & Zhu et al.\cite{7128396} & Berman et al.\cite{Berman_2016_CVPR} & Ren et al.\cite{Ren2016} & Proposed \\   
   \hline
   \rule[-1ex]{0pt}{3.5ex}roofs2   & 62.77    & 17.27    & 18.00    & 33.20    & 29.24    & 16.72    & 45.97    & 19.96 \\
   \rule[-1ex]{0pt}{3.5ex}trees2   & 74.30    & 28.45    & 28.94    & 47.80    & 41.66    & 28.79    & 46.76    & 29.47 \\
   \rule[-1ex]{0pt}{3.5ex}church   & 53.42    & 54.31    & 55.40    & 59.88    & 54.70    & 30.47    & 41.31    & 39.68 \\
   \rule[-1ex]{0pt}{3.5ex}couch    & 31.20    & 23.60    & 23.94    & 28.49    & 29.50    & 25.68    & 21.23    & 27.27 \\
   \rule[-1ex]{0pt}{3.5ex}dolls    & 29.87    & 26.83    & 27.29    & 22.75    & 26.41    & 30.39    & 25.40    & 27.97 \\
   \rule[-1ex]{0pt}{3.5ex}flower1  & 36.91    & 24.77    & 25.86    & 47.61    & 41.24    & 31.81    & 71.10    & 21.26 \\
   \rule[-1ex]{0pt}{3.5ex}flower2  & 32.75    & 28.91    & 30.05    & 44.86    & 36.35    & 41.03    & 57.63    & 31.14 \\
   \rule[-1ex]{0pt}{3.5ex}mansion  & 30.15    & 22.49    & 22.93    & 38.81    & 40.23    & 23.66    & 26.82    & 22.31 \\
   \rule[-1ex]{0pt}{3.5ex}moebius  & 42.66    & 50.62    & 51.77    & 40.67    & 37.84    & 36.37    & 33.55    & 33.11 \\
   \rule[-1ex]{0pt}{3.5ex}raindeer & 30.88    & 27.98    & 28.83    & 31.55    & 23.40    & 23.54    & 29.67    & 20.62 \\
   \hline
   \rule[-1ex]{0pt}{3.5ex}Average  & 42.49    & 30.52    & 31.30    & 39.56    & 36.06    & 28.85    & 39.94    & \textbf{27.28} \\
   \hline

       \end{tabular}%
     \label{tab:mae}%
   \end{table}%

   \begin{table}[ht]
     \centering
     \caption{Comparative analysis using the Structural Similarity Index Measure (SSIM).}
     \scriptsize
       \begin{tabular}{lcccccccc}
       \hline
       \rule[-1ex]{0pt}{3.5ex}Images &  Tarel et al.\cite{6442455} & He et al.\cite{5567108}& Pang et al. \cite{pang2011improved} & Gibson et al\cite{Gibson}. & Zhu et al.\cite{7128396} & Berman et al.\cite{Berman_2016_CVPR} & Ren et al.\cite{Ren2016} & Proposed \\   
   \hline
   \rule[-1ex]{0pt}{3.5ex}roofs2   & 0.66     & 0.89     & 0.89     & 0.69     & 0.73     & 0.91     & 0.67     & 0.86 \\
   \rule[-1ex]{0pt}{3.5ex}trees2   & 0.57     & 0.73     & 0.73     & 0.63     & 0.65     & 0.76     & 0.64     & 0.75 \\
   \rule[-1ex]{0pt}{3.5ex}church   & 0.74     & 0.76     & 0.74     & 0.71     & 0.71     & 0.87     & 0.85     & 0.86 \\
   \rule[-1ex]{0pt}{3.5ex}couch    & 0.80     & 0.84     & 0.84     & 0.78     & 0.68     & 0.80     & 0.87     & 0.80 \\
   \rule[-1ex]{0pt}{3.5ex}dolls    & 0.77     & 0.79     & 0.78     & 0.81     & 0.81     & 0.79     & 0.82     & 0.81 \\
   \rule[-1ex]{0pt}{3.5ex}flower1  & 0.85     & 0.84     & 0.82     & 0.85     & 0.86     & 0.73     & 0.43     & 0.82 \\
   \rule[-1ex]{0pt}{3.5ex}flower2  & 0.82     & 0.85     & 0.83     & 0.82     & 0.85     & 0.71     & 0.60     & 0.83 \\
   \rule[-1ex]{0pt}{3.5ex}mansion  & 0.87     & 0.91     & 0.91     & 0.76     & 0.70     & 0.90     & 0.88     & 0.91 \\
   \rule[-1ex]{0pt}{3.5ex}moebius  & 0.76     & 0.72     & 0.70     & 0.81     & 0.82     & 0.85     & 0.90     & 0.85 \\
   \rule[-1ex]{0pt}{3.5ex}raindeer & 0.76     & 0.79     & 0.76     & 0.80     & 0.82     & 0.83     & 0.80     & 0.86 \\
   \hline
   \rule[-1ex]{0pt}{3.5ex}Average  & 0.76     & 0.81     & 0.80     & 0.76     & 0.76     & 0.82     & 0.75     & \textbf{0.84} \\
   \hline
       \end{tabular}%
     \label{tab:ssim}%
   \end{table}%

   
   \begin{figure}[h]
\begin{center}
\begin{tabular}{c}
\includegraphics[width=18cm]{images/tiempo}  % fig2 includes two images 
\\
(a) \hspace{7cm} (b) 
\end{tabular}
\end{center}
\caption 
{ \label{fig:time}
Performance analysis in terms of computational time processing.} 
\end{figure} 




\section{Conclusion}
In this paper, a novel single image dehazing method was proposed. This method uses an artificial neuronal network Multi-Layer Perceptron (MLP) to estimate the transmission map of a haze image. To obtain the optimal MLP configuration a training set of 80 real-world images was used. In experiments a number of hidden layers containing different number of neurons was tested, where the best performance in terms of the Mean Squared Error (MSE) was achieved using a $256-1024$ MLP configuration with a $MSE=0.000151$. In order to evaluate the restoration quality of the proposed method, the Structural Similarity (SSIM) index and the Mean Absolute Error (MAE) were used. The experimental results have proven that the proposed method achieves a superior performance than seven state-of-art methods in terms of restoration quality, obtaining a MAE value of $27.28$ and SSIM index of $0.84$ over a test set of synthetic images. In addition a comparative analysis using the test set of 40 images between the proposed and comparative methods, reveals that the lowest computational time was obtained by the proposed method ($0.52$ seconds). Given the suitable results in terms of restoration quality and execution time of the proposed method with respect to the seven state-of-art dehazing methods, it can be highly appropriate to be used in real-time systems.

\acknowledgments 
Sebasti\'an Salazar-Colores (CVU 477758) would like to thank CONACYT (Consejo Nacional de Ciencia y Tecnolog\'ia) for the financial support of his Ph. D. studies under Scholarship 285651. In addition, the authors thank to CONACYT by the project Cátedras-CONACYT No. 3150-3097.

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\subsection{Biographies}
\vspace{2ex}\noindent\textbf{Sebasti\'an Salazar-Colores} received his B. S. degree in Computer Science from Universidad Aut\'onoma Benito Ju\'arez de Oaxaca, received his M. S. degree in Electrical Engineering at Universidad de Guanajuato. HE is a PhD candidate in Computer Science at the Universidad Aut\'onoma de Quer\'etaro. His research interest are image processing and computer vision. 

\vspace{2ex}\noindent\textbf{Ivan Cruz-Aceves } received his Ph.D. in Electrical Engineering from Universidad de Guanajuato in 2014. He works at the Mexican National Council on Science and Technology assigned to the Center for Research in Mathematics in Guanajuato, Mexico since 2014. His research areas are biomedical signal and image analysis, computational intelligence, and evolutionary computation.

\vspace{2ex}\noindent\textbf{Juan-Manuel Ramos-Arreguin} received his M. S. degree in Electrical Engineering option instrumentation and digital systems from University of Guanajuato and the Ph.D. in Mechatronics Science from Centro de Ingeniería y Desarrollo Industrial (Enginnering and Industrial Development Center). Since 2009 he is part of the Engineering Department at the UAQ where he works as Researcher and Lecturer. His research interest includes Mechatronics and embedded systems. 


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