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\begin{document}
\title{\textbf{Inpainting of images via a dictionary of models}}
\author{ Mashallah Matinfar, Ali Tavakkoli \thanks  , Mandana Tamizkar \thanks{Corresponding authors E-mail address :
 m.matinfar@umz.ac.ir, a.tavakoli@umz.ac.ir, m.tamizkar03@umail.umz.ac.ir}}
\date{}

\maketitle
\begin{center}
 University of Mazandaran, Babolsar, Iran
%\\$^b$University of Mazandaran, Babolsar, Iran\\$^b$University of Mazandaran, Babolsar, Iran
\end{center}

%------------------------------------------------------------------------------
\begin{abstract}
\noindent In this paper, we present a new scheme to inpaint an image which past of it is damaged or lost. This method is based on the dictionary of models. The method searches a model of the dictionary that ends to the damaged or lost pixel. Next, these pixels are filled out with the colour of the observed model.\\


 \textit{\textbf{Keywords:}}image inpainting, Cahn-Hilliard equation, inpainting path, path gradient .\\

\end{abstract}
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\section{Introduction}


Inpainting is the art of modifying an image such that the results of changes are not easily detectable by an ordinary observer. 
The purpose of inpainting is to reconstitute the missing or damaged portions of the image, in which the inpainted region is seamlessly merged into the image \cite{bert}. 
Propagating the image information along lines of equal gray values into the areas to be modified is expressed by the third-order nonlinear PDE \cite{bertozi1}.
In this paper, the authors explained that good inpainting algorithms should propagate sharp edges in surrounding areas into the damaged parts required to be fill in .
Furthermore, a new PDE's model based on the reformation of the Navier-Stokes equations with an isotropic term and a fidelity term is  proposed by Bertozi \cite{bertanavier}. 
Cahn and Shen proposed the well-known variational image denoising and segmentation models which was adapted to the inpainting task with a simple modification \cite{chan and shen}.
This model is based on image denoising model of Rudin, Osher, Fatemi \cite{Rudin}. 
It propogates sharp edges into the damaged domain as well.
But this model does not  keep the direction of isophotes continuous across the boundary of inpainting.
The Cahn and Shen  model \cite{Chan} have drawbacks. so later, Cahn, Kang and Shen introduced a more complete new variational image inpainting model.
Two PDE's models for image inpainting based on the Mumford-Shah model are propsosed by Esedoglu and Shen \cite{Esedoglu}, one is a simple modification of the fidelity term, another one is obtained by considering the Euler's elastica approximation.
Ballester et.al's  model \cite{Ballester} is based on joint interpolation of the image gray-levels and gradient/isophotes directions, smoothly expanding in an automatic fashion isophote lines into the missing data.
 Other PDE's inpainting models can be found in \cite{Kang}, \cite{March}, \cite{Greer1}, \cite{Greer2}, \cite{Greer3}, \cite{Masnou} and \cite{Shen}.
\cite{tam}the method contains a modification on the total variation model. It is based on a new insight into the neighborhood pixels of the damaged areas.

In this paper, our suggestion model is based on finding the paths whose the first or the end include in inpainting region. These paths involve almost the same colour.\\
The organization of the chapters are as follows:\\
In Section 2, some variety of models for image inpainting are explained.
In Section 3, our proposed model is introduced and finally in Section 4, some numerical experiments are given to confirm the advantages of proposed model.\\
\section{ Some models for image inpainting} 
Image inpainting is the process of filling a missing or a damaged image based on the information obtained from the surrounding areas. It is in fact a type of  interpolation \cite{bert,Masnou}.\\
Let $f(x)$ be a given image in a domain $\Omega$, and  $D\subset\Omega$ be the inpainting domain.
The inpainting model is to find an image $u$ in $\Omega $ such that $u\approx f$ on $\Omega \setminus D$ for a given image $f$ in  $\Omega \setminus D$ and find a suitable approximation on $D$  (Figure1).\\
\begin{figure}
\begin{center}
\centering
\includegraphics[width=5cm]{figure111}\label{pic1}
\caption{\small{a schematic of the domain $\Omega$ and the inpainting region $D$}}
\label{f1}
\end{center}
\end{figure}
  In the sequel, we peresent some variety of image inapinting models:
\subsection{ The energy method} 
In order to recover a corrupted images, the selection of suitable degradiation operator is very important.
Let $R$ be the degradation operator and 
\begin{equation}
f=Ru+\eta, \label{a}
\end{equation}
where $u$ is the original image (the unknown), $f$ is the observed image and $\eta$ is a white Gaussian noise. According to the maximum 
likelihood principle, we can find an approximation of $u$ by solving the least square problem:
\begin{equation}
\inf _{u}=\int _{\Omega} \vert  f-Ru \vert^{2} dx. \label{b}
\end{equation}
If there exists a minimum $u$ of (\ref{b}), then we will have the following equation
\begin{equation}
R^{*}f-R^{*}Ru=0, \label{c}
\end{equation}
where $R^{*}$ is the adjoint of $R$. The equation (\ref{c}) is an ill-posed problem, therefore, the idea is to add regularization term to the energy.
The idea was for Tikhonov and Arsenin \cite{Arsenin} whose proposed the minimization problem:
\begin{equation}
F(u)=\int _{\Omega} \vert  f-Ru \vert^{2} dx+\lambda \int _{\Omega} \vert \nabla u \vert ^{2} dx, \label{d}
\end{equation}
where the first term is refered to as the fidelity term while the second one is a smoothly term and the parameter $\lambda$ is a positive weightly constant. The value of $ \inf \limits_{u} {F(u)} $ is characterized by Euler-Lagrange equation,
\begin{equation}
R^{*}Ru-R^{*}f- \lambda \Delta u =0.\label{e}
\end{equation} 
The equation \eqref{e} with the Neuman boundary condition 
\begin{equation}\label{e6}
\frac{\partial u}{\partial n} =0~~ \text{on} ~~\partial \Omega, 
\end{equation} 
has a uniqe solution, where $n$ is the unit outward normal vector. Naturally if there exists smoothness in the edges of the image, the condition  \eqref{e6} will satisfies.
Adding the regularuzation term $\nabla u$ by the $L^2$ norm, will be caused to remove the noise, but unfortunately penalyzes too much the gradients corresponding to edges. So, one of the solutions can be efective is changing the norm. One of the first works in this direction is due to Rudin, Osher and Fatemi  \cite{Rudin1}, \cite{Rudin2} who proposed to use $L^{1}$ norm on the regularization term that is referred to as total variation, instead of the $L^{2}$ norm. Furthermore, the regularization term can be given in a generalized form by   \cite{Aubert}:
\begin{equation}
E(u)=\dfrac{1}{2}\int _{\Omega} \vert  f-Ru \vert^{2} dx+\lambda \int _{\Omega} \phi(\vert \nabla u \vert ) dx.\label{g}
\end{equation}
The Euler-Lagrange equation of (\ref{g}) reads:
\begin{equation}
R^{*}Ru- \lambda div(\dfrac{\phi(\vert \nabla u \vert ) }{\vert \nabla u \vert }.\nabla u)=R^{*}f. \label{h}
\end{equation} 
\subsection{Total variation} 
As we cited in the preceding section, the total variation (TV) inpainting model is represend as follows:
\begin{equation}
\min \left\lbrace \int _{\Omega}\vert \nabla u \vert +
\dfrac{\lambda}{2} \int _{\Omega \ D} \vert  u-f \vert^{2} dx \right\rbrace ,\label{i}
\end{equation}
where $f$ is a given image and $u$ is unknown.\\
The TV regularization term can be interpreted by the existence  edges of the image:
\begin{equation}
\min \limits_{u}  \int _{\Omega}\vert \nabla u \vert \Longleftrightarrow \min \limits_{\Gamma_{\lambda}} 
 \int _{-\infty}^{+\infty} L(\Gamma_{\lambda}) d \lambda ,\label{j}
\end{equation}
where $ \Gamma_ {\lambda} =  \left\lbrace  x \in \Omega : u(x)= \lambda \right\rbrace $ is the level curves for the gray value $ \lambda$. and $L(\Gamma_{\lambda})$ shows the length of the $\Gamma_{\lambda}$.
The inpainting model has drawbacks that it cannot connect edges over large distances and doesnot smoothly propogate level curves into the inpainting region. In order to decrease this deficiency, in addition to $\vert \nabla u\vert$, the curvature of the level curves are considered
\begin{equation}
\min  \int _{\Omega}(a+b \kappa^{2})\vert \nabla u \vert  dx\Longleftrightarrow \min \limits_{\Gamma_{\lambda}} 
 \int _{-\infty}^{+\infty} a L(\Gamma_{\lambda})  +b \kappa^{2}( \Gamma_ {\lambda}) d \lambda,\label{k}
\end{equation}
where $\kappa= \nabla .\dfrac{\nabla u}{\vert \nabla u \vert}$ is the curvature of level curves $\Gamma_{\lambda}$ of $u$. So, here it is expected to have a better result than the TV inpainting model \cite{main}.\
\subsection{Binary Cahn-Hilliard inpainting}
In this subsection, we explain the modified version of Cahn-Hilliard equation that is used for inpainting of binary images [4].\\
 Let $f$ be the given binary image on domain  $\Omega \subset \mathbb{R}^{d}(d=2,3)$. The aim is to construct the image in the inpainting domain $D \subset \Omega$ in an undetectable 
way by the evolution of the Cahn-Hilliard equation as
 \begin{equation} 
  \dfrac{\partial u}{\partial t}=\triangle(-\epsilon \triangle u+\dfrac{1}{\epsilon}F'(u))+ \lambda (x) (u-f), \quad in \; \Omega \label{l}
\end{equation}
where $F(u)=u^{2}(u-1)^{2}$ is a double well-potential and 
\begin{equation*}
\lambda(x)=
\begin{cases}
0, & x \in D
\\
\lambda, &x \in\Omega\setminus D.
\end{cases}
\end{equation*}
If $\lambda(x)=0$, the original Cahn-Hilliard equation will be derived by
 \begin{equation} 
  \dfrac{\partial u}{\partial t}=\triangle(-\epsilon \triangle u+\dfrac{1}{\epsilon}F'(u)).\label{m}
\end{equation}
The modified Cahn-Hilliard equation (\ref{l}) consists of two different flows. Using $H^{-1}$-norm for the first term of  (\ref{l})  concludes a gradient flow for the Cahn-Hilliard energy as follows:
\begin{equation}
 E_{1}(u)=\int _{\Omega} (\dfrac{\epsilon}{2}\vert \bigtriangledown u \vert ^{2}+\dfrac{1}{\epsilon}F(u)) dx,
\label{n}
\end{equation}
while the second term gives a gradient flow under $L^{2}$-norm for the fidelity term [4],
\begin{equation}
 E_{2}(u)=\dfrac{\lambda}{2} \int _{\Omega \setminus D}(u-f)^{2}dx.\label{o}
\end{equation}
Now, 
$$ E(u)=E_{1}(u)+ E_{2}(u),$$
is called  modified Cahn-Hilliard energy.
Thus the modified Cahn-Hilliard inpainting model (\ref{l}) gives a superposition of the $H^{-1}$-gradient flow for  $ E_{1}(u)$  and the 
 $L^{2}$-gradient flow for $ E_{2}(u)$. The evolution of $u$ can be described as follows:\\
Outside of the damaged region, $u$ stays close to the given image $f$. We can control this closeness via the fidelity parameter $\lambda$.
According to the known image information, the damaged area are filled in.
The role of $\epsilon$ in equation (\ref{l}) is important. In the original Cahn-Hilliard equation, $\epsilon$ serves as a measure of the transition region between two metals in an alloy, after heating and reaching a steady state. In image processing, $\epsilon$ is a measure of the transition
region between the two grayscale states (for example between the black and white printed text) \cite {bertozi2}. 

\section{ The proposed inpainting model}
Let $p_{0},p_{1},p_{2},...,p_{m}$ be the pixels of an image $u$. We call $\rho=\lbrace p_{0},p_{1},p_{2},...,p_{m}\rbrace $  the \textbf  {inpaintnig path}  if the following conditions hold:\\
1. $p_{0} \in D$,\\
2. $p_{1},p_{2},...,p_{m} \in \Omega \setminus D$,\\
3.$\sum_{i=1}^{m}\vert u_{p_{i}}-u_{p_{i-1}} \vert  < \epsilon$,\\
where $ u_{p_{i}}$ shows the intensity of the image u at the point $p_{i}$. The point $p_{0}$ can be at the begining or end of the inpainting path. Without loss of generality, we assume that $p_{0}$ is at the begining of the inpainting path. Let $\pi_j=\lbrace \rho_{1},\rho_{2},...,\rho_{k}\rbrace$  be inpainting pathes including the point $p_{0}$ (see Figure \ref{f22}). Now, we define the \textbf{path gradient}  at the point $p_{0} \in D$ for the path $\rho_j$ by 
\begin{equation}
\bar \nabla u_{\rho_j}(p_0)=\sum_{i=1}^{m}\vert u_{p_{i}}-u_{p_{i-1}} \vert. \label{nabla}
\end{equation}
We calculate the path gradient for each path and define:
\begin{equation}
\bar \nabla u_{\rho^*}(p)=\min \limits _{\rho_ j}\bar \nabla u_{\rho_j}(p_0).\label{hhh}
\end{equation}
Now, we set
\begin{equation}
 u(p_0):=\dfrac{1}{m} \sum_{i=1}^{m} u(p_{i}), ~~ p_i \in \rho^* ~~ \text{for}~~  i=1,\ldots,m.
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{figure21}
\caption{{\small Pixels including  zero values show the domain D and pixels including 50, 100, 150 and 200 marked in dark blue, light blue, green and orange respectively, show 4 different paths for 4 pixels in the domain $\Omega \setminus D$.}}
\label{f22}
\end{figure}
Algorithm $1$ inpaints the region $D$. Since $D$ is a closed and bounded region, it is clear that Algorithm $1$ converges.
\begin{algorithm}
	\caption{}\label{euclid}
	\begin{algorithmic}[1]
		%\Procedure{MyProcedure}{} 	
		\For{$i=1:m$}
		\If{$q_i \in D$}
		\State{$p_0=q_i$}
		\For{$j=1:k$}
	 \State \textit{Find} $\pi_j={p_1^{(j)},\ldots,p_m^{(j)}}$  \textit{Such that} ${p_0} \cup \pi_j$ \textit{is an inpainting path}
	 \State $\rho_j:={p_0}\cup \pi_j$  
	\State \textit{Compute} $\bar \nabla u_{\rho_j}(p_0)=\sum_{i=1}^{m}\vert u_{p_{i}}^{(j)}-u_{p_{i-1}}^{(j)} \vert$
	 		\EndFor
	 \State \textit{Compute} $\bar \nabla u_{\rho^*}(p_0)=\min \limits _{j}\bar \nabla u_{\rho_j}(p_0)$
	  \State $u(p_0)=\frac{1}{m}\sum_{i=1}^{m}u(p_j)$
	 	\EndIf
	 \EndFor
		%\EndProcedure
	\end{algorithmic}
\end{algorithm}
\newpage
\subsection{The mathematical model of our method } 
In this subsection, we  present a mathematical model for our proposed method. According to modified Cahn-Hilliard equation,  our proposed method can be expressed as follows:
 \begin{equation}
\min\limits_{u} J[u]=\int_{D} \vert \bar \nabla u \vert dx+\dfrac{\lambda}{2}\int_{\Omega \setminus D} \vert f- u \vert^{2} dx
, \label{12}
\end{equation}
where $\bar \nabla$ is the continuous form of (\ref{nabla}) and $\lambda $ is a positive regularization parameter. 
Therefore, the model can be rewritten by 
 \begin{equation}
\min\limits_{u} J[u]=\int_{\Omega} \chi_{\Omega} \vert  \bar \nabla u \vert dx+\dfrac{\lambda}{2}\int_{\Omega } \chi_{\Omega \setminus D}(x)\vert u-f\vert^{2} dx
, \label{111}
\end{equation}
where the characteristic function $\chi_{E}(x)$ on a domain $E$ is defined by 
\begin{equation*}
\chi_{E}(x)=
\begin{cases}
1 & x \in E
\\
0 & x \notin E.
\end{cases}
\end{equation*}

The Euler-Lagrange equation for the minimization problem (\ref{111}) reads:

\begin{equation}
- \bar\nabla.(\dfrac{ \bar \nabla u}{\vert \bar \nabla u \vert })+\lambda \chi_{\Omega \setminus D} (x)(u-f)=0. \label{112}
\end{equation}
In order to solve (\ref{112}) by an iterative process, we consider the following PDE problem:
\begin{equation}
\dfrac{\partial u}{\partial t}=\bar \nabla.(\dfrac{ \bar\nabla u}{\vert \bar \nabla u \vert })+\lambda \chi_{\Omega \setminus D}(x) (u-f). \label{113}
\end{equation}
To solve numerically (\ref{113}), let $u^{n}(i,j)$  be the value of $u$ in pixel $(i,j)$ at time $nh$ where $h$ is the  time step. By forward differences, we have
\begin{align}\label{e.23}
u^{n+1}(i,j)&=u^{n}(i,j)+h(\bar \nabla.(\dfrac{\bar\nabla u^{n}(i,j)}{\vert \bar\nabla u^{n}(i,j) \vert })+\\ 
&\lambda \chi_{\Omega \setminus D}(x) (u^{n}(i,j)-f(i,j)),  \nonumber
\end{align}
%The accelerating factor $\vert \bar\nabla u^{n}(i,j) \vert $ is approximated by the forward differencing:
%\begin{align}
%\bar\nabla u(i,j)&=u(i+1,j)-u(i,j),\nonumber \\
%\vert \bar\nabla u(i,j) \vert &=\sqrt{(u(i+1,j))^{2}+(u(i,j))^{2}}.\label{19}
%\end{align}
that is an explicit finite difference scheme. 
\section{Numerical experiments}
In this section, we present some numerical examples to show the efficiency of our proposed method.
\\
\textbf{Example 1:} For the first example, we consider Figure \ref{fig3}(a) that is a $300\times 300$ image of cross in which the inpainting region is shown by gray colour. Figure \ref{fig3}(b) denotes the implementation of the numerical scheme (\ref{e.23}) only by $2$ iterations. As is seen,  the inpainting region is inpainted well

\begin{figure}[!ht]                   
  \centering
  \begin{minipage}[t]{.45\textwidth}
    \centering
    \includegraphics[height=.4\linewidth]{fig1example1}
    \subcaption{}
  \end{minipage}
  \begin{minipage}[t]{.45\textwidth}
    \centering
    \includegraphics[height=.4\linewidth]{fig2example1}
    \subcaption{}
  \end{minipage}\\
  \caption{\small The performance of our approach on. (a) a damaged image (b) in 2 iteration.}
  \label{fig3}
\end{figure}
\\
Let us make a comparison to show how much faster our method is. We compare Figure(\ref{fig3}) with a figure "inpainting of a cross" in \cite{bertozi2}. They represented the model for inpainting based on the modified Cahn-Hilliard equation for binary image. It is observed that in spite of Figure 3(a), there exist a blurring case in Figure \ref{fig3}(b). So, we see better results in the edges of the image. 
\begin{figure}
\centering
\includegraphics[width=10cm]{fig25}
\caption{\small (a) Initial data of cross (inpainting region in gray). (b) Intermediate state at $t = 300$. (c) Steady state at $t = 1000$. (Image
domain is $128 \times 128$, stripe width is $20$ units, initial gap distance is $50$ units). }
\label{f25}
\end{figure}
\newpage
\textbf{Example 2:} Figure \ref{fig4}(a) shows a $200\times 200$ image of a double stripe whose inpainting region is cleared by gray colour. Figures \ref{fig4}(b-d) show the implementation of the numerical scheme (\ref{e.23}) in 5, 10 and 14 iterations, respectively. As is seen, the inpainting region is inpainted well. 

  \begin{figure}[!ht]  
\begin{center}
\centering
\includegraphics[width=15cm]{figfexample2}
  \caption{\small The performance of our approach on an image. (a) original image and implementation of our proposed method in  (b)  5 iterations, (c) 10 iterations and (d) 14 iterations.}
  \label{fig4}
\end{center}
\end{figure}

%\begin{figure}[!ht]                   
  %\centering
  %\begin{minipage}[t]{.45\textwidth}
  %  \centering
   % \includegraphics[height=.4\linewidth]{fig21example2}
 %   \subcaption{}
  %\end{minipage}
 % \begin{minipage}[t]{.45\textwidth}
  %  \centering
  %  \includegraphics[height=.4\linewidth]{fig22example2}
 %   \subcaption{}
 % \end{minipage}\\
  %\begin{minipage}[t]{.45\textwidth}
   % \centering
  %  \includegraphics[height=.4\linewidth]{fig23example2}
    %\subcaption{}
%  \end{minipage}
 % \begin{minipage}[t]{.45\textwidth}
   % \centering
  %  \includegraphics[height=.4\linewidth]{fig24example2}
  % \subcaption{}
 % \end{minipage}
  %\caption{\small The performance of our approach on an image. (a) original image and implementation of our proposed method in  (b)  $22$ iterations, (c) $49$ iterations and (d) $54$ iterations.}
 % \label{fig4}
%\end{figure}
As the last example, we compare our method with  \cite{bertozi2} on the double strip image (Figure \ref{fig4}(a)). Again we observed that our method works well for Figure \ref{fig4}(a).
\begin{figure}[h]
\centering
\includegraphics[width=10cm]{fig16}
\caption{{\small (a) Initial data (inpainting region in gray). (b) Intermediate state at $t = 50$. (c) steady state at $t = 700$. (Gap distance is $30$ units, Image domain is $128 \times 128$).}}
\label{f16}
\end{figure}
\newpage
\textbf{Example 3:} Figure \ref{fig5}(a) shows a $200\times 200 $ image(inpainting region is black square) .Figure \ref{fig5}(b-d) denotes the implementation of the numerical scheme (\ref{e.23}) in 5, 8 and 14 iterations, respectively. As is seen, the inpainting region is inpainted well. 
  \begin{figure}[!ht]  
\begin{center}
\centering
\includegraphics[width=15cm]{figfexample3}
  \caption{\small The performance of our approach on (a) an original image (b) in 5 iterations, (c) 8 iterations and (d) 14 iterations.}
\label{fig5}
\end{center}
\end{figure}
%\begin{figure}[!ht]                   
 % \centering
 % \begin{minipage}[t]{.45\textwidth}
  %  \centering
 %   \includegraphics[height=.4\linewidth]{fig1example3}
  %  \subcaption{}
%  \end{minipage}
 % \begin{minipage}[t]{.45\textwidth}
 %  \centering
  %  \includegraphics[height=.4\linewidth]{fig14example3}
  % \subcaption{}
%  \end{minipage}\\
 % \begin{minipage}[t]{.45\textwidth}
  %  \centering
 %   \includegraphics[height=.4\linewidth]{fig22example3}
  %  \subcaption{}
 % \end{minipage}
% \begin{minipage}[t]{.45\textwidth}
  %  \centering
  %  \includegraphics[height=.4\linewidth]{fig25example3}
    %\subcaption{}
%  \end{minipage}
%  \caption{\small The performance of our approach on (a) an original image (b) in $14$ iterations, (c) $22$ iterations and (d) $25$ iterations.}
 % \label{fig5}
%\end{figure}


\newpage
  \textbf{Example 4:} Figure \ref{fig6}(a) shows a $200\times 200 $ image(inpainting region is black square) .Figure \ref{fig6}(b-d) denotes the implementation of the numerical scheme (\ref{e.23}) in 14, 22 and 29 iterations, respectively. As is seen, the inpainting region is inpainted well. 
  \begin{figure}[!ht]  
\begin{center}
\centering
\includegraphics[width=15cm]{figfexample4}
 \caption{\small The performance of our approach on (a) an original image (b) in 14 iterations, (c) 22 iterations and (d) 29 iterations.}
  \label{fig6}
\end{center}
\end{figure}
  
  
  

\section{result}
As  we observed although the inpainting region is fairly big, the proposed method inpaintes well it. 

\newpage



%----------------------------------------------------------------------------------------









%----------------------------------------------------------------------------------------









%-------------------------------------------------------------------------
\newpage
\section{Conclusion}

%-----------------------------------------------------------
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