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<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\begin{center}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\textbf{Abstract}‎‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> Image inpainting are used to repair damaged parts of the image. There are several ways to ‎inpaint‎‎ an image. In this classification, the methods are classified based on the highest order of the derivative in the Euler-Lagrange equation, which includes the second, third and fourth order ‎variational‎  methods. Different models have been presented for each of these methods, some of which are mentioned in this ‎thesis‎ along with advantages, disadvantages and examples. In this ‎thesis‎, two second-order methods for ‎inpainting‎ are proposed. The advantages of the proposed methods include preserving the edge and their ‎fast‎ convergence. There are also several examples for each method that retrieve higher quality images compared to previous methods.</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\end{center}‎</p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"> Image inpainting are used to repair damaged parts of the image. There are several ways to ‎inpaint‎‎ an image. In this classification, the methods are classified based on the highest order of the derivative in the Euler-Lagrange equation, which includes the second, third and fourth order ‎variational‎  methods. Different models have been presented for each of these methods, some of which are mentioned in this ‎thesis‎ along with advantages, disadvantages and examples. In this ‎thesis‎, two second-order methods for ‎inpainting‎ are proposed. The advantages of the proposed methods include preserving the edge and their ‎fast‎ convergence. There are also several examples for each method that retrieve higher quality images compared to previous methods.</p>
<p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;"></p>
<p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; -qt-user-state:0;">‎\textbf{Keywords: Image inpainting, total variation, Euler-Lagrange equation‎‎}‎ </p>
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