\chapter*{چکیده    \label{chekide}}
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 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.
\textbf{کلمات کلیدی :}
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