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چكيده انگليسي :

Numerical Solution to Some Partial Di erential Equations with Application in Image Inpainting and Denoising Farinaz Mostajeran f mostajeran@math iut ac ir 14 10 2014 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Reza Mokhtari mokhtari@cc iut ac ir Advisor Dr Nader Karimi nader karimi@cc iut ac ir 2010 MSC 65M 06 68U 10 Keywords Nonlinear partial di erential equations nite di erence methods image inpainting image denoising AbstractStudying image inpainting and denoising based on some partial di erential equations PDEs is themain goal of this thesis Among several approaches have been proposed to solve the problem PDE based methods are the most powerful ones Image inpainting means to restore a damaged or corruptedimage that information of some parts of it has been lost or changed Inpainting may also be a useful toolfor some people who arti cially need either to remove some parts of an image such as overlapping texts or to implement tricks used in special e ects It is important to restore the missing parts of an imageso that the nal image looks unaltered to the naked eye Denoising is to remove noisy componentsfrom the pixels of an image We must note that in common image enhancement applications thepixels contain both information about the real data and the noise e g image plus noise for additivenoise while in image inpainting there is no signi cant information in the region to be inpainted Recently a new PDE approach to image inpainting and image denoising are presented by Barbu etal To the best of our knowledge the pioneer work of Bertalmio et al in 2000 is the rst approach ofimage inpainting that introduced a third order PDE that propagates the level lines arriving at hole And numerous nonlinear PDE denoising approaches based on di usion have been introduced sincethe early work of Perona and Malik in 1987 In this thesis at rst some preliminaries which will be used in the sequel are prepared Then we investigate a PDE based approach to image inpainting using a minimization problem It must beinteresting that the history of image inpainting returns to about recent 20 years We provide importanttheories about the resulting PDE and solve it using a classical and two non classical explicit nitedi erence FD methods Comparing these methods over various deteriorated regions is presented The results show that non classical methods can be applied over a rectangle and even more complexmissing regions Furthermore non classical methods can be used for removing some objects suchas a text over an image After that the porous media di usion ltering model for image denoisingwith explanation of important theories is described The nonlinear noise removal techniques based onPDEs have been extensively studied in the last two decades Nonlinear di usion methods can reducenoise and enhance contours in images We present an explicit version of the fast di usion lteringscheme and review the nonlinear anisotropic di usion presented by Perona and Malik The explicitversion and Perona Malik methods are tested over some images with the Gaussian noise In order toquantify the achieved performance we use mean square error MSE and the signal to mean squareerror S MSE ratio that computed based on the original and denoised images A brief descriptionof calculus of variation and the ABC s of digital image processing in MATLAB are represented too Finally we present some RGB color images with di erent deteriorated regions that restored by classicaland non classical explicit FD methods

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حل عددي برخي معادلات با مشتقات پاره اي با كاربرد در ترميم و كاهش نويز تصوير

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