معادله تلگراف كسري - مكاني براي كاهش نويز تصوير

چكيده انگليسي :

spatial fractional telegraph equation for image denoising Hoda Sarvi h saravi@math iut ac ir 10 01 2016 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 99 94A08 Keywords Partial di erential equations fractional image denoising telegraph equation Fourier trans form nite di erence methods AbstractOne of the most used branches of image processing is image improvement through restoration and denoising The goal of image denoising is to removing the noise from the observed noisy image and to get some imagesas close as possible to the original image Besides noise removal ability another important requirement forimage denoising procedures is to preserving the original image structures in the whole of the process Thereare several tools used for image denoising which we focus on one the most important approaches i e a methodbased on the partial di erential equation PDE Among various PDEs some evolution equations such asparabolic equations were more used in the denoising process However hyperbolic equations can improvequality of the revealed edges and thus quality of the image is preserved better than parabolic equations Perona and Malik were the rst ones who o ered the anisotropic di usion equation for the image denoisingprocess After a while Catt et al provided a well de ned version of that equation by changing the di usionfunction The problem with this type of equation was the staircase e ect on the image In this regard a fourth order PDE was suggested by You and Kaveh But some new problems were emerged by solvingthat equation Some small white and black spots were appeared on the image After a gap Ratner andZeevi considered the image as an elastic sheet placed in a damping environment and provided the telegraph di usion equation which is a parabolic hyperbolic one The advent of the fractional derivatives leads togenerate a new calculus entitled fractional calculus Then fractional partial di erential equations weredeveloped and applied to model some problems Bai and Feng were the rst ones used fractional PDEs forimage denoising Recently Zhang et al proposed a space fractional telegraph equation for image denoisingwhich we focused on it in this thesis We provide some preliminaries from the real analysis functional analysis and fractional calculus which weneed them in the sequel Then we investigate an initial boundary value problem on the basis of a telegraphequation i e a parabolic hyperbolic partial di erential equation Existence and uniqueness theorems relatedto the solution of that problem is represented and dealt with This problem is applied for image denoisingby using an explicit nite di erence scheme After that we deal with the e ect of the fractional derivativesby applying a space fractional telegraph di usion equation This equation interpolates between the secondand the fourth order anisotropic di usion equations using spatial fractional derivatives We represent awell posedness theorem related to the existence uniqueness and stability of the solution of the model Anumerical method based on the nite di erence method and fast Fourier transform is used for solving theproblem and denoising some images Some numerical tests in comparison with the literature imply that thespace fractional telegraph di usion equation can be considered as a valuable tool for image denoising

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معادله تلگراف كسري - مكاني براي كاهش نويز تصوير

معادلات با مشتقات جزيي (كسري) , تبديل فوريه , روش تفاضلات متناهي