Image inpainting with the complex Ginzburg-Landau equation

The topic of this thesis is a mathematical technique for image inpainting based on the Ginzburg-Landau equation (GL). This approach was first presented by Grossauer and Scherzer. The Ginzburg-Landau equation is a classical equation in the field of electromagnetism, introduced by V.L. Ginzburg and L.D. Landau in 1950. At first the Ginzburg-Landau energy is discussed, and then it is shown how to obtain the GL equation by minimizing this energy. This is followed by a brief discussion of the mathematical properties of the GL equation. It is then shown how to apply the real and complex GL equation to inpainting. Finite differences are employed to discretize the equation, and the stability and consistency of an explicit and an implicit scheme is analyzed. A comparison between the GL-based method and the one introduced by Bertalmio, Sapiro, Caselles and Ballester concludes the presentation and indicates the GL approach is superior.