Variational Numerical Methods for Solving Nonlinear Diffusion Equations Arising in Image Processing

In this paper we give a general, robust, and efficient approach for numerical solu-tions of partial differential equations (PDEs) arising in image processing and com-puter vision. The well-established variational computational techniques, namely, finite element, finite volume, and complementary volume methods, are introduced on a common base to solve nonlinear problems in image multiscale analysis. Since they are based on principles like minimization of energy (finite element method) or conservation laws (finite and complemetary volume methods), they have strong physical backgrounds. They allow clear and physically meaningful derivation of dif-ference equations that are local and easy to implement. The variational methods are combined with semi-implicit discretization in scale, which gives favorable stability and efficiency properties of computations. We show here L∞-stability without any restrictions on scale steps. Our approach leads finally to solving linear systems in every discrete scale level, which can be done efficiently by fast preconditioned it-erative solvers. We discuss such computational schemes for the regularized (in th