Locally Analytic Schemes for Diffusion Filtering of Images

Nonlinear diffusion filtering has proven its value as a versatile tool for structure-preserving image denoising. Among the most interesting methods of this class are tensor-driven anisotropic diffusion as well as singular isotropic diffusion filters like total variation flow. For different reasons, devising good numerical algorithms for these filters is challenging. A spatial discretisation transforms nonlinear diffusion partial differential equations into systems of ordinary differential equations. Their investigation yields insights into the properties of diffusion-based algorithms but leads also to the design of new algorithms with favourable stability properties which are at the same time simple to implement. Moreover, interesting links to wavelet-based denoising methods are established in this way. The talk focusses on the construction and properties of locally (semi-)analytic schemes for nonlinear isotropic and anisotropic diffusion on 2D images, with extensions to the 3D case.