A multiscale method in surface processing is presented which carries over image processing methodology based on nonlinear diffusion equations to the fairing of noisy, textured, parametric surfaces. The aim is to smooth noisy, triangulated surfaces and accompanying noisy textures-as they are delivered by new scanning technology-while enhancing geometric and texture features. For an initial textured surface a fairing method is described which simultaneously processes the texture and the surface. Considering an appropriate coupling of the two smoothing processes one can take advantage of the frequently present strong correlation between edge features in the texture and on the surface edges. The method is based on an anisotropic curvature evolution of the surface itself and an anisotropic diffusion on the processed surface applied to the texture. Here, the involved diffusion tensors depends on a regularized shape operator of the evolving surface and on regularized texture gradients. A spatial finite element discretization on arbitrary unstructured triangular grids and a semi-implicit finite difference discretization in time are the building blocks of the corresponding numerical algorithm. A normal projection is applied to the discrete propagation velocity to avoid tangential drifting in the surface evolution. Different applications underline the efficiency and flexibility of the presented surface processing tool.

A new multiscale method in surface processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth discretized surfaces while simultaneously enhancing geometric features such as edges and corners. This is obtained by an anisotropic curvature evolution, where time is the multiscale parameter. Here, the diffusion tensor depends on the shape operator of the evolving surface. A spatial finite element discretization on arbitrary unstructured triangular meshes and a semi-implicit finite difference discretization in time are the building blocks of the easy to code algorithm presented. The systems of linear equations in each timestep are solved by appropriate, preconditioned iterative solvers. Different applications underline the efficiency and flexibility of the presented type of surface processing tool.

Vector field visualization is an important topic in scientific visualization. Its aim is to graphically represent field data on two and three-dimensional domains and on surfaces in an intuitively understandable way. Here, a new approach based on anisotropic nonlinear diffusion is introduced. It enables an easy perception of vector field data and serves as an appropriate scale space method for the visualization of complicated flow pattern. The approach is closely related to nonlinear diffusion methods in image analysis where images are smoothed while still retaining and enhancing edges. Here, an initial noisy image intensity is smoothed along integral lines, whereas the image is sharpened in the orthogonal direction. The method is based on a continuous model and requires the solution of a parabolic PDE problem. It is discretized only in the final implementational step. Therefore, many important qualitative aspects can already be discussed on a continuous level. Applications are shown for flow fields in 2D and 3D, as well as for principal directions of curvature on general triangulated surfaces. Furthermore, the provisions for flow segmentation are outlined.