We present a framework for processing point-based surfaces via partial differential equations (PDEs). Our framework allows an efficient and effective way to bring well-established PDE-based surface processing techniques to the field of point-based representations. We demonstrate the method by a PDE-based surface fairing application

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.

The stable local classification of discrete surfaces with respect to features such as edges and corners or concave and convex regions, respectively, is as quite difficult as well as indispensable for many surface processing applications. Usually, the feature detection is done via a local curvature analysis. If concerned with large triangular and irregular grids, e.g., generated via a marching cube algorithm, the detectors are tedious to treat and a robust classification is hard to achieve. Here, a local classification method on surfaces is presented which avoids the evaluation of discretized curvature quantities. Moreover, it provides an indicator for smoothness of a given discrete surface and comes together with a built-in multiscale. The proposed classification tool is based on local zero and first moments on the discrete surface. The corresponding integral quantities are stable to compute and they give less noisy results compared to discrete curvature quantities. The stencil width for the integration of the moments turns out to be the scale parameter. Prospective surface processing applications are the segmentation on surfaces, surface comparison, and matching and surface modeling. Here, a method for feature preserving fairing of surfaces is discussed to underline the applicability of the presented approach.

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.