Geometric Surface Processing via Normal Mapsa

The generalization of signal and image processing to surfaces entails filtering the normals of the surface, rather than filtering the positions of points on a mesh. Using a variational framework, smooth surfaces minimize the norm of the derivative of the surface normals— i.e. total curvature. Penalty functions on the surface normals are computed using geometrybased shape metrics and minimized using gradient descent. This produces a set of partial differential equations (PDE). In this paper, we introduce a novel framework for implementing geometric processing tools for surfaces using a two step algorithm: (i) operating on the normal map of a surface, and (ii) manipulating the surface to fit the processed normals. The computational approach uses level set surface models; therefore, the processing does not depend on any underlying parameterization. Iterating this two-step process, we can implement geometric fourth-order flows efficiently by solving a set of coupled second-order PDEs. This paper will demonstrate that the framework provides for a wide range of surface processing operations, including edge-preserving smoothing and high-boost filtering. Furthermore, the generality of the implementation makes it appropriate for very complex surface models, e.g. those constructed directly from measured data.