Smooth Surface Extraction from Unstructured Point-based Volume Data Using PDEs

Smooth surface extraction using PDEs is a well-known and widely used technique for visualizing volume data. Existing approaches operate on gridded data and mainly on regular structured grids. When considering unstructured point-based volume data where sample points do not form regular patterns nor are they connected in any form, one would typically resample the data over a grid prior to applying the known PDE-based methods. As resampling inserts interpolation inaccuracies, data providers would rather have segmentation methods operate on the actual unstructured data. We propose an approach that directly extracts smooth surfaces from unstructured point-based volume data without prior resampling or mesh generation.When operating on unstructured data one needs to quickly derive neighborhood informations. The respective information is retrieved by partitioning the 3D domain into cells using a kd-tree and operating on its cells. We exploit neighborhood information to estimate gradients and mean curvature at every sample point using a four-dimensional least-squares fitting approach. Gradients and mean curvature are required for applying the chosen PDE-based method that combines hyperbolic advection to an isovalue of a given scalar field and mean curvature flow. Since we are using an explicit time-integration scheme, time steps are bounded by the Courant-Friedrichs-Lewy condition. To avoid small global time steps, we use asynchronous local integration. We extract the surface by successively fitting a smooth function to the data set. This function is initialized with a signed distance function. For each sample and for every time step we compute the respective gradient, the mean curvature, and a stable time step. With these informations the function is manipulated using an explicit Euler time integration. The process continues with the next sample point in time. If the norm of the function gradient in a sample exceeds a given threshold at some time the function is reinitializ