Surface approximation with elevation maps and observations thinning in numerical weather prediction

The rapidly increasing amount of 3D content that is generated through geometry acquisition, simulation, and modeling tools, demands methods that produce efficient data representations. This thesis is concerned with the approximation of two different kinds of densely sampled data sets. Firstly, surfaces from geometry processing in computer graphics, and secondly, observational data in numerical weather prediction. The first part of the thesis introduces a generic framework for approximating densely sampled 3D models by using an image-based data structure. In our approach, we decompose a given surface into a set of patches that are parameterized as height fields over planar domains and resampled on regular grids. The resulting surface representation relies on a set of irregularly shaped images. Our framework differs vastly from previous approaches in that it introduces a model representation, which is independent of the underlying surface primitive. Both, the input model as well as the output model can be processed and reconstructed as a mesh or as a point set. This approach offers a variety of applications. The first application of the elevation map representation addresses an elementary problem in geometry processing that is the efficient compression of 3D content for storage and transmission. Here, a surface encoder transforms an explicit surface representation into a compact bit-stream, which is then decoded at the receiver to generate a surface reconstruction. We adopt this procedure to establish a surface compression method that relies on the elevation map representation. Our results show that our method outperforms current point-based compression methods and provides competitive results in comparison to current mesh compression systems. Moreover, our elementary parameterization yields an implementation with a simple algorithmic design and fast encoding and decoding behavior. In the second application, we introduce a novel real-time rendering pipeline that relies on components of current point and mesh rendering methods to improve the performance of model rendering. Utilizing our elevation maps, we propose to use a single base data set to render each patch in the common vertex and fragment shader pipeline. We show the benefits of this method for point rendering by replacing attribute blending through a simplified and fast attribute interpolation. Compared to previous approaches, we achieve faster renderings and a better visual quality. In the last part of the thesis, we focus on the problem of simplification of observations in numerical weather prediction, where measurements of various observation systems are combined with background data to define the initial states for the forecasts. Current satellite instruments produce large amounts of observations with high spatial and temporal density. Moreover, the availability of future meteorological measurement systems induces a significant increase in the amount and complexity of the data. We propose two thinning algorithms using simplification methods from geometry processing in computer graphics and clustering algorithms, which produce reduced data sets as approximations of the original data. We apply the two methods to ATOVS satellite data, which are processed to retrieve profiles of atmospheric temperature and humidity. These profiles are input data for the meteorological forecasts at the German Weather Service (Deutscher Wetter Dienst, DWD).