Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad-hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting ("probabilistic quadrics") where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system - 50x faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop-in replacement which we demonstrate with mesh smoothing via filtered quadrics and non-linear subdivision surfaces.
We present a city reconstruction and visualization framework that integrates geometric models reconstructed with a range of different techniques. The framework generates the vast majority of buildings procedurally, which yields plausible visualizations for structurally simple buildings, e.g. residential buildings. For structurally complex landmarks, e.g. churches, a procedural approach does not achieve satisfactory visual fidelity. Thus, we also employ image-based techniques to reconstruct the latter in a more realistic, recognizable way. As the manual acquisition of data required for the procedural and image-based reconstructions is practically infeasible for whole cities, we rely on publicly available data as well as crowd sourcing projects. This enables our framework to render views from cities without any dedicated data acquisition as long as there are sufficient public data sources available. To obtain a more lively impression of a city, we also visualize dynamic features like weather conditions and traffic based on publicly available real-time data.