Vector field topology has been successfully applied to represent the structure of steady vector fields. Critical points, one of the essential components of vector field topology, play an important role in describing the complexity of the extracted structure. Simplifying vector fields via critical point cancellation has practical merit for interpreting the behaviors of complex vector fields such as turbulence. However, there is no effective technique that allows direct cancellation of critical points in 3D. This work fills this gap and introduces the first framework to directly cancel pairs or groups of 3D critical points in a hierarchical manner with a guaranteed minimum amount of perturbation based on their robustness, a quantitative measure of their stability. In addition, our framework does not require the extraction of the entire 3D topology, which contains non-trivial separation structures, and thus is computationally effective. Furthermore, our algorithm can remove critical points in any subregion of the domain whose degree is zero and handle complex boundary configurations, making it capable of addressing challenging scenarios that may not be resolved otherwise. We apply our method to synthetic and simulation datasets to demonstrate its effectiveness.
Critical Point Cancellation in 3D Vector Fields: Robustness and Discussion
P Skraba, P Rosen, B Wang, G Chen, H Bhatia, V Pascucci
Transactions on Visualization and Computer Graphics
Three dimensional printing has gained considerable interest lately due to the proliferation of inexpensive devices as well as open source software that drive those devices. Public interest is often followed by media coverage that tends to sensationalize technology. Based on popular articles, the public may create the impression that 3D printing is the Holy Grail; we are going to print everything as one piece, traditional manufacturing is at the brink of collapse, and exotic applications, such as cloning a human body by 3D bio-printing, are just around the corner. The purpose of this paper is to paint a more realistic picture by identifying ten challenges that clearly illustrate the limitations of this technology, which makes it just as vulnerable as anything else that had been touted before as the next game changer.
Ten challenges in 3D printing
W Oropallo, LA Piegl
Engineering with Computers 32 (1), 135-148
Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.
Robustness-based simplification of 2d steady and unsteady vector fields
P Skraba, B Wang, G Chen, P Rosen
IEEE transactions on visualization and computer graphics 21 (8), 930-944