FLOW VISUALIZATION WITH QUANTIFIED SPATIAL AND TEMPORAL ERRORS USING EDGE MAPS, VOL. X, NO. Y, ZZZZ 2011 1 Flow Visualization with Quantified Spatial and Temporal Errors using Edge Maps Harsh Bhatia, Student Member, IEEE, Shreeraj Jadhav, Student Member, IEEE, Peer-Timo Bremer, Member, IEEE, Guoning Chen, Member, IEEE, Joshua A. Levine, Member, IEEE, Luis Gustavo Nonato, Member, IEEE and Valerio Pascucci, Member, IEEE Abstract—Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures. Index Terms—Vector Fields, Error Quantification, Edge Maps. ✦ 1 MOTIVATIONS V ECTOR fields are a common form of simulation data appearing in a wide variety of applications ranging from computational fluid dynamics (CFD) and weather prediction to engineering design. Visualizing and analyzing the flow behavior of these fields can help provide critical insights into simulated physical processes. However, achieving a consistent and rigorous interpretation of vector fields is difficult, in part because traditional numerical techniques for integration do not preserve the expected invariants of vector fields. To better understand this challenge inherent in traditional numerical techniques, we reconsider the most common way to store vector fields. Both a discretization of the domain of the field (often in the form of a triangulated mesh) as well as a set of sample vectors (defined at the vertices of the mesh) are required. The vector field on the interior of a triangle is approximated by interpolating vector values from the samples at the triangle’s corners. Subsequently, computing properties that require integrating these vector values presents a significant computational challenge. For example, consider computing the flow paths (streamlines) of massless particles that travel using the instantaneous velocity defined by the field. Naive integration techniques may violate the property that every two of these paths are expected to be pairwise disjoint • H. Bhatia, S. Jadhav, G. Chen, J.A. Levine and V. Pascucci are with the SCI Institute, University of Utah, USA. Emails: {hbhatia, jadhav, chengu, jlevine, pascucci}@sci.utah.edu • P-T Bremer is with Lawrence Livermore National Lab, CA, USA. Email: bremer5@llnl.gov • L.G. Nonato is with Universidade de S˜ ao Paulo, Brazil. Email: gnonato@icmc.usp.br (i.e. the uniqueness of the solution of an ordinary differential equation). Fig. 1 gives one such example, where a fourth- order Runge-Kutta integration technique creates two crossing streamlines. Fig. 1. Left: Two streamlines are seeded traveling clock- wise around this sink (red ball) in a domain [−1, 1] × [−1, 1]. Right bottom: Initially, the magenta streamline is seeded outside of the blue streamline with respect to the center of the domain. Right top: After integration with a step size of 0.025 the streamlines cross, now the magenta streamline is inside the blue streamline with respect to the center. Despite these problems, many of the standard techniques used for vector fields rely on variants of Runge-Kutta methods. Consequently, robustly computing flow becomes a formidable task. Integration is confounded by numerical errors at each Digital Object Indentifier 10.1109/TVCG.2011.265 1077-2626/11/$26.00 © 2011 IEEE IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.