Eurographics / IEEE Symposium on Visualization 2011 (EuroVis 2011) H. Hauser, H. Pfister, and J. J. van Wijk (Guest Editors) Volume 30 (2011), Number 3 Efficient Parallel Vectors Feature Extraction from Higher-Order Data C. Pagot 1 , D. Osmari 1 , F. Sadlo 2 , D. Weiskopf 2 , T. Ertl 2 , J. Comba 1 1 Instituto de Informática, UFRGS, Brazil 2 VISUS, Universität Stuttgart, Germany Abstract The parallel vectors (PV) operator is a feature extraction approach for defining line-type features such as creases (ridges and valleys) in scalar fields, as well as separation, attachment, and vortex core lines in vector fields. In this work, we extend PV feature extraction to higher-order data represented by piecewise analytical functions defined over grid cells. The extraction uses PV in two distinct stages. First, seed points on the feature lines are placed by evaluating the inclusion form of the PV criterion with reduced affine arithmetic. Second, a feature flow field is derived from the higher-order PV expression where the features can be extracted as streamlines starting at the seeds. Our approach allows for guaranteed bounds regarding accuracy with respect to existence, position, and topology of the features obtained. The method is suitable for parallel implementation and we present results obtained with our GPU-based prototype. We apply our method to higher-order data obtained from discontinuous Galerkin fluid simulations. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation—Line and curve generation 1. Introduction Feature extraction is becoming increasingly important in scientific visualization for capturing meaningful structures out of large and intricate scalar and vector fields [PVH ∗ 03, LHZP07]. Extending feature extraction methods to other data representations is an active area of research. One ex- ample is higher-order data generated by newer discretization schemes such as discontinuous Galerkin methods [CKS00, GLM08]. In these methods, the solution is represented by piecewise analytic basis functions, often polynomials over grid cells. These methods have recently drawn great atten- tion in the simulation community due to their capability to generate accurate results through less refined grids, and high applicability for parallelization. Most visualization and analysis approaches handle such data indirectly by resam- pling into lower degree approximations that can be pro- cessed by existing feature extraction techniques. However, this approach incurs severe drawbacks in terms of accuracy and efficiency. To fill this gap, we introduce a method to efficiently ex- tract line-type features from higher-order data based on two concepts: the parallel vectors (PV ) operator [PR99], which defines line-type features as the loci where two (derived) vector fields become parallel or anti-parallel, and the fea- ture flow field (FFF)[TS03], a derived vector field where features are represented as streamlines. In the original PV method, features are extracted from trilinearly interpolated data by finding intersection points with the faces of grid cells that later are connected by straight line segments. This method is local (solutions are found per cell), robust, and comparably fast. However, it might not be accurate enough since it approximates features by straight segments and suf- fers from topological ambiguity problems when connecting more than two intersections per cell. In contrast, the origi- nal FFF method provides a more accurate and smooth fea- ture extraction. However, FFF streamlines are typically C 0 continuous at cell boundaries [TS03], seed points for ob- taining features are not trivial to find in general, and crit- ical points may emerge, imposing problems during feature integration. The FFF has been applied to PV feature extrac- tion [TSW ∗ 05] using a subdivision method for finding seeds per cell in trilinearly interpolated data. Our approach can be seen as an extension of [TSW ∗ 05] to higher-order data. For efficient feature extraction, seed re- c  2012 The Author(s) Journal compilation c 2012 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.