In flow visualization, vortex extraction is a long-standing and unsolved problem. For decades, scientists developed numerous definitions that characterize vortex regions and their corelines in different ways, but none emerged as ultimate solution. One reason is that almost all techniques have a fundamental weakness: they are not invariant under changes of the reference frame, i.e., they are not objective. This has two severe implications: First, the result depends on the movement of the observer, and second, they cannot track vortices that are moving on arbitrary paths, which limits their reliability and usefulness in practice. Objective measures are rare, but recently gained more attention in the literature. Instead of only introducing a new objective measure, we show in this paper how all existing measures that are based on velocity and its derivatives can be made objective. We achieve this by observing the vector field in optimal local reference frames, in which the temporal derivative of the flow vanishes, i.e., reference frames in which the flow appears steady. The central contribution of our paper is to show that these optimal local reference frames can be found by a simple and elegant linear optimization. We prove that in the optimal frame, all local vortex extraction methods that are based on velocity and its derivatives become objective. We demonstrate our approach with objective counterparts to λ 2 , vorticity and Sujudi-Haimes.

For the visualization of dense line fields, the careful selection of lines to be rendered is a vital aspect. In this paper, we present a global line selection approach that is based on an optimization process. Starting with an initial set of lines that covers the domain, all lines are rendered with a varying opacity, which is subject to the minimization of a bounded-variable least-squares problem. The optimization strives to keep a balance between information presentation and occlusion avoidance. This way, we obtain view-dependent opacities of the line segments, allowing a real-time free navigation while minimizing the danger of missing important structures in the visualization. We compare our technique with existing local and greedy approaches and apply it to data sets in flow visualization, medical imaging, physics, and computer graphics.

Vortices are commonly understood as rotating motions in fluid flows. The analysis of vortices plays an important role in numerous scientific applications, such as in engineering, meteorology, oceanology, medicine and many more. The successful analysis consists of three steps: vortex definition, extraction and visualization. All three have a long history, and the early themes and topics from the 1970s survived to this day, namely, the identification of vortex cores, their extent and the choice of suitable reference frames. This paper provides an overview over the advances that have been made in the last 40 years. We provide sufficient background on differential vector field calculus, extraction techniques like critical point search and the parallel vectors operator, and we introduce the notion of reference frame invariance. We explain the most important region‐based and line‐based methods, integration‐based and geometry‐based approaches, recent objective techniques, the selection of reference frames by means of flow decompositions, as well as a recent local optimization‐based technique. We point out relationships between the various approaches, classify the literature and identify open problems and challenges for future work.

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Figure 1: Our method splits the previous opacity optimization technique [GRT13, GSM * 14] into two smaller problems, which accelerates the optimization and allows us to combine different geometry types (points, lines and surfaces) in a single unified framework. Compared to previous work our method is completely GPU-based, runs the optimization per pixel, and has view-independent parameters. Left: atmospheric trace gas pathways in an air flow provided by the European Centre for Medium-Range Weather Forecasts (ECMWF) and right: streamlines and rain clouds (isosurfaces) at the boundary between troposphere and stratosphere in the cloud-topped boundary layer (CTBL) flow. AbstractDisplaying geometry in flow visualization is often accompanied by occlusion problems, making it difficult to perceive information that is relevant in the respective application. In a recent technique, named opacity optimization, the balance of occlusion avoidance and the selection of meaningful geometry was recognized to be a view-dependent, global optimization problem. The method solves a bounded-variable least-squares problem, which minimizes energy terms for the reduction of occlusion, background clutter, adding smoothness and regularization. The original technique operates on an object-space discretization and was shown for line and surface geometry. Recently, it has been extended to volumes, where it was solved locally per ray by dropping the smoothness energy term and replacing it by pre-filtering the importance measure. In this paper, we pick up the idea of splitting the opacity optimization problem into two smaller problems. The first problem is a minimization with analytic solution, and the second problem is a smoothing of the obtained minimizer in object-space. Thereby, the minimization problem can be solved locally per pixel, making it possible to combine all geometry types (points, lines and surfaces) consistently in a single optimization framework. We call this decoupled opacity optimization and apply it to a number of steady 3D vector fields.