Abstract-This paper presents a method for filtered ridge extraction based on adaptive mesh refinement. It is applicable in situations where the underlying scalar field can be refined during ridge extraction. This requirement is met by the concept of Lagrangian coherent structures which is based on trajectories started at arbitrary sampling grids that are independent of the underlying vector field. The Lagrangian coherent structures are extracted as ridges in finite Lyapunov exponent fields computed from these grids of trajectories. The method is applied to several variants of finite Lyapunov exponents, one of which is newly introduced. High computation time due to the high number of required trajectories is a main drawback when computing Lyapunov exponents of 3-dimensional vector fields. The presented method allows a substantial speed-up by avoiding the seeding of trajectories in regions where no ridges are present or do not satisfy the prescribed filter criteria such as a minimum finite Lyapunov exponent.

In this paper we present a scalable 3D video framework for capturing and rendering dynamic scenes. The acquisition system is based on multiple sparsely placed 3D video bricks, each comprising a projector, two grayscale cameras, and a color camera. Relying on structured light with complementary patterns, texture images and pattern-augmented views of the scene are acquired simultaneously by time-multiplexed projections and synchronized camera exposures. Using space-time stereo on the acquired pattern images, high-quality depth maps are extracted, whose corresponding surface samples are merged into a view-independent, point-based 3D data structure. This representation allows for effective photo-consistency enforcement and outlier removal, leading to a significant decrease of visual artifacts and a high resulting rendering quality using EWA volume splatting. Our framework and its view-independent representation allow for simple and straightforward editing of 3D video. In order to demonstrate its flexibility, we show compositing techniques and spatiotemporal effects.

It was shown recently how the 2D vector field topology concept, directly applicable to stationary vector fields only, can be generalized to time-dependent vector fields by replacing the role of stream lines by streak lines. The present paper extends this concept to 3D vector fields. In traditional 3D vector field topology separatrices can be obtained by integrating stream lines from 0D seeds corresponding to critical points. We show that in our new concept, in contrast, 1D seeding constructs are required for computing streak-based separatrices. In analogy to the 2D generalization we show that invariant manifolds can be obtained by seeding streak surfaces along distinguished path surfaces emanating from intersection curves between codimension-1 ridges in the forward and reverse finite-time Lyapunov exponent (FTLE) fields. These path surfaces represent a time-dependent generalization of critical points and convey further structure in time-dependent topology of vector fields. Compared to the traditional approach based on FTLE ridges, the resulting streak manifolds ease the analysis of Lagrangian coherent structures (LCS) with respect to visual quality and computational cost, especially when time series of LCS are computed. We exemplify validity and utility of the new approach using both synthetic examples and computational fluid dynamics results.

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics.

Motivated by the growing interest in the use of ridges in scientific visualization, we analyze the two height ridge definitions by Eberly and Lindeberg. We propose a raw feature definition leading to a superset of the ridge points as obtained by these two definitions. The set of raw feature points has the correct dimensionality, and it can be narrowed down to either Eberly's or Lindeberg's ridges by using Boolean filters which we formulate. While the straight-forward computation of height ridges requires explicit eigenvalue calculation, this can be avoided by using an equivalent definition of the raw feature set, for which we give a derivation. We describe efficient algorithms for two special cases, height ridges of dimension one and of co-dimension one. As an alternative to the aforementioned filters, we propose a new criterion for filtering raw features based on the distance between contours which generally makes better decisions, as we demonstrate on a few synthetic fields, a topographical dataset, and a fluid flow simulation dataset. The same set of test data shows that it is unavoidable to use further filters to eliminate false positives. For this purpose, we use the angle between feature tangent and slope line as a quality measure and, based on this, formalize a previously published filter.

Jet-streams, their core lines and their role in atmospheric dynamics have been subject to considerable meteorological research since the first half of the twentieth century. Yet, until today no consistent automated feature detection approach has been proposed to identify jet-stream core lines from 3D wind fields. Such 3D core lines can facilitate meteorological analyses previously not possible. Although jet-stream cores can be manually analyzed by meteorologists in 2D as height ridges in the wind speed field, to the best of our knowledge no automated ridge detection approach has been applied to jet-stream core detection. In this work, we -a team of visualization scientists and meteorologists-propose a method that exploits directional information in the wind field to extract core lines in a robust and numerically less involved manner than traditional 3D ridge detection. For the first time, we apply the extracted 3D core lines to meteorological analysis, considering real-world case studies and demonstrating our method's benefits for weather forecasting and meteorological research.

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This paper presents an acceleration scheme for the numerical computation of sets of trajectories in vector fields or iterated solutions in maps, possibly with simultaneous evaluation of quantities along the curves such as integrals or extrema. It addresses cases with a dense evaluation on the domain, where straightforward approaches are subject to redundant calculations. These are avoided by first calculating short solutions for the whole domain. From these, longer solutions are then constructed in a hierarchical manner until the designated length is achieved. While the computational complexity of the straightforward approach depends linearly on the length of the solutions, the computational cost with the proposed scheme grows only logarithmically with increasing length. Due to independence of subtasks and memory locality, our algorithm is suitable for parallel execution on many-core architectures like GPUs. The trade-offs of the method--lower accuracy and increased memory consumption--are analyzed, including error order as well as numerical error for discrete computation grids. The usefulness and flexibility of the scheme are demonstrated with two example applications: line integral convolution and the computation of the finite-time Lyapunov exponent. Finally, results and performance measurements of our GPU implementation are presented for both synthetic and simulated vector fields from computational fluid dynamics.