Visualizing and analyzing 3D unsteady flow fields is a very challenging task. We approach this problem by leveraging the mathematical foundations of 3D observer fields to explore and analyze 3D flows in reference frames that are more suitable to visual analysis than the input reference frame. We design novel interactive tools for determining, filtering, and combining reference frames for observer-aware 3D unsteady flow visualization. We represent the space of reference frame motions in a 3D spatial domain via a 6D parameter space, in which every observer is a time-dependent curve. Our framework supports operations in this 6D observer space by separately focusing on two 3D subspaces, for 3D translations, and 3D rotations, respectively. We show that this approach facilitates a variety of interactions with 3D flow fields. Building on the interactive selection of observers, we furthermore introduce novel techniques such as observer-aware streamline- and pathline-filtering as well as observer-aware isosurface animations of scalar fluid properties for the enhanced visualization and analysis of 3D unsteady flows. We discuss the theoretical underpinnings as well as practical implementation considerations of our approach, and demonstrate the benefits of its 6+1D observer-based methodology on several 3D unsteady flow datasets.

Abstract We present a novel technique for the extraction, visualization, and analysis of material boundaries and Lagrangian coherent structures (LCS) in 2D unsteady flow fields relative to local reference frame transformations. In addition to the input flow field, we leverage existing methods for computing reference frames adapted to local fluid features, in particular those that minimize the observed time derivative. Although, by definition, transforming objective tensor fields between reference frames does not change the tensor field, we show that transforming objective tensors, such as the finite‐time Lyapunov exponent (FTLE) or Lagrangian‐averaged vorticity deviation (LAVD), or the second‐order rate‐of‐strain tensor, into local reference frames that are naturally adapted to coherent fluid structures has several advantages: (1) The transformed fields enable analyzing LCS in space‐time visualizations that are adapted to each structure; (2) They facilitate extracting geometric features, such as iso‐surfaces and ridge lines, in a straightforward manner with high accuracy. The resulting visualizations are characterized by lower geometric complexity and enhanced topological fidelity. To demonstrate the effectiveness of our technique, we measure geometric complexity and compare it with iso‐surfaces extracted in the conventional reference frame. We show that the decreased geometric complexity of the iso‐surfaces in the local reference frame, not only leads to improved geometric and topological results, but also to a decrease in computation time.

This paper describes a novel method for detecting and visualizing vortex structures in unsteady 2D fluid flows. The method is based on an interactive local reference frame estimation that minimizes the observed time derivative of the input flow field v(x,t). A locally optimal reference frame w($x,t$) assists the user in the identification of physically observable vortex structures inObserved Line Integral Convolution(LIC) visualizations. The observed LIC visualizations are interactively computed and displayed in a user-steered vortex lens region, embedded in the context of a conventional LIC visualization outside the lens. The locally optimal reference frame is then used to detect observed critical points, where v = w, which are used to seed vortex core lines. Each vortex core line is computed as a solution of the ordinary differential equation (ODE) ˙$w(t)=w(w(t),t)$, with an observed critical point as initial condition$(w(t_0),t_0)$. During integration, we enforce a strict error bound on the difference between the extracted core line and the integration of a path line of the input vector field, i.e., a solution to the ODE ˙$v(t)=v(v(t),t)$. We experimentally verify that this error depends on the step size of the core line integration. This ensures that our method extracts Lagrangian vortex core lines that are the simultaneous solution of both ODEs with a numerical error that is controllable by the integration step size. We show the usability of our method in the context of an interactive system using a lens metaphor, and evaluate the results in comparison to state-of-the-art vortex core line extraction methods

State-of-the-art computation and visualization of vortices in unsteady fluid flow employ objective vortex criteria, which makes them independent of reference frames or observers. However, objectivity by itself, although crucial, is not sufficient to guarantee that one can identify physically-realizable observers that would perceive or detect the same vortices. Moreover, a significant challenge is that a single reference frame is often not sufficient to accurately observe multiple vortices that follow different motions. This paper presents a novel framework for the exploration and use of an interactively-chosen set of observers, of the resulting relative velocity fields, and of objective vortex structures. We show that our approach facilitates the objective detection and visualization of vortices relative to well-adapted reference frame motions, while at the same time guaranteeing that these observers are in fact physically realizable. In order to represent and manipulate observers efficiently, we make use of the low-dimensional vector space structure of the Lie algebra of physically-realizable observer motions. We illustrate that our framework facilitates the efficient choice and guided exploration of objective vortices in unsteady 2D flow, on planar as well as on spherical domains, using well-adapted reference frames.

Computing and visualizing features in fluid flow often depends on the observer, or reference frame, relative to which the input velocity field is given. A desired property of feature detectors is therefore that they are objective, meaning independent of the input reference frame. However, the standard definition of objectivity is only given for Euclidean domains and cannot be applied in curved spaces. We build on methods from mathematical physics and Riemannian geometry to generalize objectivity to curved spaces, using the powerful notion of symmetry groups as the basis for definition. From this, we develop a general mathematical framework for the objective computation of observer fields for curved spaces, relative to which other computed measures become objective. An important property of our framework is that it works intrinsically in 2D, instead of in the 3D ambient space. This enables a direct generalization of the 2D computation via optimization of observer fields in flat space to curved domains, without having to perform optimization in 3D. We specifically develop the case of unsteady 2D geophysical flows given on spheres, such as the Earth. Our observer fields in curved spaces then enable objective feature computation as well as the visualization of the time evolution of scalar and vector fields, such that the automatically computed reference frames follow moving structures like vortices in a way that makes them appear to be steady.

Flow fields are usually visualized relative to a global observer, i.e., a single frame of reference. However, often no global frame can depict all flow features equally well. Likewise, objective criteria for detecting features such as vortices often use either a global reference frame, or compute a separate frame for each point in space and time. We propose the first general framework that enables choosing a smooth trade-off between these two extremes. Using global optimization to minimize specific differential geometric properties, we compute a time-dependent observer velocity field that describes the motion of a continuous field of observers adapted to the input flow. This requires developing the novel notion of an observed time derivative. While individual observers are restricted to rigid motions, overall we compute an approximate Killing field, corresponding to almost-rigid motion. This enables continuous transitions between different observers. Instead of focusing only on flow features, we furthermore develop a novel general notion of visualizing how all observers jointly perceive the input field. This in fact requires introducing the concept of an observation time, with respect to which a visualization is computed. We develop the corresponding notions of observed stream, path, streak, and time lines. For efficiency, these characteristic curves can be computed using standard approaches, by first transforming the input field accordingly. Finally, we prove that the input flow perceived by the observer field is objective. This makes derived flow features, such as vortices, objective as well.

We report on using computed tomography (CT) as a model acquisition tool for complex objects in computer graphics. Unlike other modeling and scanning techniques the complexity of the object is irrelevant in CT, which naturally enables to model objects with, for example, concavities, holes, twists or fine surface details. Once the data is scanned, one can apply post-processing techniques for data enhancement, modification or presentation. For demonstration purposes we chose to scan a Christmas tree which exhibits high complexity which is difficult or even impossible to handle with other techniques. However, care has to be taken to achieve good scanning results with CT. Further, we illustrate post-processing by means of data segmentation and photorealistic as well as non-photorealistic surface and volume rendering techniques.

The classification of volumetric data sets as well as their rendering algorithms are typically based on the representation of the underlying grid. Grid structures based on a Cartesian lattice are the de-facto standard for regular representations of volumetric data. In this paper we introduce a more general concept of regular grids for the representation of volumetric data. We demonstrate that a specific type of regular lattice-the so-called body-centered cubic-is able to represent the same data set as a Cartesian grid to the same accuracy but with 29.3% fewer samples. This speeds up traditional volume rendering algorithms by the same ratio, which we demonstrate by adopting a splatting implementation for these new lattices. We investigate different filtering methods required for computing the normals on this lattice. The lattice representation results also in lossless compression ratios that are better than previously reported. Although other regular grid structures achieve the same sample efficiency, the body-centered cubic is particularly easy to use. The only assumption necessary is that the underlying volume is isotropic and band-limited-an assumption that is valid for most practical data sets.