We generalize the framework of moments and introduce a definition of invariants for three-dimensional vector fields. To do so, we use the method of moment normalization that has been shown to be useful in the two dimensions. Using invariant moments, we show how to search for patterns in these fields independent from their position, orientation and scale. From the first order vector moment tensor, we construct a complete and independent set of descriptors. We test the invariants in queries on synthetic and real world flow fields.

This paper proposes a vector field visualization, which mimics a sketch-like representation. The visualization combines two major perspectives: Large scale trends based on a strongly simplified field as background visualization and a local visualization highlighting strongly expressed features at their exact position. Each component considers the vector field itself and its spatial derivatives. The derivate is an asymmetric tensor field, which allows the deduction of scalar quantities reflecting distinctive field properties like strength of rotation or shear. The basis of the background visualization is a vector and scalar clustering approach. The local features are defined as the extrema of the respective scalar fields. Applying scalar field topology provides a profound mathematical basis for the feature extraction. All design decisions are guided by the goal of generating a simple to read visualization. To demonstrate the effectiveness of our approach, we show results for three different data sets with different complexity and characteristics.

Linear approximation plays an important role in many areas employing numerical algorithms. Particularly in the field of vector field visualization, it is the basis of widely used techniques. In this paper, we introduce two methods to extract areas in two- and three-dimensional vector fields that are connected to linear flow behavior. We propose a region-growing algorithm that extracts the linear neighborhood for a certain position. The region is characterized by linear flow behavior up to a user-defined approximation threshold. While this first method computes the size of a region given the mentioned threshold, our second method computes the quality of a linear approximation given a user-defined n-ring neighborhood. The scalar field resulting from the second method is, therefore, called affine linear approximation error. Isosurfaces of this field show regions of close-to-linear and non-linear flow behavior. We demonstrate the expressiveness and discuss the properties of the extracted regions using analytical examples and several datasets from the domain of computational fluid dynamics (CFD).

Abstract Particle swarm optimization (PSO) is a metaheuristic that has been applied successfully to many continuous and combinatorial optimization problems, e.g., in the fields of economics, engineering, and natural sciences. In PSO, a swarm of particles moves within a search space in order to find an optimal solution. Unfortunately, it is hard to understand in detail why and how changes in the design of PSO algorithms affect the optimization behavior. Visualizing the particle states could provide substantially better insight into PSO algorithms. Though in case of combinatorial optimization problems, it often raises the problem of illustrating the states within the discrete search space that cannot be embedded spatially. We propose a visualization approach to depict the optimization problem topologically using a landscape metaphor. This visualization is augmented by an illustration of the time‐dependent states of the particles. Thus, the user of dPSO‐Vis is able to analyze the swarm's behavior within the search space. In principle, our method can be used for any optimization algorithm where a swarm of individuals searches within a discrete search space. Our approach is verified with a case study for the PSO algorithm HelixPSO that predicts the secondary structure of RNA molecules.

We propose a combinatorial algorithm to track critical points of 2D time-dependent scalar fields. Existing tracking algorithms such as Feature Flow Fields apply numerical schemes utilizing derivatives of the data, which makes them prone to noise and involve a large number of computational parameters. In contrast, our method is robust against noise since it does not require derivatives, interpolation, and numerical integration. Furthermore, we propose an importance measure that combines the spatial persistence of a critical point with its temporal evolution. This leads to a time-aware feature hierarchy, which allows us to discriminate important from spurious features. Our method requires only a single, easy-to-tune computational parameter and is naturally formulated in an out-of-core fashion, which enables the analysis of large data sets. We apply our method to synthetic data and data sets from computational fluid dynamics and compare it to the stabilized continuous Feature Flow Field tracking algorithm.

Acceleration is a fundamental quantity of flow fields that captures Galilean invariant properties of particle motion. Considering the magnitude of this field, minima represent characteristic structures of the flow that can be classified as saddle- or vortex-like. We made the interesting observation that vortex-like minima are enclosed by particularly pronounced ridges. This makes it possible to define boundaries of vortex regions in a parameter-free way. Utilizing scalar field topology, a robust algorithm can be designed to extract such boundaries. They can be arbitrarily shaped. An efficient tracking algorithm allows us to display the temporal evolution of vortices. Various vortex models are used to evaluate the method. We apply our method to two-dimensional model systems from computational fluid dynamics and compare the results to those arising from existing definitions.

This paper introduces a novel importance measure for critical points in 2D scalar fields. This measure is based on a combination of the deep structure of the scale space with the well-known concept of homological persistence. We enhance the noise robust persistence measure by implicitly taking the hill-, ridge- and outlier-like spatial extent of maxima and minima into account. This allows for the distinction between different types of extrema based on their persistence at multiple scales. Our importance measure can be computed efficiently in an out-of-core setting. To demonstrate the practical relevance of our method we apply it to a synthetic and a real-world data set and evaluate its performance and scalability.

Abstract We propose a new criterion to characterize hierarchical two‐dimensional vortex regions induced by swirling motion. Central to the definition are closed loops that intersect the flow field at a constant angle. The union of loops belonging to the same area of swirling motion defines a vortex region. These regions are disjunct but may be nested, thus introducing a spatial hierarchy of vortex regions. We present a parameter free algorithm for the identification of these regions. Since they are not restricted to star‐ or convex‐shaped geometries, we are able to identify also intricate regions, e.g., of elongated vortices. Computing an integrated value for each loop and mapping these values to a vortex region, introduces new ways for visualizing or filtering the vortex regions. Exemplary, an application based on the Rankine vortex model is presented. We apply our method to several CFD datasets and compare our results to existing approaches.