Abstract Magnetic fields exhibit higher‐order, nonlinear singularities in the form of point‐dipole singularities. In addition, due to absence of divergence, they feature only a subset of invariant structures from traditional vector field topology. For magnetic fields of sets of point dipoles—widely present in physics and often used as an approximation—we present a technique revealing the topology of magnetic flux. The flux topology is identified with areas covered by field lines that directly connect pairs of dipoles. We introduce the dipole connectrix as a reduced one‐manifold representation of those areas. The set of connectrices serves as our concise visualization of the global structure of magnetic flux. In addition, the quantitative values of flux are displayed by the thickness of the connectrices. We evaluate our technique for simulations of ferroparticle monolayers and magnetic gels.

Abstract Continuous scatterplots and parallel coordinates are used to visualize multivariate data defined on a continuous domain. With the existing techniques, rendering such plots becomes prohibitively slow, especially for large scientific datasets. This paper presents a scalable and progressive rendering algorithm for continuous data plots that allows exploratory analysis of large datasets at interactive framerates. The algorithm employs splatting to produce a series of plots that are combined using alpha blending to achieve a progressively improving image. For each individual frame, splats are obtained by transforming Gaussian density kernels from the 3‐D domain of the input dataset to the respective data domain. A closed‐form analytic description of the resulting splat footprints is derived to allow pre‐computation of splat textures for efficient GPU rendering. The plotting method is versatile because it supports arbitrary reconstruction or interpolation schemes for the input data and the splatting technique is scalable because it chooses splat samples independently from the size of the input dataset. Finally, the effectiveness of the method is compared to existing techniques regarding rendering performance and quality.

Abstract We extend the rendering technique for continuous scatterplots to allow for a broad class of interpolation methods within the spatial grid instead of only linear interpolation. To do this, we propose an approach that projects the image of a cell from the spatial domain to the scatterplot domain. We approximate this image using either the convex hull or an axis‐aligned rectangle that forms a tight fit of the projected points. In both cases, the approach relies on subdivision in the spatial domain to control the approximation error introduced in the scatterplot domain. Acceleration of this algorithm in homogeneous regions of the spatial domain is achieved using an octree hierarchy. The algorithm is scalable and adaptive since it allows us to balance computation time and scatterplot quality. We evaluate and discuss the results with respect to accuracy and computational speed. Our methods are applied to examples of 2‐D transfer function design.

This paper introduces orthogonal vector field visualization on 2D manifolds: a representation by lines that are perpendicular to the input vector field. Line patterns are generated by line integral convolution (LIC). This visualization is combined with animation based on motion along the vector field. This decoupling of the line direction from the direction of animation allows us to choose the spatial frequencies along the direction of motion independently from the length scales along the LIC line patterns. Vision research indicates that local motion detectors are tuned to certain spatial frequencies of textures, and the above decoupling enables us to generate spatial frequencies optimized for motion perception. Furthermore, we introduce a combined visualization that employs orthogonal LIC patterns together with conventional, tangential streamline LIC patterns in order to benefit from the advantages of these two visualization approaches. In addition, a filtering process is described to achieve a consistent and temporally coherent animation of orthogonal vector field visualization. Different filter kernels and filter methods are compared and discussed in terms of visualization quality and speed. We present respective visualization algorithms for 2D planar vector fields and tangential vector fields on curved surfaces, and demonstrate that those algorithms lend themselves to efficient and interactive GPU implementations.

Scatterplots are well established means of visualizing discrete data values with two data variables as a collection of discrete points. We aim at generalizing the concept of scatterplots to the visualization of spatially continuous input data by a continuous and dense plot. An example of a continuous input field is data defined on an n-D spatial grid with respective interpolation or reconstruction of in-between values. We propose a rigorous, accurate, and generic mathematical model of continuous scatterplots that considers an arbitrary density defined on an input field on an n-D domain and that maps this density to m-D scatterplots. Special cases are derived from this generic model and discussed in detail: scatterplots where the n-D spatial domain and the m-D data attribute domain have identical dimension, 1-D scatterplots as a way to define continuous histograms, and 2-D scatterplots of data on 3-D spatial grids. We show how continuous histograms are related to traditional discrete histograms and to the histograms of isosurface statistics. Based on the mathematical model of continuous scatterplots, respective visualization algorithms are derived, in particular for 2-D scatterplots of data from 3-D tetrahedral grids. For several visualization tasks, we show the applicability of continuous scatterplots. Since continuous scatterplots do not only sample data at grid points but interpolate data values within cells, a dense and complete visualization of the data set is achieved that scales well with increasing data set size. Especially for irregular grids with varying cell size, improved results are obtained when compared to conventional scatterplots. Therefore, continuous scatterplots are a suitable extension of a statistics visualization technique to be applied to typical data from scientific computation.

One challenge in video processing is to detect actions and events, known or unknown, in video streams dynamically. This paper proposes a visualization solution, where a video stream is depicted as a series of snapshots at a relatively sparse interval, and detected actions are highlighted with continuous abstract illustrations. The combined imagery and illustrative visualization conveys multi-field information in a manner similar to electrocardiograms (ECG) and seismographs. We thus name this type of video visualization as VideoPerpetuoGram (VPG). In this paper, we describe a system that handles the aw and processed information of the video stream in a multi-field visualization pipeline. As examples, we consider the needs for highlighting several types of processed information, including detected actions in video streams, and estimated relationship between recognized objects. We examine the effective means for depicting multi-field information in VPG, and support our choice of visual mappings through a survey. Our GPU implementation facilitates the VPG-specific viewing specification through a sheared object space, as well as volume bricking and combinational rendering of volume data and glyphs.