Adaptive Sampling of 3D Spatial Correlations for Focus+Context Visualization.

Visualizing spatial correlations in 3D ensembles is challenging due to the vast amounts of information that need to be conveyed. Memory and time constraints make it unfeasible to pre-compute and store the correlations between all pairs of domain points. We propose the embedding of adaptive correlation sampling into chord diagrams with hierarchical edge bundling to alleviate these constraints. Entities representing spatial regions are arranged along the circular chord layout via a space-filling curve, and Bayesian optimal sampling is used to efficiently estimate the maximum occurring correlation between any two points from different regions. Hierarchical edge bundling reduces visual clutter and emphasizes the major correlation structures. By selecting an edge, the user triggers a focus diagram in which only the two regions connected via this edge are refined and arranged in a specific way in a second chord layout. For visualizing correlations between two different variables, which are not symmetric anymore, we switch to showing a full correlation matrix. This avoids drawing the same edges twice with different correlation values. We introduce GPU implementations of both linear and non-linear correlation measures to further reduce the time that is required to generate the context and focus views, and to even enable the analysis of correlations in a 1000-member ensemble.

Spatio-Temporal Visual Analysis of Turbulent Superstructures in Unsteady Flow.

The large-scale motions in 3D turbulent channel flows, known as Turbulent Superstructures (TSS), play an essential role in the dynamics of small-scale structures within the turbulent boundary layer. However, as of today, there is no common agreement on the spatial and temporal relationships between these multiscale structures. We propose a novel space-time visualization technique for analyzing the temporal evolution of these multiscale structures in their spatial context and, thus, to further shed light on the conceptually different explanations of their dynamics. Since the temporal dynamics of TSS are believed to influence the structures in the turbulent boundary layer, we propose a combination of a 2D space-time velocity plot with an orthogonal 2D plot of projected 3D flow structures, which can interactively span the time and the space axis. Besides flow structures indicating the fluid motion, we propose showing the variations in derived fields as an additional source of explanation. The relationships between the structures in different spatial and temporal scales can be more effectively resolved by using various filtering operations and image registration algorithms. To reduce the information loss due to the non-injective nature of projection, spatial information is encoded into transparency or color. Since the proposed visualization is heavily demanding computational resources and memory bandwidth to stream unsteady flow fields and instantly compute derived 3D flow structures, the implementation exploits data compression, parallel computation capabilities, and high memory bandwidth on recent GPUs via the CUDA compute library.

Learning Adaptive Sampling and Reconstruction for Volume Visualization.

A central challenge in data visualization is to understand which data samples are required to generate an image of a data set in which the relevant information is encoded. In this article, we make a first step towards answering the question of whether an artificial neural network can predict where to sample the data with higher or lower density, by learning of correspondences between the data, the sampling patterns and the generated images. We introduce a novel neural rendering pipeline, which is trained end-to-end to generate a sparse adaptive sampling structure from a given low-resolution input image, and reconstructs a high-resolution image from the sparse set of samples. For the first time, to the best of our knowledge, we demonstrate that the selection of structures that are relevant for the final visual representation can be jointly learned together with the reconstruction of this representation from these structures. Therefore, we introduce differentiable sampling and reconstruction stages, which can leverage back-propagation based on supervised losses solely on the final image. We shed light on the adaptive sampling patterns generated by the network pipeline and analyze its use for volume visualization including isosurface and direct volume rendering.

Visual Analysis of Multi-Parameter Distributions Across Ensembles of 3D Fields.

For an ensemble of 3D multi-parameter fields, we present a visual analytics workflow to analyse whether and which parts of a selected multi-parameter distribution is present in all ensemble members. Supported by a parallel coordinate plot, a multi-parameter brush is applied to all ensemble members to select data points with similar multi-parameter distribution. By a combination of spatial sub-division and a covariance analysis of partitioned sub-sets of data points, a tight partition in multi-parameter space with reduced number of selected data points is obtained. To assess the representativeness of the selected multi-parameter distribution across the ensemble, we propose a novel extension of violin plots that can show multiple parameter distributions simultaneously. We investigate the visual design that effectively conveys (dis-)similarities in multi-parameter distributions, and demonstrate that users can quickly comprehend parameter-specific differences regarding distribution shape and representativeness from a side-by-side view of these plots. In a 3D spatial view, users can analyse and compare the spatial distribution of selected data points in different ensemble members via interval-based isosurface raycasting. In two real-world application cases we show how our approach is used to analyse the multi-parameter distributions across an ensemble of 3D fields.

Differentiable Direct Volume Rendering.

We present a differentiable volume rendering solution that provides differentiability of all continuous parameters of the volume rendering process. This differentiable renderer is used to steer the parameters towards a setting with an optimal solution of a problem-specific objective function. We have tailored the approach to volume rendering by enforcing a constant memory footprint via analytic inversion of the blending functions. This makes it independent of the number of sampling steps through the volume and facilitates the consideration of small-scale changes. The approach forms the basis for automatic optimizations regarding external parameters of the rendering process and the volumetric density field itself. We demonstrate its use for automatic viewpoint selection using differentiable entropy as objective, and for optimizing a transfer function from rendered images of a given volume. Optimization of per-voxel densities is addressed in two different ways: First, we mimic inverse tomography and optimize a 3D density field from images using an absorption model. This simplification enables comparisons with algebraic reconstruction techniques and state-of-the-art differentiable path tracers. Second, we introduce a novel approach for tomographic reconstruction from images using an emission-absorption model with post-shading via an arbitrary transfer function.

Interactive Focus+Context Rendering for Hexahedral Mesh Inspection.

The visual inspection of a hexahedral mesh with respect to element quality is difficult due to clutter and occlusions that are produced when rendering all element faces or their edges simultaneously. Current approaches overcome this problem by using focus on specific elements that are then rendered opaque, and carving away all elements occluding their view. In this work, we make use of advanced GPU shader functionality to generate a focus+context rendering that highlights the elements in a selected region and simultaneously conveys the global mesh structure and deformation field. To achieve this, we propose a gradual transition from edge-based focus rendering to volumetric context rendering, by combining fragment shader-based edge and face rendering with per-pixel fragment lists. A fragment shader smoothly transitions between wireframe and face-based rendering, including focus-dependent rendering style and depth-dependent edge thickness and halos, and per-pixel fragment lists are used to blend fragments in correct visibility order. To maintain the global mesh structure in the context regions, we propose a new method to construct a sheet-based level-of-detail hierarchy and smoothly blend it with volumetric information. The user guides the exploration process by moving a lens-like hotspot. Since all operations are performed on the GPU, interactive frame rates are achieved even for large meshes.

A Comparison of Rendering Techniques for 3D Line Sets With Transparency.

This article presents a comprehensive study of rendering techniques for 3D line sets with transparency. The rendering of transparent lines is widely used for visualizing trajectories of tracer particles in flow fields. Transparency is then used to fade out lines deemed unimportant, based on, for instance, geometric properties or attributes defined along with them. Accurate blending of transparent lines requires rendering the lines in back-to-front or front-to-back order, yet enforcing this order for space-filling 3D line sets with extremely high-depth complexity becomes challenging. In this article, we study CPU and GPU rendering techniques for transparent 3D line sets. We compare accurate and approximate techniques using optimized implementations and several benchmark data sets. We discuss the effects of data size and transparency on quality, performance, and memory consumption. Based on our study, we propose two improvements to per-pixel fragment lists and multi-layer alpha blending. The first improves the rendering speed via an improved GPU sorting operation, and the second improves rendering quality via transparency-based bucketing.

Volumetric Isosurface Rendering with Deep Learning-Based Super-Resolution.

Rendering an accurate image of an isosurface in a volumetric field typically requires large numbers of data samples. Reducing this number lies at the core of research in volume rendering. With the advent of deep learning networks, a number of architectures have been proposed recently to infer missing samples in multidimensional fields, for applications such as image super-resolution. In this article, we investigate the use of such architectures for learning the upscaling of a low resolution sampling of an isosurface to a higher resolution, with reconstruction of spatial detail and shading. We introduce a fully convolutional neural network, to learn a latent representation generating smooth, edge-aware depth and normal fields as well as ambient occlusions from a low resolution depth and normal field. By adding a frame-to-frame motion loss into the learning stage, upscaling can consider temporal variations and achieves improved frame-to-frame coherence. We assess the quality of inferred results and compare it to bi-linear and cubic upscaling. We do this for isosurfaces which were never seen during training, and investigate the improvements when the network can train on the same or similar isosurfaces. We discuss remote visualization and foveated rendering as potential applications.

A Voxel-Based Rendering Pipeline for Large 3D Line Sets.

We present a voxel-based rendering pipeline for large 3D line sets that employs GPU ray-casting to achieve scalable rendering including transparency and global illumination effects. Even for opaque lines we demonstrate superior rendering performance compared to GPU rasterization of lines, and when transparency is used we can interactively render amounts of lines that are infeasible to be rendered via rasterization. We propose a direction-preserving encoding of lines into a regular voxel grid, along with the quantization of directions using face-to-face connectivity in this grid. On the regular grid structure, parallel GPU ray-casting is used to determine visible fragments in correct visibility order. To enable interactive rendering of global illumination effects like low-frequency shadows and ambient occlusions, illumination simulation is performed during ray-casting on a level-of-detail (LoD) line representation that considers the number of lines and their lengths per voxel. In this way we can render effects which are very difficult to render via GPU rasterization. A detailed performance and quality evaluation compares our approach to rasterization-based rendering of lines.

Visual Analysis of the Temporal Evolution of Ensemble Forecast Sensitivities.

Ensemble sensitivity analysis (ESA) has been established in the atmospheric sciences as a correlation-based approach to determine the sensitivity of a scalar forecast quantity computed by a numerical weather prediction model to changes in another model variable at a different model state. Its applications include determining the origin of forecast errors and placing targeted observations to improve future forecasts. We-a team of visualization scientists and meteorologists-present a visual analysis framework to improve upon current practice of ESA. We support the user in selecting regions to compute a meaningful target forecast quantity by embedding correlation-based grid-point clustering to obtain statistically coherent regions. The evolution of sensitivity features computed via ESA are then traced through time, by integrating a quantitative measure of feature matching into optical-flow-based feature assignment, and displayed by means of a swipe-path showing the geo-spatial evolution of the sensitivities. Visualization of the internal correlation structure of computed features guides the user towards those features robustly predicting a certain weather event. We demonstrate the use of our method by application to real-world 2D and 3D cases that occurred during the 2016 NAWDEX field campaign, showing the interactive generation of hypothesis chains to explore how atmospheric processes sensitive to each other are interrelated.

Interactive 3D Visual Analysis of Atmospheric Fronts.

Atmospheric fronts play a central role in meteorology, as the boundaries between different air masses and as fundamental features of extra-tropical cyclones. They appear in numerous conceptual model depictions of extra-tropical weather systems. Conceptually, fronts are three-dimensional surfaces in space possessing an innate structural complexity, yet in meteorology, both manual and objective identification and depiction have historically focused on the structure in two dimensions. In this work, we -a team of visualization scientists and meteorologists- propose a novel visualization approach to analyze the three-dimensional structure of atmospheric fronts and related physical and dynamical processes. We build upon existing approaches to objectively identify fronts as lines in two dimensions and extend these to obtain frontal surfaces in three dimensions, using the magnitude of temperature change along the gradient of a moist potential temperature field as the primary identifying factor. We introduce the use of normal curves in the temperature gradient field to visualize a frontal zone (i.e., the transitional zone between the air masses) and the distribution of atmospheric variables in such zones. To enable for the first time a statistical analysis of frontal zones, we present a new approach to obtain the volume enclosed by a zone, by classifying grid boxes that intersect with normal curves emanating from a selected front. We introduce our method by means of an idealized numerical simulation and demonstrate its use with two real-world cases using numerical weather prediction data.

Visualization in Meteorology-A Survey of Techniques and Tools for Data Analysis Tasks.

This article surveys the history and current state of the art of visualization in meteorology, focusing on visualization techniques and tools used for meteorological data analysis. We examine characteristics of meteorological data and analysis tasks, describe the development of computer graphics methods for visualization in meteorology from the 1960s to today, and visit the state of the art of visualization techniques and tools in operational weather forecasting and atmospheric research. We approach the topic from both the visualization and the meteorological side, showing visualization techniques commonly used in meteorological practice, and surveying recent studies in visualization research aimed at meteorological applications. Our overview covers visualization techniques from the fields of display design, 3D visualization, flow dynamics, feature-based visualization, comparative visualization and data fusion, uncertainty and ensemble visualization, interactive visual analysis, efficient rendering, and scalability and reproducibility. We discuss demands and challenges for visualization research targeting meteorological data analysis, highlighting aspects in demonstration of benefit, interactive visual analysis, seamless visualization, ensemble visualization, 3D visualization, and technical issues.

Infill Optimization for Additive Manufacturing-Approaching Bone-Like Porous Structures.

Porous structures such as trabecular bone are widely seen in nature. These structures are lightweight and exhibit strong mechanical properties. In this paper, we present a method to generate bone-like porous structures as lightweight infill for additive manufacturing. Our method builds upon and extends voxel-wise topology optimization. In particular, for the purpose of generating sparse yet stable structures distributed in the interior of a given shape, we propose upper bounds on the localized material volume in the proximity of each voxel in the design domain. We then aggregate the local per-voxel constraints by their p-norm into an equivalent global constraint, in order to facilitate an efficient optimization process. Implemented on a high-resolution topology optimization framework, our results demonstrate mechanically optimized, detailed porous structures which mimic those found in nature. We further show variants of the optimized structures subject to different design specifications, and we analyze the optimality and robustness of the obtained structures.

Robust Detection and Visualization of Jet-Stream Core Lines in Atmospheric Flow.

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.

Visualizing Confidence in Cluster-Based Ensemble Weather Forecast Analyses.

In meteorology, cluster analysis is frequently used to determine representative trends in ensemble weather predictions in a selected spatio-temporal region, e.g., to reduce a set of ensemble members to simplify and improve their analysis. Identified clusters (i.e., groups of similar members), however, can be very sensitive to small changes of the selected region, so that clustering results can be misleading and bias subsequent analyses. In this article, we - a team of visualization scientists and meteorologists-deliver visual analytics solutions to analyze the sensitivity of clustering results with respect to changes of a selected region. We propose an interactive visual interface that enables simultaneous visualization of a) the variation in composition of identified clusters (i.e., their robustness), b) the variability in cluster membership for individual ensemble members, and c) the uncertainty in the spatial locations of identified trends. We demonstrate that our solution shows meteorologists how representative a clustering result is, and with respect to which changes in the selected region it becomes unstable. Furthermore, our solution helps to identify those ensemble members which stably belong to a given cluster and can thus be considered similar. In a real-world application case we show how our approach is used to analyze the clustering behavior of different regions in a forecast of "Tropical Cyclone Karl", guiding the user towards the cluster robustness information required for subsequent ensemble analysis.

Time-Hierarchical Clustering and Visualization of Weather Forecast Ensembles.

We propose a new approach for analyzing the temporal growth of the uncertainty in ensembles of weather forecasts which are started from perturbed but similar initial conditions. As an alternative to traditional approaches in meteorology, which use juxtaposition and animation of spaghetti plots of iso-contours, we make use of contour clustering and provide means to encode forecast dynamics and spread in one single visualization. Based on a given ensemble clustering in a specified time window, we merge clusters in time-reversed order to indicate when and where forecast trajectories start to diverge. We present and compare different visualizations of the resulting time-hierarchical grouping, including space-time surfaces built by connecting cluster representatives over time, and stacked contour variability plots. We demonstrate the effectiveness of our visual encodings with forecast examples of the European Centre for Medium-Range Weather Forecasts, which convey the evolution of specific features in the data as well as the temporally increasing spatial variability.

Solving the Fluid Pressure Poisson Equation Using Multigrid-Evaluation and Improvements.

In many numerical simulations of fluids governed by the incompressible Navier-Stokes equations, the pressure Poisson equation needs to be solved to enforce mass conservation. Multigrid solvers show excellent convergence in simple scenarios, yet they can converge slowly in domains where physically separated regions are combined at coarser scales. Moreover, existing multigrid solvers are tailored to specific discretizations of the pressure Poisson equation, and they cannot easily be adapted to other discretizations. In this paper we analyze the convergence properties of existing multigrid solvers for the pressure Poisson equation in different simulation domains, and we show how to further improve the multigrid convergence rate by using a graph-based extension to determine the coarse grid hierarchy. The proposed multigrid solver is generic in that it can be applied to different kinds of discretizations of the pressure Poisson equation, by using solely the specification of the simulation domain and pre-assembled computational stencils. We analyze the proposed solver in combination with finite difference and finite volume discretizations of the pressure Poisson equation. Our evaluations show that, despite the common assumption, multigrid schemes can exploit their potential even in the most complicated simulation scenarios, yet this behavior is obtained at the price of higher memory consumption.

A System for High-Resolution Topology Optimization.

A key requirement in 3D fabrication is to generate objects with individual exterior shapes and their interior being optimized to application-specific force constraints and low material consumption. Accomplishing this task is challenging on desktop computers, due to the extreme model resolutions that are required to accurately predict the physical shape properties, requiring memory and computational capacities going beyond what is currently available. Moreover, fabrication-specific constraints need to be considered to enable printability. To address these challenges, we present a scalable system for generating 3D objects using topology optimization, which allows to efficiently evolve the topology of high-resolution solids towards printable and light-weight-high-resistance structures. To achieve this, the system is equipped with a high-performance GPU solver which can efficiently handle models comprising several millions of elements. A minimum thickness constraint is built into the optimization process to automatically enforce printability of the resulting shapes. We further shed light on the question how to incorporate geometric shape constraints, such as symmetry and pattern repetition, in the optimization process. We analyze the performance of the system and demonstrate its potential by a variety of different shapes such as interior structures within closed surfaces, exposed support structures, and surface models.

Streamline Variability Plots for Characterizing the Uncertainty in Vector Field Ensembles.

We present a new method to visualize from an ensemble of flow fields the statistical properties of streamlines passing through a selected location. We use principal component analysis to transform the set of streamlines into a low-dimensional Euclidean space. In this space the streamlines are clustered into major trends, and each cluster is in turn approximated by a multivariate Gaussian distribution. This yields a probabilistic mixture model for the streamline distribution, from which confidence regions can be derived in which the streamlines are most likely to reside. This is achieved by transforming the Gaussian random distributions from the low-dimensional Euclidean space into a streamline distribution that follows the statistical model, and by visualizing confidence regions in this distribution via iso-contours. We further make use of the principal component representation to introduce a new concept of streamline-median, based on existing median concepts in multidimensional Euclidean spaces. We demonstrate the potential of our method in a number of real-world examples, and we compare our results to alternative clustering approaches for particle trajectories as well as curve boxplots.

Real-time haptic cutting of high-resolution soft tissues.

We present our systematic efforts in advancing the computational performance of physically accurate soft tissue cutting simulation, which is at the core of surgery simulators in general. We demonstrate a real-time performance of 15 simulation frames per second for haptic soft tissue cutting of a deformable body at an effective resolution of 170,000 finite elements. This is achieved by the following innovative components: (1) a linked octree discretization of the deformable body, which allows for fast and robust topological modifications of the simulation domain, (2) a composite finite element formulation, which thoroughly reduces the number of simulation degrees of freedom and thus enables to carefully balance simulation performance and accuracy, (3) a highly efficient geometric multigrid solver for solving the linear systems of equations arising from implicit time integration, (4) an efficient collision detection algorithm that effectively exploits the composition structure, and (5) a stable haptic rendering algorithm for computing the feedback forces. Considering that our method increases the finite element resolution for physically accurate real-time soft tissue cutting simulation by an order of magnitude, our technique has a high potential to significantly advance the realism of surgery simulators.