An information-theoretic framework for flow visualization

FlowNL: Asking the Flow Data in Natural Languages

Flow visualization is essentially a tool to answer domain experts' questions about flow fields using rendered images. Static flow visualization approaches require domain experts to raise their questions to visualization experts, who develop specific techniques to extract and visualize the flow structures of interest. Interactive visualization approaches allow domain experts to ask the system directly through the visual analytic interface, which provides flexibility to support various tasks. However, in practice, the visual analytic interface may require extra learning effort, which often discourages domain experts and limits its usage in real-world scenarios. In this paper, we propose FlowNL, a novel interactive system with a natural language interface. FlowNL allows users to manipulate the flow visualization system using plain English, which greatly reduces the learning effort. We develop a natural language parser to interpret user intention and translate textual input into a declarative language. We design the declarative language as an intermediate layer between the natural language and the programming language specifically for flow visualization. The declarative language provides selection and composition rules to derive relatively complicated flow structures from primitive objects that encode various kinds of information about scalar fields, flow patterns, regions of interest, connectivities, etc. We demonstrate the effectiveness of FlowNL using multiple usage scenarios and an empirical evaluation.

A Bounded Measure for Estimating the Benefit of Visualization (Part I): Theoretical Discourse and Conceptual Evaluation

Information theory can be used to analyze the cost–benefit of visualization processes. However, the current measure of benefit contains an unbounded term that is neither easy to estimate nor intuitive to interpret. In this work, we propose to revise the existing cost–benefit measure by replacing the unbounded term with a bounded one. We examine a number of bounded measures that include the Jenson–Shannon divergence, its square root, and a new divergence measure formulated as part of this work. We describe the rationale for proposing a new divergence measure. In the first part of this paper, we focus on the conceptual analysis of the mathematical properties of these candidate measures. We use visualization to support the multi-criteria comparison, narrowing the search down to several options with better mathematical properties. The theoretical discourse and conceptual evaluation in this part provides the basis for further data-driven evaluation based on synthetic and experimental case studies that are reported in the second part of this paper.

Reconstructing Unsteady Flow Data From Representative Streamlines via Diffusion and Deep-Learning-Based Denoising

We propose VFR-UFD, a new deep learning framework that performs vector field reconstruction (VFR) for unsteady flow data (UFD). Given integral flow lines (i.e., streamlines), we first generate low-quality UFD via diffusion. VFR-UFD then leverages a convolutional neural network to reconstruct spatiotemporally coherent, high-quality UFD. The core of VFR-UFD lies in recurrent residual blocks that iteratively refine and denoise the input vector fields at different scales, both locally and globally. We take consecutive time steps as input to capture temporal coherence and apply streamline-based optimization to preserve spatial coherence. To show the effectiveness of VFR-UFD, we experiment with several vector field data sets to report quantitative and qualitative results and compare VFR-UFD with two VFR methods and one compression algorithm.

Integral Curve Clustering and Simplification for Flow Visualization: A Comparative Evaluation

Unsupervised clustering techniques have been widely applied to flow simulation data to alleviate clutter and occlusion in the resulting visualization. However, there is an absence of systematic guidelines for users to evaluate (both quantitatively and visually) the appropriate clustering technique and similarity measures for streamline and pathline curves. In this work, we provide an overview of a number of prevailing curve clustering techniques. We then perform a comprehensive experimental study to qualitatively and quantitatively compare these clustering techniques coupled with popular similarity measures used in the flow visualization literature. Based on our experimental results, we derive empirical guidelines for selecting the appropriate clustering technique and similarity measure given the requirements of the visualization task. We believe our work will inform the task of generating meaningful reduced representations for large-scale flow data and inspire the continuous investigation of a more refined guidance on clustering technique selection.

Fourier Opacity Optimization for Scalable Exploration

Over the past decades, scientific visualization became a fundamental aspect of modern scientific data analysis. Across all data-intensive research fields, ranging from structural biology to cosmology, data sizes increase rapidly. Dealing with the growing large-scale data is one of the top research challenges of this century. For the visual exploratory data analysis, interactivity, a view-dependent visibility optimization and frame coherence are indispensable. In this work, we extend the recent decoupled opacity optimization framework to enable a navigation without occlusion of important features through large geometric data. By expressing the accumulation of importance and optical depth in Fourier basis, the computation, evaluation and rendering of optimized transparent geometry become not only order-independent, but also operate within a fixed memory bound. We study the quality of our Fourier approximation in terms of accuracy, memory requirements and efficiency for both the opacity computation, as well as the order-independent compositing. We apply the method to different point, line and surface data sets originating from various research fields, including meteorology, health science, astrophysics and organic chemistry.

FlowNet: A Deep Learning Framework for Clustering and Selection of Streamlines and Stream Surfaces

For effective flow visualization, identifying representative flow lines or surfaces is an important problem which has been studied. However, no work can solve the problem for both lines and surfaces. In this paper, we present FlowNet, a single deep learning framework for clustering and selection of streamlines and stream surfaces. Given a collection of streamlines or stream surfaces generated from a flow field data set, our approach converts them into binary volumes and then employs an autoencoder to learn their respective latent feature descriptors. These descriptors are used to reconstruct binary volumes for error estimation and network training. Once converged, the feature descriptors can well represent flow lines or surfaces in the latent space. We perform dimensionality reduction of these feature descriptors and cluster the projection results accordingly. This leads to a visual interface for exploring the collection of flow lines or surfaces via clustering, filtering, and selection of representatives. Intuitive user interactions are provided for visual reasoning of the collection with ease. We validate and explain our deep learning framework from multiple perspectives, demonstrate the effectiveness of FlowNet using several flow field data sets of different characteristics, and compare our approach against state-of-the-art streamline and stream surface selection algorithms.

A Model of Spatial Directness in Interactive Visualization

We discuss the concept of directness in the context of spatial interaction with visualization. In particular, we propose a model that allows practitioners to analyze and describe the spatial directness of interaction techniques, ultimately to be able to better understand interaction issues that may affect usability. To reach these goals, we distinguish between different types of directness. Each type of directness depends on a particular mapping between different spaces, for which we consider the data space, the visualization space, the output space, the user space, the manipulation space, and the interaction space. In addition to the introduction of the model itself, we also show how to apply it to several real-world interaction scenarios in visualization, and thus discuss the resulting types of spatial directness, without recommending either more direct or more indirect interaction techniques. In particular, we will demonstrate descriptive and evaluative usage of the proposed model, and also briefly discuss its generative usage.

Flow Field Reduction Via Reconstructing Vector Data From 3-D Streamlines Using Deep Learning

We present a new approach for streamline-based flow field representation and reduction. Our method can work in the in situ visualization setting by tracing streamlines from each time step of the simulation and storing compressed streamlines for post hoc analysis where users can afford longer reconstruction time for higher reconstruction quality using decompressed streamlines. At the heart of our approach is a deep learning method for vector field reconstruction that takes the streamlines traced from the original vector fields as input and applies a two-stage process to reconstruct high-quality vector fields. To demonstrate the effectiveness of our approach, we show qualitative and quantitative results with several data sets and compare our method against the de facto method of gradient vector flow in terms of speed and quality tradeoff.

Two closely-spaced Aneurysms of the Supraclinoid Internal Carotid Artery: How Does One Influence the Other?

The objective of this study was to use image-based CFD simulation techniques to analyze the impact that multiple closely spaced IAs of the supra-clinioid segment of the ICA have on each other's hemodynamic characteristics. The vascular geometry of fifteen (15) subjects with 2 IAs were gathered using a 3D clinical system. Two groups of computer models were created for each subject's vascular geometry: both IAs present (Model A) and after removal of one IA (Model B). Models were separated into two groups based on IA separation: tandem (one proximal and one distal) and tandem (aneurysms directly opposite on a vessel). Simulations using a pulsatile velocity waveform were solved by a commercial CFD solver. Proximal IAs altered flow into distal IAs (5 of 7), increasing flow energy and spatial-temporally averaged wall shear stress (STA-WSS: 3-50\% comparing Model A to B) while decreasing flow stability within distal IAs. Thus, proximal IAs may ``protect" a distal aneurysm from destructive remodeling due to flow stagnation. Among adjacent IAs, the presence of both IAs decreased each other's flow characteristics, lowering WSS (Model A to B) and increasing flow stability: all changes statistically significant (t-test p < 0.05). A negative relationship exists between the mean percent change in flow stability in relation to adjacent IA volume and ostium area. Closely spaced IAs impact hemodynamic alterations onto each other concerning flow energy, stressors and stability. Understanding these alterations may improve clinical management of closely-spaced IAs.

Visual Exploration of Air Quality Data with a Time-correlation-partitioning Tree Based on Information Theory

<?tight?>Discovering the correlations among variables of air quality data is challenging, because the correlation time series are long-lasting, multi-faceted, and information-sparse. In this article, we propose a novel visual representation, called Time-correlation-partitioning (TCP) tree, that compactly characterizes correlations of multiple air quality variables and their evolutions. A TCP tree is generated by partitioning the information-theoretic correlation time series into pieces with respect to the variable hierarchy and temporal variations, and reorganizing these pieces into a hierarchically nested structure. The visual exploration of a TCP tree provides a sparse data traversal of the correlation variations and a situation-aware analysis of correlations among variables. This can help meteorologists understand the correlations among air quality variables better. We demonstrate the efficiency of our approach in a real-world air quality investigation scenario.

Tensor Decompositions for Integral Histogram Compression and Look-Up

Histograms are a fundamental tool for multidimensional data analysis and processing, and many applications in graphics and visualization rely on computing histograms over large regions of interest (ROI). Integral histograms (IH) greatly accelerate the calculation in the case of rectangular regions, but come at a large extra storage cost. Based on the tensor train decomposition model, we propose a new compression and approximate retrieval algorithm to reduce the overall IH memory usage by several orders of magnitude at a user-defined accuracy. To this end we propose an incremental tensor decomposition algorithm that allows us to compress integral histograms of hundreds of gigabytes. We then encode the borders of any desired rectangular ROI in the IH tensor-compressed domain and reconstruct the target histogram at a high speed which is independent of the region size. We furthermore generalize the algorithm to support regions of arbitrary shape rather than only rectangles, as well as histogram field computation, i.e., recovering many histograms at once. We test our method with several multidimensional data sets and demonstrate that it radically speeds up costly histogram queries while avoiding storing massive, uncompressed IHs.

Efficient Local Statistical Analysis via Point-Wise Histograms in Tetrahedral Meshes and Curvilinear Grids

Local histograms (i.e., point-wise histograms computed from local regions of mesh vertices) have been used in many data analysis and visualization applications. Previous methods for computing local histograms mainly work for regular or rectilinear grids only. In this paper, we develop theory and novel algorithms for computing local histograms in tetrahedral meshes and curvilinear grids. Our algorithms are theoretically sound and efficient, and work effectively and fast in practice. Our main focus is on scalar fields, but the algorithms also work for vector fields as a by-product with small, easy modifications. Our methods can benefit information theoretic and other distribution-driven analysis. The experiments demonstrate the efficacy of our new techniques, including a utility case study on tetrahedral vector field visualization.

Dissecting Deep Learning Networks-Visualizing Mutual Information

Deep Learning (DL) networks are recent revolutionary developments in artificial intelligence research. Typical networks are stacked by groups of layers that are further composed of many convolutional kernels or neurons. In network design, many hyper-parameters need to be defined heuristically before training in order to achieve high cross-validation accuracies. However, accuracy evaluation from the output layer alone is not sufficient to specify the roles of the hidden units in associated networks. This results in a significant knowledge gap between DL's wider applications and its limited theoretical understanding. To narrow the knowledge gap, our study explores visualization techniques to illustrate the mutual information (MI) in DL networks. The MI is a theoretical measurement, reflecting the relationship between two sets of random variables even if their relationship is highly non-linear and hidden in high-dimensional data. Our study aims to understand the roles of DL units in classification performance of the networks. Via a series of experiments using several popular DL networks, it shows that the visualization of MI and its change patterns between the input/output with the hidden layers and basic units can facilitate a better understanding of these DL units' roles. Our investigation on network convergence suggests a more objective manner to potentially evaluate DL networks. Furthermore, the visualization provides a useful tool to gain insights into the network performance, and thus to potentially facilitate the design of better network architectures by identifying redundancy and less-effective network units.

Relative movement on the articular surfaces of the tibiotalar and subtalar joints during walking

Objectives

The objective of this study was to quantify the relative movement between the articular surfaces in the tibiotalar and subtalar joints during normal walking in asymptomatic individuals.

Methods

3D movement data of the ankle joint complex were acquired from 18 subjects using a biplanar fluoroscopic system and 3D-to-2D registration of bone models obtained from CT images. Surface relative velocity vectors (SRVVs) of the articular surfaces of the tibiotalar and subtalar joints were calculated. The relative movement of the articulating surfaces was quantified as the mean relative speed (RS) and synchronization index (SI_ENT) of the SRVVs.

Results

SI_ENT and mean RS data showed that the tibiotalar joint exhibited translational movement throughout the stance, with a mean SI_ENT of 0.54 (sd 0.21). The mean RS of the tibiotalar joint during the 0% to 20% post heel-strike phase was 36.0 mm/s (sd 14.2), which was higher than for the rest of the stance period. The subtalar joint had a mean SI_ENT value of 0.43 (sd 0.21) during the stance phase and exhibited a greater degree of rotational movement than the tibiotalar joint. The mean relative speeds of the subtalar joint in early (0% to 10%) and late (80% to 90%) stance were 23.9 mm/s (sd 11.3) and 25.1 mm/s (sd 9.5), respectively, which were significantly higher than the mean RS during mid-stance (10% to 80%).

Conclusion

The tibiotalar and subtalar joints exhibited significant translational and rotational movement in the initial stance, whereas only the subtalar joint exhibited significant rotational movement during the late stance. The relative movement on the articular surfaces provided deeper insight into the interactions between articular surfaces, which are unobtainable using the joint coordinate system.

Cite this article: C-B. Phan, D-P. Nguyen, K. M. Lee, S. Koo. Relative movement on the articular surfaces of the tibiotalar and subtalar joints during walking. Bone Joint Res 2018;7:501–507. DOI: 10.1302/2046-3758.78.BJR-2018-0014.R1.

Semi-Automatic Generation of Stream Surfaces via Sketching

We present a semi-automatic approach for stream surface generation. Our approach is based on the conjecture that good seeding curves can be inferred from a set of streamlines. Given a set of densely traced streamlines over the flow field, we design a sketch-based interface that allows users to describe their perceived flow patterns through drawing simple strokes directly on top of the streamline visualization results. Based on the 2D stroke, we identify a 3D seeding curve and generate a stream surface that captures the flow pattern of streamlines at the outermost layer. Then, we remove the streamlines whose patterns are covered by the stream surface. Repeating this process, users can peel the flow by replacing the streamlines with customized surfaces layer by layer. Furthermore, we propose an optimization scheme to identify the optimal seeding curve in the neighborhood of an original seeding curve based on surface quality measures. To support interactive optimization, we design a parallel surface quality estimation strategy that estimates the quality of a seeding curve without generating the surface. Our sketch-based interface leverages an intuitive painting metaphor which most users are familiar with. We present results using multiple data sets to show the effectiveness of our approach.

Visual Analysis of Spatio-temporal Relations of Pairwise Attributes in Unsteady Flow

Despite significant advances in the analysis and visualization of unsteady flow, the interpretation of it's behavior still remains a challenge. In this work, we focus on the linear correlation and non-linear dependency of different physical attributes of unsteady flows to aid their study from a new perspective. Specifically, we extend the existing spatial correlation quantification, i.e. the Local Correlation Coefficient (LCC), to the spatio-temporal domain to study the correlation of attribute-pairs from both the Eulerian and Lagrangian views. To study the dependency among attributes, which need not be linear, we extend and compute the mutual information (MI) among attributes over time. To help visualize and interpret the derived correlation and dependency among attributes associated with a particle, we encode the correlation and dependency values on individual pathlines. Finally, to utilize the correlation and MI computation results to identify regions with interesting flow behavior, we propose a segmentation strategy of the flow domain based on the ranking of the strength of the attributes relations. We have applied our correlation and dependency metrics to a number of 2D and 3D unsteady flows with varying spatio-temporal kernel sizes to demonstrate and assess their effectiveness.

An Information-Theoretic Approach to the Cost-benefit Analysis of Visualization in Virtual Environments

Visualization and virtual environments (VEs) have been two interconnected parallel strands in visual computing for decades. Some VEs have been purposely developed for visualization applications, while many visualization applications are exemplary showcases in general-purpose VEs. Because of the development and operation costs of VEs, the majority of visualization applications in practice have yet to benefit from the capacity of VEs. In this paper, we examine this status quo from an information-theoretic perspective. Our objectives are to conduct cost-benefit analysis on typical VE systems (including augmented and mixed reality, theater-based systems, and large powerwalls), to explain why some visualization applications benefit more from VEs than others, and to sketch out pathways for the future development of visualization applications in VEs. We support our theoretical propositions and analysis using theories and discoveries in the literature of cognitive sciences and the practical evidence reported in the literatures of visualization and VEs.

Information Guided Exploration of Scalar Values and Isocontours in Ensemble Datasets

Uncertainty of scalar values in an ensemble dataset is often represented by the collection of their corresponding isocontours. Various techniques such as contour-boxplot, contour variability plot, glyphs and probabilistic marching-cubes have been proposed to analyze and visualize ensemble isocontours. All these techniques assume that a scalar value of interest is already known to the user. Not much work has been done in guiding users to select the scalar values for such uncertainty analysis. Moreover, analyzing and visualizing a large collection of ensemble isocontours for a selected scalar value has its own challenges. Interpreting the visualizations of such large collections of isocontours is also a difficult task. In this work, we propose a new information-theoretic approach towards addressing these issues. Using specific information measures that estimate the predictability and surprise of specific scalar values, we evaluate the overall uncertainty associated with all the scalar values in an ensemble system. This helps the scientist to understand the effects of uncertainty on different data features. To understand in finer details the contribution of individual members towards the uncertainty of the ensemble isocontours of a selected scalar value, we propose a conditional entropy based algorithm to quantify the individual contributions. This can help simplify analysis and visualization for systems with more members by identifying the members contributing the most towards overall uncertainty. We demonstrate the efficacy of our method by applying it on real-world datasets from material sciences, weather forecasting and ocean simulation experiments.

What May Visualization Processes Optimize?

In this paper, we present an abstract model of visualization and inference processes, and describe an information-theoretic measure for optimizing such processes. In order to obtain such an abstraction, we first examined six classes of workflows in data analysis and visualization, and identified four levels of typical visualization components, namely disseminative, observational, analytical and model-developmental visualization. We noticed a common phenomenon at different levels of visualization, that is, the transformation of data spaces (referred to as alphabets) usually corresponds to the reduction of maximal entropy along a workflow. Based on this observation, we establish an information-theoretic measure of cost-benefit ratio that may be used as a cost function for optimizing a data visualization process. To demonstrate the validity of this measure, we examined a number of successful visualization processes in the literature, and showed that the information-theoretic measure can mathematically explain the advantages of such processes over possible alternatives.

Modality-Driven Classification and Visualization of Ensemble Variance

Advances in computational power now enable domain scientists to address conceptual and parametric uncertainty by running simulations multiple times in order to sufficiently sample the uncertain input space. While this approach helps address conceptual and parametric uncertainties, the ensemble datasets produced by this technique present a special challenge to visualization researchers as the ensemble dataset records a distribution of possible values for each location in the domain. Contemporary visualization approaches that rely solely on summary statistics (e.g., mean and variance) cannot convey the detailed information encoded in ensemble distributions that are paramount to ensemble analysis; summary statistics provide no information about modality classification and modality persistence. To address this problem, we propose a novel technique that classifies high-variance locations based on the modality of the distribution of ensemble predictions. Additionally, we develop a set of confidence metrics to inform the end-user of the quality of fit between the distribution at a given location and its assigned class. Finally, for the special application of evaluating the stability of bimodal regions, we develop local and regional metrics.

View-Dependent Streamline Deformation and Exploration

Occlusion presents a major challenge in visualizing 3D flow and tensor fields using streamlines. Displaying too many streamlines creates a dense visualization filled with occluded structures, but displaying too few streams risks losing important features. We propose a new streamline exploration approach by visually manipulating the cluttered streamlines by pulling visible layers apart and revealing the hidden structures underneath. This paper presents a customized view-dependent deformation algorithm and an interactive visualization tool to minimize visual clutter in 3D vector and tensor fields. The algorithm is able to maintain the overall integrity of the fields and expose previously hidden structures. Our system supports both mouse and direct-touch interactions to manipulate the viewing perspectives and visualize the streamlines in depth. By using a lens metaphor of different shapes to select the transition zone of the targeted area interactively, the users can move their focus and examine the vector or tensor field freely.

An Automated Approach for Slicing Plane Placement in Visual Data Analysis

Effective display and visual analysis of complex 3D data is a challenging task. Occlusions, overlaps, and projective distortions-as frequently caused by typical 3D rendering techniques-can be major obstacles to unambiguous and robust data analysis. Slicing planes are a ubiquitous tool to resolve several of these issues. They act as simple clipping geometry to provide clear cut-away views of the data. We propose to enhance the visualization and analysis process by providing methods for automatic placement of such slicing planes based on local optimization of gradient vector flow. The final obtained slicing planes maximize the total amount of information displayed with respect to a pre-specified importance function. We demonstrate how such automated slicing plane placement is able to support and enrich 3D data visualization and analysis in multiple scenarios, such as volume or surface rendering, and evaluate its performance in several benchmark data sets.

FLDA: Latent Dirichlet Allocation Based Unsteady Flow Analysis

In this paper, we present a novel feature extraction approach called FLDA for unsteady flow fields based on Latent Dirichlet allocation (LDA) model. Analogous to topic modeling in text analysis, in our approach, pathlines and features in a given flow field are defined as documents and words respectively. Flow topics are then extracted based on Latent Dirichlet allocation. Different from other feature extraction methods, our approach clusters pathlines with probabilistic assignment, and aggregates features to meaningful topics at the same time. We build a prototype system to support exploration of unsteady flow field with our proposed LDA-based method. Interactive techniques are also developed to explore the extracted topics and to gain insight from the data. We conduct case studies to demonstrate the effectiveness of our proposed approach.

A deformation framework for focus+context flow visualization

Striking a careful balance among coverage, occlusion, and complexity is a resounding theme in the visual understanding of large and complex three-dimensional flow fields. In this paper, we present a novel deformation framework for focus+context streamline visualization that reduces occlusion and clutter around the focal regions while compacting the context region in a full view. Unlike existing techniques that vary streamline densities, we advocate a different approach that manipulates streamline positions. This is achieved by partitioning the flow field's volume space into blocks and deforming the blocks to guide streamline repositioning. We formulate block expansion and block smoothing into energy terms and solve for a deformed grid that minimizes the objective function under the volume boundary and edge flipping constraints. Leveraging a GPU linear system solver, we demonstrate interactive focus+context visualization with 3D flow field data of various characteristics. Compared to the fisheye focus+context technique, our method can magnify multiple streamlines of focus in different regions simultaneously while minimizing the distortion through optimized deformation. Both automatic and manual feature specifications are provided for flexible focus selection and effective visualization.

An information-aware framework for exploring multivariate data sets

Information theory provides a theoretical framework for measuring information content for an observed variable, and has attracted much attention from visualization researchers for its ability to quantify saliency and similarity among variables. In this paper, we present a new approach towards building an exploration framework based on information theory to guide the users through the multivariate data exploration process. In our framework, we compute the total entropy of the multivariate data set and identify the contribution of individual variables to the total entropy. The variables are classified into groups based on a novel graph model where a node represents a variable and the links encode the mutual information shared between the variables. The variables inside the groups are analyzed for their representativeness and an information based importance is assigned. We exploit specific information metrics to analyze the relationship between the variables and use the metrics to choose isocontours of selected variables. For a chosen group of points, parallel coordinates plots (PCP) are used to show the states of the variables and provide an interface for the user to select values of interest. Experiments with different data sets reveal the effectiveness of our proposed framework in depicting the interesting regions of the data sets taking into account the interaction among the variables.

Efficient local statistical analysis via integral histograms with discrete wavelet transform

Histograms computed from local regions are commonly used in many visualization applications, and allowing the user to query histograms interactively in regions of arbitrary locations and sizes plays an important role in feature identification and tracking. Computing histograms in regions with arbitrary location and size, nevertheless, can be time consuming for large data sets since it involves expensive I/O and scan of data elements. To achieve both performance- and storage-efficient query of local histograms, we present a new algorithm called WaveletSAT, which utilizes integral histograms, an extension of the summed area tables (SAT), and discrete wavelet transform (DWT). Similar to SAT, an integral histogram is the histogram computed from the area between each grid point and the grid origin, which can be be pre-computed to support fast query. Nevertheless, because one histogram contains multiple bins, it will be very expensive to store one integral histogram at each grid point. To reduce the storage cost for large integral histograms, WaveletSAT treats the integral histograms of all grid points as multiple SATs, each of which can be converted into a sparse representation via DWT, allowing the reconstruction of axis-aligned region histograms of arbitrary sizes from a limited number of wavelet coefficients. Besides, we present an efficient wavelet transform algorithm for SATs that can operate on each grid point separately in logarithmic time complexity, which can be extended to parallel GPU-based implementation. With theoretical and empirical demonstration, we show that WaveletSAT can achieve fast preprocessing and smaller storage overhead than the conventional integral histogram approach with close query performance.

Visualizing the variability of gradients in uncertain 2D scalar fields

In uncertain scalar fields where data values vary with a certain probability, the strength of this variability indicates the confidence in the data. It does not, however, allow inferring on the effect of uncertainty on differential quantities such as the gradient, which depend on the variability of the rate of change of the data. Analyzing the variability of gradients is nonetheless more complicated, since, unlike scalars, gradients vary in both strength and direction. This requires initially the mathematical derivation of their respective value ranges, and then the development of effective analysis techniques for these ranges. This paper takes a first step into this direction: Based on the stochastic modeling of uncertainty via multivariate random variables, we start by deriving uncertainty parameters, such as the mean and the covariance matrix, for gradients in uncertain discrete scalar fields. We do not make any assumption about the distribution of the random variables. Then, for the first time to our best knowledge, we develop a mathematical framework for computing confidence intervals for both the gradient orientation and the strength of the derivative in any prescribed direction, for instance, the mean gradient direction. While this framework generalizes to 3D uncertain scalar fields, we concentrate on the visualization of the resulting intervals in 2D fields. We propose a novel color diffusion scheme to visualize both the absolute variability of the derivative strength and its magnitude relative to the mean values. A special family of circular glyphs is introduced to convey the uncertainty in gradient orientation. For a number of synthetic and real-world data sets, we demonstrate the use of our approach for analyzing the stability of certain features in uncertain 2D scalar fields, with respect to both local derivatives and feature orientation.

Parallel streamline placement for 2D flow fields

Parallel streamline placement is still an open problem in flow visualization. In this paper, we propose an innovative method to place streamlines in parallel for 2D flow fields. This method is based on our proposed concept of local tracing areas (LTAs). An LTA is defined as a subdomain enclosed by streamlines and/or field borders, where the tracing of streamlines are localized. Given a flow field, it is initialized as an LTA, which is later recursively partitioned into hierarchical LTAs. Streamlines are placed within different LTAs simultaneously and independently. At the same time, to control the density of streamlines, each streamline is associated with an isolation zone and a saturation zone, both of which are center aligned with the streamline but have different widths. None of streamlines can trace into isolation zones of others. And new streamlines are only seeded within valid seeding areas (VSAs) that are enclosed by saturation zones and/or field borders. To implement the parallel strategy and the density control, a cell-based modeling is devised to describe isolation zones and LTAs as well as saturation zones and VSAs. With the help of these cell-based models, a heuristic seeding strategy is proposed to seed streamlines within irregular LTAs, and a cell-marking technique is used to control the seeding and tracing of streamlines. Test results show that the placement method can achieve highly parallel performance on shared memory systems without losing the quality of placements.

A unified approach to streamline selection and viewpoint selection for 3D flow visualization

We treat streamline selection and viewpoint selection as symmetric problems which are formulated into a unified information-theoretic framework. This is achieved by building two interrelated information channels between a pool of candidate streamlines and a set of sample viewpoints. We define the streamline information to select best streamlines and in a similar manner, define the viewpoint information to select best viewpoints. Furthermore, we propose solutions to streamline clustering and viewpoint partitioning based on the representativeness of streamlines and viewpoints, respectively. Finally, we define a camera path that passes through all selected viewpoints for automatic flow field exploration. We demonstrate the robustness of our approach by showing experimental results with different flow data sets, and conducting rigorous comparisons between our algorithm and other seed placement or streamline selection algorithms based on information theory.

Multimodal Data Fusion Based on Mutual Information

Multimodal visualization aims at fusing different data sets so that the resulting combination provides more information and understanding to the user. To achieve this aim, we propose a new information-theoretic approach that automatically selects the most informative voxels from two volume data sets. Our fusion criteria are based on the information channel created between the two input data sets that permit us to quantify the information associated with each intensity value. This specific information is obtained from three different ways of decomposing the mutual information of the channel. In addition, an assessment criterion based on the information content of the fused data set can be used to analyze and modify the initial selection of the voxels by weighting the contribution of each data set to the final result. The proposed approach has been integrated in a general framework that allows for the exploration of volumetric data models and the interactive change of some parameters of the fused data set. The proposed approach has been evaluated on different medical data sets with very promising results.

Hierarchical streamline bundles

Effective 3D streamline placement and visualization play an essential role in many science and engineering disciplines. The main challenge for effective streamline visualization lies in seed placement, i.e., where to drop seeds and how many seeds should be placed. Seeding too many or too few streamlines may not reveal flow features and patterns either because it easily leads to visual clutter in rendering or it conveys little information about the flow field. Not only does the number of streamlines placed matter, their spatial relationships also play a key role in understanding the flow field. Therefore, effective flow visualization requires the streamlines to be placed in the right place and in the right amount. This paper introduces hierarchical streamline bundles, a novel approach to simplifying and visualizing 3D flow fields defined on regular grids. By placing seeds and generating streamlines according to flow saliency, we produce a set of streamlines that captures important flow features near critical points without enforcing the dense seeding condition. We group spatially neighboring and geometrically similar streamlines to construct a hierarchy from which we extract streamline bundles at different levels of detail. Streamline bundles highlight multiscale flow features and patterns through clustered yet not cluttered display. This selective visualization strategy effectively reduces visual clutter while accentuating visual foci, and therefore is able to convey the desired insight into the flow data.

Automatic transfer functions based on informational divergence

In this paper we present a framework to define transfer functions from a target distribution provided by the user. A target distribution can reflect the data importance, or highly relevant data value interval, or spatial segmentation. Our approach is based on a communication channel between a set of viewpoints and a set of bins of a volume data set, and it supports 1D as well as 2D transfer functions including the gradient information. The transfer functions are obtained by minimizing the informational divergence or Kullback-Leibler distance between the visibility distribution captured by the viewpoints and a target distribution selected by the user. The use of the derivative of the informational divergence allows for a fast optimization process. Different target distributions for 1D and 2D transfer functions are analyzed together with importance-driven and view-based techniques.

Volume analysis using multimodal surface similarity

The combination of volume data acquired by multiple modalities has been recognized as an important but challenging task. Modalities often differ in the structures they can delineate and their joint information can be used to extend the classification space. However, they frequently exhibit differing types of artifacts which makes the process of exploiting the additional information non-trivial. In this paper, we present a framework based on an information-theoretic measure of isosurface similarity between different modalities to overcome these problems. The resulting similarity space provides a concise overview of the differences between the two modalities, and also serves as the basis for an improved selection of features. Multimodal classification is expressed in terms of similarities and dissimilarities between the isosurfaces of individual modalities, instead of data value combinations. We demonstrate that our approach can be used to robustly extract features in applications such as dual energy computed tomography of parts in industrial manufacturing.