Continuous scatterplots

De-Cluttering Scatterplots With Integral Images

Scatterplots provide a visual representation of bivariate data (or 2D embeddings of multivariate data) that allows for effective analyses of data dependencies, clusters, trends, and outliers. Unfortunately, classical scatterplots suffer from scalability issues, since growing data sizes eventually lead to overplotting and visual clutter on a screen with a fixed resolution, which hinders the data analysis process. We propose an algorithm that compensates for irregular sample distributions by a smooth transformation of the scatterplot's visual domain. Our algorithm evaluates the scatterplot's density distribution to compute a regularization mapping based on integral images of the rasterized density function. The mapping preserves the samples' neighborhood relations. Few regularization iterations suffice to achieve a nearly uniform sample distribution that efficiently uses the available screen space. We further propose approaches to visually convey the transformation that was applied to the scatterplot and compare them in a user study. We present a novel parallel algorithm for fast GPU-based integral-image computation, which allows for integrating our de-cluttering approach into interactive visual data analysis systems.

Visualization-Driven Illumination for Density Plots

We present a novel visualization-driven illumination model for density plots, a new technique to enhance density plots by effectively revealing the detailed structures in high- and medium-density regions and outliers in low-density regions, while avoiding artifacts in the density field's colors. When visualizing large and dense discrete point samples, scatterplots and dot density maps often suffer from overplotting, and density plots are commonly employed to provide aggregated views while revealing underlying structures. Yet, in such density plots, existing illumination models may produce color distortion and hide details in low-density regions, making it challenging to look up density values, compare them, and find outliers. The key novelty in this work includes (i) a visualization-driven illumination model that inherently supports density-plot-specific analysis tasks and (ii) a new image composition technique to reduce the interference between the image shading and the color-encoded density values. To demonstrate the effectiveness of our technique, we conducted a quantitative study, an empirical evaluation of our technique in a controlled study, and two case studies, exploring twelve datasets with up to two million data point samples.

TTK is Getting MPI-Ready

This system paper documents the technical foundations for the extension of the Topology ToolKit (TTK) to distributed-memory parallelism with the Message Passing Interface (MPI). While several recent papers introduced topology-based approaches for distributed-memory environments, these were reporting experiments obtained with tailored, mono-algorithm implementations. In contrast, we describe in this paper a versatile approach (supporting both triangulated domains and regular grids) for the support of topological analysis pipelines, i.e., a sequence of topological algorithms interacting together, possibly on distinct numbers of processes. While developing this extension, we faced several algorithmic and software engineering challenges, which we document in this paper. Specifically, we describe an MPI extension of TTK's data structure for triangulation representation and traversal, a central component to the global performance and generality of TTK's topological implementations. We also introduce an intermediate interface between TTK and MPI, both at the global pipeline level, and at the fine-grain algorithmic level. We provide a taxonomy for the distributed-memory topological algorithms supported by TTK, depending on their communication needs and provide examples of hybrid MPI+thread parallelizations. Detailed performance analyses show that parallel efficiencies range from 20% to 80% (depending on the algorithms), and that the MPI-specific preconditioning introduced by our framework induces a negligible computation time overhead. We illustrate the new distributed-memory capabilities of TTK with an example of advanced analysis pipeline, combining multiple algorithms, run on the largest publicly available dataset we have found (120 billion vertices) on a standard cluster with 64 nodes (for a total of 1536 cores). Finally, we provide a roadmap for the completion of TTK's MPI extension, along with generic recommendations for each algorithm communication category.

Continuous Scatterplot Operators for Bivariate Analysis and Study of Electronic Transitions

Electronic transitions in molecules due to the absorption or emission of light is a complex quantum mechanical process. Their study plays an important role in the design of novel materials. A common yet challenging task in the study is to determine the nature of electronic transitions, namely which subgroups of the molecule are involved in the transition by donating or accepting electrons, followed by an investigation of the variation in the donor-acceptor behavior for different transitions or conformations of the molecules. In this article, we present a novel approach for the analysis of a bivariate field and show its applicability to the study of electronic transitions. This approach is based on two novel operators, the continuous scatterplot (CSP) lens operator and the CSP peel operator, that enable effective visual analysis of bivariate fields. Both operators can be applied independently or together to facilitate analysis. The operators motivate the design of control polygon inputs to extract fiber surfaces of interest in the spatial domain. The CSPs are annotated with a quantitative measure to further support the visual analysis. We study different molecular systems and demonstrate how the CSP peel and CSP lens operators help identify and study donor and acceptor characteristics in molecular systems.

Automatic Scatterplot Design Optimization for Clustering Identification

Scatterplots are among the most widely used visualization techniques. Compelling scatterplot visualizations improve understanding of data by leveraging visual perception to boost awareness when performing specific visual analytic tasks. Design choices in scatterplots, such as graphical encodings or data aspects, can directly impact decision-making quality for low-level tasks like clustering. Hence, constructing frameworks that consider both the perceptions of the visual encodings and the task being performed enables optimizing visualizations to maximize efficacy. In this article, we propose an automatic tool to optimize the design factors of scatterplots to reveal the most salient cluster structure. Our approach leverages the merge tree data structure to identify the clusters and optimize the choice of subsampling algorithm, sampling rate, marker size, and marker opacity used to generate a scatterplot image. We validate our approach with user and case studies that show it efficiently provides high-quality scatterplot designs from a large parameter space.

Fiber Uncertainty Visualization for Bivariate Data With Parametric and Nonparametric Noise Models

Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of level-set visualization for univariate data to multivariate data. In this paper, we present a statistical framework to quantify positional probabilities of fibers extracted from uncertain bivariate fields. Specifically, we extend the state-of-the-art Gaussian models of uncertainty for bivariate data to other parametric distributions (e.g., uniform and Epanechnikov) and more general nonparametric probability distributions (e.g., histograms and kernel density estimation) and derive corresponding spatial probabilities of fibers. In our proposed framework, we leverage Green's theorem for closed-form computation of fiber probabilities when bivariate data are assumed to have independent parametric and nonparametric noise. Additionally, we present a nonparametric approach combined with numerical integration to study the positional probability of fibers when bivariate data are assumed to have correlated noise. For uncertainty analysis, we visualize the derived probability volumes for fibers via volume rendering and extracting level sets based on probability thresholds. We present the utility of our proposed techniques via experiments on synthetic and simulation datasets.

Uncertainty Visualization: Concepts, Methods, and Applications in Biological Data Visualization

This paper provides an overview of uncertainty visualization in general, along with specific examples of applications in bioinformatics. Starting from a processing and interaction pipeline of visualization, components are discussed that are relevant for handling and visualizing uncertainty introduced with the original data and at later stages in the pipeline, which shows the importance of making the stages of the pipeline aware of uncertainty and allowing them to propagate uncertainty. We detail concepts and methods for visual mappings of uncertainty, distinguishing between explicit and implict representations of distributions, different ways to show summary statistics, and combined or hybrid visualizations. The basic concepts are illustrated for several examples of graph visualization under uncertainty. Finally, this review paper discusses implications for the visualization of biological data and future research directions.

Attribute-based Explanation of Non-Linear Embeddings of High-Dimensional Data

Embeddings of high-dimensional data are widely used to explore data, to verify analysis results, and to communicate information. Their explanation, in particular with respect to the input attributes, is often difficult. With linear projects like PCA the axes can still be annotated meaningfully. With non-linear projections this is no longer possible and alternative strategies such as attribute-based color coding are required. In this paper, we review existing augmentation techniques and discuss their limitations. We present the Non-Linear Embeddings Surveyor (NoLiES) that combines a novel augmentation strategy for projected data (rangesets) with interactive analysis in a small multiples setting. Rangesets use a set-based visualization approach for binned attribute values that enable the user to quickly observe structure and detect outliers. We detail the link between algebraic topology and rangesets and demonstrate the utility of NoLiES in case studies with various challenges (complex attribute value distribution, many attributes, many data points) and a real-world application to understand latent features of matrix completion in thermodynamics.

Uncertainty in Continuous Scatterplots, Continuous Parallel Coordinates, and Fibers

In this paper, we introduce uncertainty to continuous scatterplots and continuous parallel coordinates. We derive respective models, validate them with sampling-based brute-force schemes, and present acceleration strategies for their computation. At the same time, we show that our approach lends itself as well for introducing uncertainty into the definition of fibers in bivariate data. Finally, we demonstrate the properties and the utility of our approach using specifically designed synthetic cases and simulated data.

Visual Analysis of Large Multivariate Scattered Data using Clustering and Probabilistic Summaries

Rapidly growing data sizes of scientific simulations pose significant challenges for interactive visualization and analysis techniques. In this work, we propose a compact probabilistic representation to interactively visualize large scattered datasets. In contrast to previous approaches that represent blocks of volumetric data using probability distributions, we model clusters of arbitrarily structured multivariate data. In detail, we discuss how to efficiently represent and store a high-dimensional distribution for each cluster. We observe that it suffices to consider low-dimensional marginal distributions for two or three data dimensions at a time to employ common visual analysis techniques. Based on this observation, we represent high-dimensional distributions by combinations of low-dimensional Gaussian mixture models. We discuss the application of common interactive visual analysis techniques to this representation. In particular, we investigate several frequency-based views, such as density plots in 1D and 2D, density-based parallel coordinates, and a time histogram. We visualize the uncertainty introduced by the representation, discuss a level-of-detail mechanism, and explicitly visualize outliers. Furthermore, we propose a spatial visualization by splatting anisotropic 3D Gaussians for which we derive a closed-form solution. Lastly, we describe the application of brushing and linking to this clustered representation. Our evaluation on several large, real-world datasets demonstrates the scaling of our approach.

Continuous Histograms for Anisotropy of 2D Symmetric Piece-Wise Linear Tensor Fields

In this chapter we present an accurate derivation of the distribution of scalar invariants with quadratic behavior represented as continuous histograms. The anisotropy field, computed from a two-dimensional piece-wise linear tensor field, is used as an example and is discussed in all details. Histograms visualizing an approximation of the distribution of scalar values play an important role in visualization. They are used as an interface for the design of transfer-functions for volume rendering or feature selection in interactive interfaces. While there are standard algorithms to compute continuous histograms for piece-wise linear scalar fields, they are not directly applicable to tensor invariants with non-linear, often even non-convex behavior in cells when applying linear tensor interpolation. Our derivation is based on a sub-division of the mesh in triangles that exhibit a monotonic behavior. We compare the results to a naïve approach based on linear interpolation on the original mesh or the subdivision.

A Recursive Subdivision Technique for Sampling Multi-class Scatterplots

We present a non-uniform recursive sampling technique for multi-class scatterplots, with the specific goal of faithfully presenting relative data and class densities, while preserving major outliers in the plots. Our technique is based on a customized binary kd-tree, in which leaf nodes are created by recursively subdividing the underlying multi-class density map. By backtracking, we merge leaf nodes until they encompass points of all classes for our subsequently applied outlier-aware multi-class sampling strategy. A quantitative evaluation shows that our approach can better preserve outliers and at the same time relative densities in multi-class scatterplots compared to the previous approaches, several case studies demonstrate the effectiveness of our approach in exploring complex and real world data.

Data Sampling in Multi-view and Multi-class Scatterplots via Set Cover Optimization

We present a method for data sampling in scatterplots by jointly optimizing point selection for different views or classes. Our method uses space-filling curves (Z-order curves) that partition a point set into subsets that, when covered each by one sample, provide a sampling or coreset with good approximation guarantees in relation to the original point set. For scatterplot matrices with multiple views, different views provide different space-filling curves, leading to different partitions of the given point set. For multi-class scatterplots, the focus on either per-class distribution or global distribution provides two different partitions of the given point set that need to be considered in the selection of the coreset. For both cases, we convert the coreset selection problem into an Exact Cover Problem (ECP), and demonstrate with quantitative and qualitative evaluations that an approximate solution that solves the ECP efficiently is able to provide high-quality samplings.

Relaxing Dense Scatter Plots with Pixel-Based Mappings

Scatter plots are the most commonly employed technique for the visualization of bivariate data. Despite their versatility and expressiveness in showing data aspects, such as clusters, correlations, and outliers, scatter plots face a main problem. For large and dense data, the representation suffers from clutter due to overplotting. This is often partially solved with the use of density plots. Yet, data overlap may occur in certain regions of a scatter or density plot, while other regions may be partially, or even completely empty. Adequate pixel-based techniques can be employed for effectively filling the plotting space, giving an additional notion of the numerosity of data motifs or clusters. We propose the Pixel-Relaxed Scatter Plots, a new and simple variant, to improve the display of dense scatter plots, using pixel-based, space-filling mappings. Our Pixel-Relaxed Scatter Plots make better use of the plotting canvas, while avoiding data overplotting, and optimizing space coverage and insight in the presence and size of data motifs. We have employed different methods to map scatter plot points to pixels and to visually present this mapping. We demonstrate our approach on several synthetic and realistic datasets, and we discuss the suitability of our technique for different tasks. Our conducted user evaluation shows that our Pixel-Relaxed Scatter Plots can be a useful enhancement to traditional scatter plots.

Stippling of 2D Scalar Fields

We propose a technique to represent two-dimensional data using stipples. While stippling is often regarded as an illustrative method, we argue that it is worth investigating its suitability for the visualization domain. For this purpose, we generalize the Linde-Buzo-Gray stippling algorithm for information visualization purposes to encode continuous and discrete 2D data. Our proposed modifications provide more control over the resulting distribution of stipples for encoding additional information into the representation, such as contours. We show different approaches to depict contours in stipple drawings based on locally adjusting the stipple distribution. Combining stipple-based gradients and contours allows for simultaneous assessment of the overall structure of the data while preserving important local details. We discuss the applicability of our technique using datasets from different domains and conduct observation-validating studies to assess the perception of stippled representations.

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.

GAN Lab: Understanding Complex Deep Generative Models using Interactive Visual Experimentation

Recent success in deep learning has generated immense interest among practitioners and students, inspiring many to learn about this new technology. While visual and interactive approaches have been successfully developed to help people more easily learn deep learning, most existing tools focus on simpler models. In this work, we present GAN Lab, the first interactive visualization tool designed for non-experts to learn and experiment with Generative Adversarial Networks (GANs), a popular class of complex deep learning models. With GAN Lab, users can interactively train generative models and visualize the dynamic training process's intermediate results. GAN Lab tightly integrates an model overview graph that summarizes GAN's structure, and a layered distributions view that helps users interpret the interplay between submodels. GAN Lab introduces new interactive experimentation features for learning complex deep learning models, such as step-by-step training at multiple levels of abstraction for understanding intricate training dynamics. Implemented using TensorFlow.js, GAN Lab is accessible to anyone via modern web browsers, without the need for installation or specialized hardware, overcoming a major practical challenge in deploying interactive tools for deep learning.

Image-Based Aspect Ratio Selection

Selecting a good aspect ratio is crucial for effective 2D diagrams. There are several aspect ratio selection methods for function plots and line charts, but only few can handle general, discrete diagrams such as 2D scatter plots. However, these methods either lack a perceptual foundation or heavily rely on intermediate isoline representations, which depend on choosing the right isovalues and are time-consuming to compute. This paper introduces a general image-based approach for selecting aspect ratios for a wide variety of 2D diagrams, ranging from scatter plots and density function plots to line charts. Our approach is derived from Federer's co-area formula and a line integral representation that enable us to directly construct image-based versions of existing selection methods using density fields. In contrast to previous methods, our approach bypasses isoline computation, so it is faster to compute, while following the perceptual foundation to select aspect ratios. Furthermore, this approach is complemented by an anisotropic kernel density estimation to construct density fields, allowing us to more faithfully characterize data patterns, such as the subgroups in scatterplots or dense regions in time series. We demonstrate the effectiveness of our approach by quantitatively comparing to previous methods and revisiting a prior user study. Finally, we present extensions for ROI banking, multi-scale banking, and the application to image data.

Indexed-Points Parallel Coordinates Visualization of Multivariate Correlations

We address the problem of visualizing multivariate correlations in parallel coordinates. We focus on multivariate correlation in the form of linear relationships between multiple variables. Traditional parallel coordinates are well prepared to show negative correlations between two attributes by distinct visual patterns. However, it is difficult to recognize positive correlations in parallel coordinates. Furthermore, there is no support to highlight multivariate correlations in parallel coordinates. In this paper, we exploit the indexed point representation of p -flats (planes in multidimensional data) to visualize local multivariate correlations in parallel coordinates. Our method yields clear visual signatures for negative and positive correlations alike, and it supports large datasets. All information is shown in a unified parallel coordinates framework, which leads to easy and familiar user interactions for analysts who have experience with traditional parallel coordinates. The usefulness of our method is demonstrated through examples of typical multidimensional datasets.

DSPCP: A Data Scalable Approach for Identifying Relationships in Parallel Coordinates

Parallel coordinates plots (PCPs) are a well-studied technique for exploring multi-attribute datasets. In many situations, users find them a flexible method to analyze and interact with data. Unfortunately, using PCPs becomes challenging as the number of data items grows large or multiple trends within the data mix in the visualization. The resulting overdraw can obscure important features. A number of modifications to PCPs have been proposed, including using color, opacity, smooth curves, frequency, density, and animation to mitigate this problem. However, these modified PCPs tend to have their own limitations in the kinds of relationships they emphasize. We propose a new data scalable design for representing and exploring data relationships in PCPs. The approach exploits the point/line duality property of PCPs and a local linear assumption of data to extract and represent relationship summarizations. This approach simultaneously shows relationships in the data and the consistency of those relationships. Our approach supports various visualization tasks, including mixed linear and nonlinear pattern identification, noise detection, and outlier detection, all in large data. We demonstrate these tasks on multiple synthetic and real-world datasets.

Scatterplots: Tasks, Data, and Designs

Traditional scatterplots fail to scale as the complexity and amount of data increases. In response, there exist many design options that modify or expand the traditional scatterplot design to meet these larger scales. This breadth of design options creates challenges for designers and practitioners who must select appropriate designs for particular analysis goals. In this paper, we help designers in making design choices for scatterplot visualizations. We survey the literature to catalog scatterplot-specific analysis tasks. We look at how data characteristics influence design decisions. We then survey scatterplot-like designs to understand the range of design options. Building upon these three organizations, we connect data characteristics, analysis tasks, and design choices in order to generate challenges, open questions, and example best practices for the effective design of scatterplots.

The Topology ToolKit

This system paper presents the Topology ToolKit (TTK), a software platform designed for the topological analysis of scalar data in scientific visualization. While topological data analysis has gained in popularity over the last two decades, it has not yet been widely adopted as a standard data analysis tool for end users or developers. TTK aims at addressing this problem by providing a unified, generic, efficient, and robust implementation of key algorithms for the topological analysis of scalar data, including: critical points, integral lines, persistence diagrams, persistence curves, merge trees, contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots, Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due to a tight integration with ParaView. It is also easily accessible to developers through a variety of bindings (Python, VTK/C++) for fast prototyping or through direct, dependency-free, C++, to ease integration into pre-existing complex systems. While developing TTK, we faced several algorithmic and software engineering challenges, which we document in this paper. In particular, we present an algorithm for the construction of a discrete gradient that complies to the critical points extracted in the piecewise-linear setting. This algorithm guarantees a combinatorial consistency across the topological abstractions supported by TTK, and importantly, a unified implementation of topological data simplification for multi-scale exploration and analysis. We also present a cached triangulation data structure, that supports time efficient and generic traversals, which self-adjusts its memory usage on demand for input simplicial meshes and which implicitly emulates a triangulation for regular grids with no memory overhead. Finally, we describe an original software architecture, which guarantees memory efficient and direct accesses to TTK features, while still allowing for researchers powerful and easy bindings and extensions. TTK is open source (BSD license) and its code, online documentation and video tutorials are available on TTK's website [108].

Fast and Exact Fiber Surfaces for Tetrahedral Meshes

Isosurfaces are fundamental geometrical objects for the analysis and visualization of volumetric scalar fields. Recent work has generalized them to bivariate volumetric fields with fiber surfaces, the pre-image of polygons in range space. However, the existing algorithm for their computation is approximate, and is limited to closed polygons. Moreover, its runtime performance does not allow instantaneous updates of the fiber surfaces upon user edits of the polygons. Overall, these limitations prevent a reliable and interactive exploration of the space of fiber surfaces. This paper introduces the first algorithm for the exact computation of fiber surfaces in tetrahedral meshes. It assumes no restriction on the topology of the input polygon, handles degenerate cases and better captures sharp features induced by polygon bends. The algorithm also allows visualization of individual fibers on the output surface, better illustrating their relationship with data features in range space. To enable truly interactive exploration sessions, we further improve the runtime performance of this algorithm. In particular, we show that it is trivially parallelizable and that it scales nearly linearly with the number of cores. Further, we study acceleration data-structures both in geometrical domain and range space and we show how to generalize interval trees used in isosurface extraction to fiber surface extraction. Experiments demonstrate the superiority of our algorithm over previous work, both in terms of accuracy and running time, with up to two orders of magnitude speedups. This improvement enables interactive edits of range polygons with instantaneous updates of the fiber surface for exploration purpose. A VTK-based reference implementation is provided as additional material to reproduce our results.

Visualizing High-Dimensional Data: Advances in the Past Decade

Massive simulations and arrays of sensing devices, in combination with increasing computing resources, have generated large, complex, high-dimensional datasets used to study phenomena across numerous fields of study. Visualization plays an important role in exploring such datasets. We provide a comprehensive survey of advances in high-dimensional data visualization that focuses on the past decade. We aim at providing guidance for data practitioners to navigate through a modular view of the recent advances, inspiring the creation of new visualizations along the enriched visualization pipeline, and identifying future opportunities for visualization research.

Jacobi Fiber Surfaces for Bivariate Reeb Space Computation

This paper presents an efficient algorithm for the computation of the Reeb space of an input bivariate piecewise linear scalar function f defined on a tetrahedral mesh. By extending and generalizing algorithmic concepts from the univariate case to the bivariate one, we report the first practical, output-sensitive algorithm for the exact computation of such a Reeb space. The algorithm starts by identifying the Jacobi set of f, the bivariate analogs of critical points in the univariate case. Next, the Reeb space is computed by segmenting the input mesh along the new notion of Jacobi Fiber Surfaces, the bivariate analog of critical contours in the univariate case. We additionally present a simplification heuristic that enables the progressive coarsening of the Reeb space. Our algorithm is simple to implement and most of its computations can be trivially parallelized. We report performance numbers demonstrating orders of magnitude speedups over previous approaches, enabling for the first time the tractable computation of bivariate Reeb spaces in practice. Moreover, unlike range-based quantization approaches (such as the Joint Contour Net), our algorithm is parameter-free. We demonstrate the utility of our approach by using the Reeb space as a semi-automatic segmentation tool for bivariate data. In particular, we introduce continuous scatterplot peeling, a technique which enables the reduction of the cluttering in the continuous scatterplot, by interactively selecting the features of the Reeb space to project. We provide a VTK-based C++ implementation of our algorithm that can be used for reproduction purposes or for the development of new Reeb space based visualization techniques.

Direct Multifield Volume Ray Casting of Fiber Surfaces

Multifield data are common in visualization. However, reducing these data to comprehensible geometry is a challenging problem. Fiber surfaces, an analogy of isosurfaces to bivariate volume data, are a promising new mechanism for understanding multifield volumes. In this work, we explore direct ray casting of fiber surfaces from volume data without any explicit geometry extraction. We sample directly along rays in domain space, and perform geometric tests in range space where fibers are defined, using a signed distance field derived from the control polygons. Our method requires little preprocess, and enables real-time exploration of data, dynamic modification and pixel-exact rendering of fiber surfaces, and support for higher-order interpolation in domain space. We demonstrate this approach on several bivariate datasets, including analysis of multi-field combustion data.

A Survey of Colormaps in Visualization

Colormaps are a vital method for users to gain insights into data in a visualization. With a good choice of colormaps, users are able to acquire information in the data more effectively and efficiently. In this survey, we attempt to provide readers with a comprehensive review of colormap generation techniques and provide readers a taxonomy which is helpful for finding appropriate techniques to use for their data and applications. Specifically, we first briefly introduce the basics of color spaces including color appearance models. In the core of our paper, we survey colormap generation techniques, including the latest advances in the field by grouping these techniques into four classes: procedural methods, user-study based methods, rule-based methods, and data-driven methods; we also include a section on methods that are beyond pure data comprehension purposes. We then classify colormapping techniques into a taxonomy for readers to quickly identify the appropriate techniques they might use. Furthermore, a representative set of visualization techniques that explicitly discuss the use of colormaps is reviewed and classified based on the nature of the data in these applications. Our paper is also intended to be a reference of colormap choices for readers when they are faced with similar data and/or tasks.

Interactive Visualization for Singular Fibers of Functions f : R³ → R²

Scalar topology in the form of Morse theory has provided computational tools that analyze and visualize data from scientific and engineering tasks. Contracting isocontours to single points encapsulates variations in isocontour connectivity in the Reeb graph. For multivariate data, isocontours generalize to fibers-inverse images of points in the range, and this area is therefore known as fiber topology. However, fiber topology is less fully developed than Morse theory, and current efforts rely on manual visualizations. This paper presents how to accelerate and semi-automate this task through an interface for visualizing fiber singularities of multivariate functions R³ → R². This interface exploits existing conventions of fiber topology, but also introduces a 3D view based on the extension of Reeb graphs to Reeb spaces. Using the Joint Contour Net, a quantized approximation of the Reeb space, this accelerates topological visualization and permits online perturbation to reduce or remove degeneracies in functions under study. Validation of the interface is performed by assessing whether the interface supports the mathematical workflow both of experts and of less experienced mathematicians.

Visual Abstraction and Exploration of Multi-class Scatterplots

Scatterplots are widely used to visualize scatter dataset for exploring outliers, clusters, local trends, and correlations. Depicting multi-class scattered points within a single scatterplot view, however, may suffer from heavy overdraw, making it inefficient for data analysis. This paper presents a new visual abstraction scheme that employs a hierarchical multi-class sampling technique to show a feature-preserving simplification. To enhance the density contrast, the colors of multiple classes are optimized by taking the multi-class point distributions into account. We design a visual exploration system that supports visual inspection and quantitative analysis from different perspectives. We have applied our system to several challenging datasets, and the results demonstrate the efficiency of our approach.

Joint Contour Nets

Contour Trees and Reeb Graphs are firmly embedded in scientific visualization for analysing univariate (scalar) fields. We generalize this analysis to multivariate fields with a data structure called the Joint Contour Net that quantizes the variation of multiple variables simultaneously. We report the first algorithm for constructing the Joint Contour Net, and demonstrate some of the properties that make it practically useful for visualisation, including accelerating computation by exploiting a relationship with rasterisation in the range of the function.

Regression Cube

Scatterplots are commonly used to visualize multidimensional data; however, 2D projections of data offer limited understanding of the high-dimensional interactions between data points. We introduce an interactive 3D extension of scatterplots called the Regression Cube (RC), which augments a 3D scatterplot with three facets on which the correlations between the two variables are revealed by sensitivity lines and sensitivity streamlines. The sensitivity visualization of local regression on the 2D projections provides insights about the shape of the data through its orientation and continuity cues. We also introduce a series of visual operations such as clustering, brushing, and selection supported in RC. By iteratively refining the selection of data points of interest, RC is able to reveal salient local correlation patterns that may otherwise remain hidden with a global analysis. We have demonstrated our system with two examples and a user-oriented evaluation, and we show how RCs enable interactive visual exploration of multidimensional datasets via a variety of classification and information retrieval tasks. A video demo of RC is available.

The generalized sensitivity scatterplot

Scatterplots remain a powerful tool to visualize multidimensional data. However, accurately understanding the shape of multidimensional points from 2D projections remains challenging due to overlap. Consequently, there are a lot of variations on the scatterplot as a visual metaphor for this limitation. An important aspect often overlooked in scatterplots is the issue of sensitivity or local trend, which may help in identifying the type of relationship between two variables. However, it is not well known how or what factors influence the perception of trends from 2D scatterplots. To shed light on this aspect, we conducted an experiment where we asked people to directly draw the perceived trends on a 2D scatterplot. We found that augmenting scatterplots with local sensitivity helps to fill the gaps in visual perception while retaining the simplicity and readability of a 2D scatterplot. We call this augmentation the generalized sensitivity scatterplot (GSS). In a GSS, sensitivity coefficients are visually depicted as flow lines, which give a sense of continuity and orientation of the data that provide cues about the way data points are scattered in a higher dimensional space. We introduce a series of glyphs and operations that facilitate the analysis of multidimensional data sets using GSS, and validate with a number of well-known data sets for both regression and classification tasks.

Bristle Maps: a multivariate abstraction technique for geovisualization

We present Bristle Maps, a novel method for the aggregation, abstraction, and stylization of spatiotemporal data that enables multiattribute visualization, exploration, and analysis. This visualization technique supports the display of multidimensional data by providing users with a multiparameter encoding scheme within a single visual encoding paradigm. Given a set of geographically located spatiotemporal events, we approximate the data as a continuous function using kernel density estimation. The density estimation encodes the probability that an event will occur within the space over a given temporal aggregation. These probability values, for one or more set of events, are then encoded into a bristle map. A bristle map consists of a series of straight lines that extend from, and are connected to, linear map elements such as roads, train, subway lines, and so on. These lines vary in length, density, color, orientation, and transparency—creating the multivariate attribute encoding scheme where event magnitude, change, and uncertainty can be mapped as various bristle parameters. This approach increases the amount of information displayed in a single plot and allows for unique designs for various information schemes. We show the application of our bristle map encoding scheme using categorical spatiotemporal police reports. Our examples demonstrate the use of our technique for visualizing data magnitude, variable comparisons, and a variety of multivariate attribute combinations. To evaluate the effectiveness of our bristle map, we have conducted quantitative and qualitative evaluations in which we compare our bristle map to conventional geovisualization techniques. Our results show that bristle maps are competitive in completion time and accuracy of tasks with various levels of complexity.

Splatterplots: overcoming overdraw in scatter plots

We introduce Splatterplots, a novel presentation of scattered data that enables visualizations that scale beyond standard scatter plots. Traditional scatter plots suffer from overdraw (overlapping glyphs) as the number of points per unit area increases. Overdraw obscures outliers, hides data distributions, and makes the relationship among subgroups of the data difficult to discern. To address these issues, Splatterplots abstract away information such that the density of data shown in any unit of screen space is bounded, while allowing continuous zoom to reveal abstracted details. Abstraction automatically groups dense data points into contours and samples remaining points. We combine techniques for abstraction with perceptually based color blending to reveal the relationship between data subgroups. The resulting visualizations represent the dense regions of each subgroup of the data set as smooth closed shapes and show representative outliers explicitly. We present techniques that leverage the GPU for Splatterplot computation and rendering, enabling interaction with massive data sets. We show how Splatterplots can be an effective alternative to traditional methods of displaying scatter data communicating data trends, outliers, and data set relationships much like traditional scatter plots, but scaling to data sets of higher density and up to millions of points on the screen.

Integrating Isosurface Statistics and Histograms

Many data sets are sampled on regular lattices in two, three or more dimensions, and recent work has shown that statistical properties of these data sets must take into account the continuity of the underlying physical phenomena. However, the effects of quantization on the statistics have not yet been accounted for. This paper therefore reconciles the previous papers to the underlying mathematical theory, develops a mathematical model of quantized statistics of continuous functions, and proves convergence of geometric approximations to continuous statistics for regular sampling lattices. In addition, the computational cost of various approaches is considered, and recommendations made about when to use each type of statistic.

Abstracting Attribute Space for Transfer Function Exploration and Design

Currently, user centered transfer function design begins with the user interacting with a one or two-dimensional histogram of the volumetric attribute space. The attribute space is visualized as a function of the number of voxels, allowing the user to explore the data in terms of the attribute size/magnitude. However, such visualizations provide the user with no information on the relationship between various attribute spaces (e.g., density, temperature, pressure, x, y, z) within the multivariate data. In this work, we propose a modification to the attribute space visualization in which the user is no longer presented with the magnitude of the attribute; instead, the user is presented with an information metric detailing the relationship between attributes of the multivariate volumetric data. In this way, the user can guide their exploration based on the relationship between the attribute magnitude and user selected attribute information as opposed to being constrained by only visualizing the magnitude of the attribute. We refer to this modification to the traditional histogram widget as an abstract attribute space representation. Our system utilizes common one and two-dimensional histogram widgets where the bins of the abstract attribute space now correspond to an attribute relationship in terms of the mean, standard deviation, entropy, or skewness. In this manner, we exploit the relationships and correlations present in the underlying data with respect to the dimension(s) under examination. These relationships are often times key to insight and allow us to guide attribute discovery as opposed to automatic extraction schemes which try to calculate and extract distinct attributes a priori. In this way, our system aids in the knowledge discovery of the interaction of properties within volumetric data.

Tuner: principled parameter finding for image segmentation algorithms using visual response surface exploration

In this paper we address the difficult problem of parameter-finding in image segmentation. We replace a tedious manual process that is often based on guess-work and luck by a principled approach that systematically explores the parameter space. Our core idea is the following two-stage technique: We start with a sparse sampling of the parameter space and apply a statistical model to estimate the response of the segmentation algorithm. The statistical model incorporates a model of uncertainty of the estimation which we use in conjunction with the actual estimate in (visually) guiding the user towards areas that need refinement by placing additional sample points. In the second stage the user navigates through the parameter space in order to determine areas where the response value (goodness of segmentation) is high. In our exploration we rely on existing ground-truth images in order to evaluate the "goodness" of an image segmentation technique. We evaluate its usefulness by demonstrating this technique on two image segmentation algorithms: a three parameter model to detect microtubules in electron tomograms and an eight parameter model to identify functional regions in dynamic Positron Emission Tomography scans.

TransGraph: hierarchical exploration of transition relationships in time-varying volumetric data

A fundamental challenge for time-varying volume data analysis and visualization is the lack of capability to observe and track data change or evolution in an occlusion-free, controllable, and adaptive fashion. In this paper, we propose to organize a timevarying data set into a hierarchy of states. By deriving transition probabilities among states, we construct a global map that captures the essential transition relationships in the time-varying data. We introduce the TransGraph, a graph-based representation to visualize hierarchical state transition relationships. The TransGraph not only provides a visual mapping that abstracts data evolution over time in different levels of detail, but also serves as a navigation tool that guides data exploration and tracking. The user interacts with the TransGraph and makes connection to the volumetric data through brushing and linking. A set of intuitive queries is provided to enable knowledge extraction from time-varying data. We test our approach with time-varying data sets of different characteristics and the results show that the TransGraph can effectively augment our ability in understanding time-varying data.

Features in Continuous Parallel Coordinates

Continuous Parallel Coordinates (CPC) are a contemporary visualization technique in order to combine several scalar fields, given over a common domain. They facilitate a continuous view for parallel coordinates by considering a smooth scalar field instead of a finite number of straight lines. We show that there are feature curves in CPC which appear to be the dominant structures of a CPC. We present methods to extract and classify them and demonstrate their usefulness to enhance the visualization of CPCs. In particular, we show that these feature curves are related to discontinuities in Continuous Scatterplots (CSP). We show this by exploiting a curve-curve duality between parallel and Cartesian coordinates, which is a generalization of the well-known point-line duality. Furthermore, we illustrate the theoretical considerations. Concluding, we discuss relations and aspects of the CPC's/CSP's features concerning the data analysis.

Visibility histograms and visibility-driven transfer functions

Direct volume rendering is an important tool for visualizing complex data sets. However, in the process of generating 2D images from 3D data, information is lost in the form of attenuation and occlusion. The lack of a feedback mechanism to quantify the loss of information in the rendering process makes the design of good transfer functions a difficult and time consuming task. In this paper, we present the general notion of visibility histograms, which are multidimensional graphical representations of the distribution of visibility in a volume-rendered image. In this paper, we explore the 1D and 2D transfer functions that result from intensity values and gradient magnitude. With the help of these histograms, users can manage a complex set of transfer function parameters that maximize the visibility of the intervals of interest and provide high quality images of volume data. We present a semiautomated method for generating transfer functions, which progressively explores the transfer function space toward the goal of maximizing visibility of important structures. Our methodology can be easily deployed in most visualization systems and can be used together with traditional 1D and 2D opacity transfer functions based on scalar values, as well as with other more sophisticated rendering algorithms.

Relation-Aware Isosurface Extraction in Multifield Data

We introduce a variation density function that profiles the relationship between multiple scalar fields over isosurfaces of a given scalar field. This profile serves as a valuable tool for multifield data exploration because it provides the user with cues to identify interesting isovalues of scalar fields. Existing isosurface-based techniques for scalar data exploration like Reeb graphs, contour spectra, isosurface statistics, etc., study a scalar field in isolation. We argue that the identification of interesting isovalues in a multifield data set should necessarily be based on the interaction between the different fields. We demonstrate the effectiveness of our approach by applying it to explore data from a wide variety of applications.