Revisiting histograms and isosurface statistics

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.

The Strehl ratio as a phase histogram

It is shown that the Strehl ratio can always be written as an integral over an apodization-weighted phase histogram. The corresponding mathematical formalism, based on Federer's co-area formula, is enumerated, and a practical numerical method to quickly and accurately calculate apodization-weighted phase histograms is detailed and compared with similar methods. Conditions for expressing the Strehl ratio as a product S=S 1 S 2 are investigated.

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.

Volumetric Feature-Based Classification and Visibility Analysis for Transfer Function Design

Transfer function (TF) design is a central topic in direct volume rendering. The TF fundamentally translates data values into optical properties to reveal relevant features present in the volumetric data. We propose a semi-automatic TF design scheme which consists of two steps: First, we present a clustering process within 1D/2D TF domain based on the proximities of the respective volumetric features in the spatial domain. The presented approach provides an interactive tool that aids users in exploring clusters and identifying features of interest (FOI). Second, our method automatically generates a TF by iteratively refining the optical properties for the selected features using a novel feature visibility measurement. The proposed visibility measurement leverages the similarities of features to enhance their visibilities in DVR images. Compared to the conventional visibility measurement, the proposed feature visibility is able to efficiently sense opacity changes and precisely evaluate the impact of selected features on resulting visualizations. Our experiments validate the effectiveness of the proposed approach by demonstrating the advantages of integrating feature similarity into the visibility computations. We examine a number of datasets to establish the utility of our approach for semi-automatic TF design.

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.

Visual Analysis of Multi-Run Spatio-Temporal Simulations Using Isocontour Similarity for Projected Views

Multi-run simulations are widely used to investigate how simulated processes evolve depending on varying initial conditions. Frequently, such simulations model the change of spatial phenomena over time. Isocontours have proven to be effective for the visual representation and analysis of 2D and 3D spatial scalar fields. We propose a novel visualization approach for multi-run simulation data based on isocontours. By introducing a distance function for isocontours, we generate a distance matrix used for a multidimensional scaling projection. Multiple simulation runs are represented by polylines in the projected view displaying change over time. We propose a fast calculation of isocontour differences based on a quasi-Monte Carlo approach. For interactive visual analysis, we support filtering and selection mechanisms on the multi-run plot and on linked views to physical space visualizations. Our approach can be effectively used for the visual representation of ensembles, for pattern and outlier detection, for the investigation of the influence of simulation parameters, and for a detailed analysis of the features detected. The proposed method is applicable to data of any spatial dimensionality and any spatial representation (gridded or unstructured). We validate our approach by performing a user study on synthetic data and applying it to different types of multi-run spatio-temporal simulation data.

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.

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.

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.

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.

Positional uncertainty of isocontours: condition analysis and probabilistic measures

Uncertainty is ubiquitous in science, engineering and medicine. Drawing conclusions from uncertain data is the normal case, not an exception. While the field of statistical graphics is well established, only a few 2D and 3D visualization and feature extraction methods have been devised that consider uncertainty. We present mathematical formulations for uncertain equivalents of isocontours based on standard probability theory and statistics and employ them in interactive visualization methods. As input data, we consider discretized uncertain scalar fields and model these as random fields. To create a continuous representation suitable for visualization we introduce interpolated probability density functions. Furthermore, we introduce numerical condition as a general means in feature-based visualization. The condition number-which potentially diverges in the isocontour problem-describes how errors in the input data are amplified in feature computation. We show how the average numerical condition of isocontours aids the selection of thresholds that correspond to robust isocontours. Additionally, we introduce the isocontour density and the level crossing probability field; these two measures for the spatial distribution of uncertain isocontours are directly based on the probabilistic model of the input data. Finally, we adapt interactive visualization methods to evaluate and display these measures and apply them to 2D and 3D data sets.

Efficient Probabilistic and Geometric Anatomical Mapping Using Particle Mesh Approximation on GPUs

Deformable image registration in the presence of considerable contrast differences and large size and shape changes presents significant research challenges. First, it requires a robust registration framework that does not depend on intensity measurements and can handle large nonlinear shape variations. Second, it involves the expensive computation of nonlinear deformations with high degrees of freedom. Often it takes a significant amount of computation time and thus becomes infeasible for practical purposes. In this paper, we present a solution based on two key ideas: a new registration method that generates a mapping between anatomies represented as a multicompartment model of class posterior images and geometries and an implementation of the algorithm using particle mesh approximation on Graphical Processing Units (GPUs) to fulfill the computational requirements. We show results on the registrations of neonatal to 2-year old infant MRIs. Quantitative validation demonstrates that our proposed method generates registrations that better maintain the consistency of anatomical structures over time and provides transformations that better preserve structures undergoing large deformations than transformations obtained by standard intensity-only registration. We also achieve the speedup of three orders of magnitudes compared to a CPU reference implementation, making it possible to use the technique in time-critical applications.

An application of multivariate statistical analysis for Query-Driven Visualization

Driven by the ability to generate ever-larger, increasingly complex data, there is an urgent need in the scientific community for scalable analysis methods that can rapidly identify salient trends in scientific data. Query-Driven Visualization (QDV) strategies are among the small subset of techniques that can address both large and highly complex data sets. This paper extends the utility of QDV strategies with a statistics-based framework that integrates nonparametric distribution estimation techniques with a new segmentation strategy to visually identify statistically significant trends and features within the solution space of a query. In this framework, query distribution estimates help users to interactively explore their query's solution and visually identify the regions where the combined behavior of constrained variables is most important, statistically, to their inquiry. Our new segmentation strategy extends the distribution estimation analysis by visually conveying the individual importance of each variable to these regions of high statistical significance. We demonstrate the analysis benefits these two strategies provide and show how they maybe used to facilitate the refinement of constraints over variables expressed in a user's query. We apply our method to data sets from two different scientific domains to demonstrate its broad applicability.

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.

Direct interval volume visualization

We extend direct volume rendering with a unified model for generalized isosurfaces, also called interval volumes, allowing a wider spectrum of visual classification. We generalize the concept of scale-invariant opacity—typical for isosurface rendering—to semi-transparent interval volumes. Scale-invariant rendering is independent of physical space dimensions and therefore directly facilitates the analysis of data characteristics. Our model represents sharp isosurfaces as limits of interval volumes and combines them with features of direct volume rendering. Our objective is accurate rendering, guaranteeing that all isosurfaces and interval volumes are visualized in a crack-free way with correct spatial ordering. We achieve simultaneous direct and interval volume rendering by extending preintegration and explicit peak finding with data-driven splitting of ray integration and hybrid computation in physical and data domains. Our algorithm is suitable for efficient parallel processing for interactive applications as demonstrated by our CUDA implementation.

On the fractal dimension of isosurfaces

A (3D) scalar grid is a regular n1 x n2 x n3 grid of vertices where each vertex v is associated with some scalar value sv. Applying trilinear interpolation, the scalar grid determines a scalar function g where g(v) = sv for each grid vertex v. An isosurface with isovalue σ is a triangular mesh which approximates the level set g(-1)(σ). The fractal dimension of an isosurface represents the growth ;in the isosurface as the number of grid cubes increases. We define and discuss the fractal isosurface dimension. Plotting the fractal ;dimension as a function of the isovalues in a data set provides information about the isosurfaces determined by the data set. We present statistics on the average fractal dimension of 60 publicly available benchmark data sets. We also show the fractal dimension is highly correlated with topological noise in the benchmark data sets, measuring the topological noise by the number of connected components in the isosurface. Lastly, we present a formula predicting the fractal dimension as a function of noise and validate the formula with experimental results.

Continuous parallel coordinates

Typical scientific data is represented on a grid with appropriate interpolation or approximation schemes,defined on a continuous domain. The visualization of such data in parallel coordinates may reveal patterns latently contained in the data and thus can improve the understanding of multidimensional relations. In this paper, we adopt the concept of continuous scatterplots for the visualization of spatially continuous input data to derive a density model for parallel coordinates. Based on the point-line duality between scatterplots and parallel coordinates, we propose a mathematical model that maps density from a continuous scatterplot to parallel coordinates and present different algorithms for both numerical and analytical computation of the resulting density field. In addition, we show how the 2-D model can be used to successively construct continuous parallel coordinates with an arbitrary number of dimensions. Since continuous parallel coordinates interpolate data values within grid cells, a scalable and dense visualization is achieved, which will be demonstrated for typical multi-variate scientific data.

Continuous scatterplots

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