Research Team

A. Entezari (PI)

E. Sakhaee (RA)

B. Ma (RA)

K. Zhang (RA)

Z. Shu (RA)

Project Support

This material is based upon work supported by the National Science Foundation: IIS-1617101

Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Project Sumary

Uncertainties in observational data compounded with errors introduced as a part of modeling (e.g., truncation, quantization) are common sources of uncertainty that adversely affect the reliability of simulation results. Modern UQ techniques provide probability distributions that describe the variability of simulation results. The integration of this variability to the data analysis and visualization algorithms allows for decision making in presence of uncertainty. However, these algorithms, themselves, introduce non-linear transformations to the uncertainty in the data (e.g., error bars or distributions) that further complicate the quantification of uncertainty at the end of the visual analysis process. Moreover, common operations such as data filtering, contouring and classification need to be redefined in a probabilistic setting to allow for the integration and propagation of data uncertainty. This project addresses fundamental challenges that arise in the analysis and quantification of uncertainty as the data propagates throughout various stages of the visualization pipeline.

We present a study of linear interpolation when applied to uncertain data. Linear interpolation is a key step for isosurface extraction algorithms, and the uncertainties in the data lead to non-linear variations in the geometry of the extracted isosurface. We present an approach for deriving the probability density function of a random variable modeling the positional uncertainty in the isosurface extraction. When the uncertainty is quantified by a uniform distribution, our approach provides a closed-form characterization of the mentioned random variable. This allows us to derive, in closed form, the expected value as well as the variance of the level-crossing position. While the former quantity is used for constructing a stable isosurface for uncertain data, the latter is used for visualizing the positional uncertainties in the expected isosurface level crossings on the underlying grid [18], [2].

Figure 1: Teardrop dataset visualized at the isovalue of c = -0.002. The leftmost image shows the result of linear interpolation in the original sampled data, which serves as the ground truth. Uniform noise of δ = 0.01 is injected in the dataset, and linear interpolation is shown in the second image. The isosurface extracted using expected crossing is shown in the third image, which provides a more stable isosurface when noise level is high. The rightmost image visualizes the positional uncertainty of the vertices from triangulation in MC, found using expected crossing, by color mapping the ratio variance, var(Z). Green indicates small variance, blue indicates moderate, while red indicates high variance; hence, high spatial uncertainty (quantiles after histogram equalization).

Figure 2: Fuel dataset visualized at the isovalue of c = 56. The image on the top shows the result of linear interpolation in the original sampled data, which we consider as the ground truth. Uniform noise of δ = 21 is injected in the dataset, and again the result of linear interpolation is shown in the next image. Isosurface extracted using expected crossing is shown in the third image. The bottom image visualizes the positional uncertainty of expected level crossing on the underlying grid by color mapping the ratio variance, var(Z). Green indicates small variance, blue indicates moderate, while red indicates high variance; hence, high spatial uncertainty (quantiles formed after histogram equalization).

Classification based on mean field

Vertex-based classification method

Edge-based classification method

Figure 3: A Comparison of vertex-classification techniques for the isosurface visualization (c = 24^∘C) of an ensemble representing uncertain temperature field [20]. Boxes mark differences in isosurface topology. Edge-based classification extracts the isocontour along a coastline of the Gulf of Mexico unlike the other vertex classification techniques. Positional uncertainties in expected isosurface are color mapped (quantiles formed after histogram equalization).

The research goal of the project “A Statistical Direct Volume Rendering Framework for Visualization of Uncertain Data” is to develop a statistical framework for quantification of uncertainty and its propagation in the main stages of the visualization pipeline. This project is published in [4]. We introduce a probabilistic transfer function classification model that allows for incorporating probability density functions into the volume rendering integral. Our statistical framework allows for incorporating distributions from various sources of uncertainty which makes it suitable in a wide range of visualization applications. We demonstrate effectiveness of our approach in visualization of ensemble data, visualizing large datasets at reduced scale, iso-surface extraction, and visualization of noisy data.

Figure 4: Rendering of iso-surface corresponding to iso-value .30 for the Tangle function sampled on a grid of size 64 64 64. When the function is visualized on a low-resolution grid (shown in a), the iso- surface is disconnected and reveals no information about the critical points connecting the lobes. The statistical rendering (shown in b) presents a faithful depiction of the high resolution data, visualizing existence of the critical points with uncertainty.

Software

The research goal of our project “Volumetric Feature-Based Classification and Visibility Analysis for Transfer Function Design” is to ease the laborious transfer function (TF) design process in direct volume rendering. This project is published in  [9]. The source code is publicly available on Github (https://github.com/scumabo/TransferFunctionDVR). The proposed research approach is mainly composed of three parts: 1. Hierarchical cell-based feature similarity map: we propose an efficient feature similarity method for the analysis of isosurfaces (1D) and isovalue-gradient features (2D) present in volumetric datasets. We provide source code and shell scripts to reproduce all the experiments in the paper. 2. Feature classification: we provide an interactive interface to aid the identification of distinct volumetric structures in the data. 3. Feature visibility for TF design: we improve the conventional visibility measurement and propose feature visibility for TF specification. We have integrated our approach in an open source visualization package, Voreen [19], for automatic TF generation.

The research goal of our project, titled “Quality Assessment of Volume Compression Approaches Using Isovalue Clustering”, is to provide a new method to assess the quality of compressed volumetric data. This project is published in [3]. The proposed approach uses representative isosurfaces as benchmark structures to evaluate the visual quality of compressed 3D scalar fields. We examine a number of widely used compression approaches, namely, discrete wavelet transform, discrete cosine transform, and tensor approximation, to establish the utility of our volume quality assessment approach. The source code is available on Github (https://github.com/scumabo/VQA), which includes scripts for compressing volumetric data and evaluating the quality of volume compressions. This project provides means for selecting representative isovalues from a volumetric dataset. These representative isovalues are used to design transfer functions (e.g., as in [9] in above).

Figure 5: Psychometric function for the proposed quality metric plotted with VQ score and p pairs of the compressions (from left to right) at level 100%,25%, 8%, 1.5%, and 0.5%. The RIs at isovalue 52 serve as references for the visual degradation of different compressions.

An Interactive Visualization Framework

Numerical Weather Prediction (NWP) ensembles are commonly used to assess the uncertainty and confidence in weather forecasts. Spaghetti plots are conventional tools for meteorologists to directly examine the uncertainty exhibited by ensembles, where they simultaneously visualize isocontours of all ensemble members. To avoid visual clutter in practical usages, one needs to select a small number of informative isovalues for visual analysis. Moreover, due to the complex topology and variation of ensemble isocontours, it is often a challenging task to interpret the spaghetti plot for even a single isovalue in large ensembles. In this paper, we propose an interactive framework for uncertainty visualization of weather forecast ensembles that significantly improves and expands the utility of spaghetti plots in ensemble analysis. Complementary to state-of-the-art methods, our approach provides a complete framework for visual exploration of ensemble isocontours, including isovalue selection, interactive isocontour variability exploration, and interactive sub-region selection and re-analysis.

Figure 6: Interactive exploration of isocontours across multiple isovalues in a weather forecast ensemble. (a) The conventional spaghetti plot suffers from significant visual clutter. Based on our high-density clustering for ensemble isocontours, we present several summary plots and interactions (right box). Users can select isovalues of interest from (b) the simplified spaghetti plot, and the selected isocontours are visualized in (c) the linked mode and spaghetti plots. Users can use the mode plot to select interesting subsets of isocontours (bottom). Users can also interactively select sub-regions for re-analysis and further exploration.

Our framework is built upon the high-density clustering paradigm, where the mode structure of the density function is represented as a hierarchy of nested subsets of the data. We generalize the high-density clustering for isocontours and propose a bandwidth selection method for estimating the density function of ensemble isocontours. We present novel visualizations based on high-density clustering results, called the mode plot and the simplified spaghetti plot. The proposed mode plot visually encodes the structure provided by the high-density clustering result and summarizes the distribution of ensemble isocontours. It also enables the selection of subsets of interesting isocontours, which are interactively highlighted in a linked spaghetti plot for providing spatial context. To provide an interpretable overview of the positional variability of isocontours, our system allows for selection of informative isovalues from the simplified spaghetti plot. Due to the spatial variability of ensemble isocontours, the system allows for interactive selection and focus on sub-regions for local uncertainty and clustering re-analysis. We examine a number of ensemble datasets to establish the utility of our approach and discuss its advantages over state-of-the-art visual analysis tools for ensemble data.

Here is a brief overview of our system.

Publications

T. Athawale, E. Sakhaee, and A. Entezari. Isosurface visualization of data with nonparametric models for uncertainty. IEEE transactions on visualization and computer graphics, 22(1):777–786, 2016

T. Athawale and A. Entezari. Uncertainty quantification in linear interpolation for isosurface extraction. IEEE transactions on visualization and computer graphics, 19(12):2723–2732, 2016

B. Ma, S. K. Suter, and A. Entezari. Quality assessment of volume compression approaches using isovalue clustering. Computers & Graphics, 63:18–27, 2017

B. Ma, E. Jain, and A. Entezari. 3d saliency from eye tracking with tomography. In Eye Tracking and Visualization, pages 185–198. Springer International Publishing, 2017

E. Sakhaee and A. Entezari. A statistical direct volume rendering framework for visualization of uncertain data. IEEE transactions on visualization and computer graphics, 23(12):2509–2520, 2017

E. Sakhaee and A. Entezari. Joint inverse problems for signal reconstruction via dictionary splitting. IEEE signal processing letters, 24(8):1203–1207, 2017

E. Sakhaee and A. Entezari. Variable splitting and cycle spinning for sparse signal recovery. In Signal Processing with Adaptive Sparse Structured Representations (SPARS). IEEE, 2017

K. Zhang and A. Entezari. On concentration inequalities for sparse vectors. In Signal Processing with Adaptive Sparse Structured Representations (SPARS). IEEE, 2017

D. Zwick, E. Sakhaee, S. Balachandar, and A. Entezari. Accurate signal reconstruction for higher order lagrangian–eulerian back-coupling in multiphase turbulence. Fluid Dynamics Research, 49(5):055507, 2017

E. Sakhaee. Joint Linear Inverse Problems with Sparse Solutions: Theory and Applications. PhD thesis, University of Florida, 2017

B. Ma and A. Entezari. Volumetric feature-based classification and visibility analysis for transfer function design. IEEE transactions on visualization and computer graphics, 24(12):3253–3267, 2017

B. Ma and A. Entezari. An interactive framework for visualization of weather forecast ensembles. IEEE transactions on visualization and computer graphics, 25(1):1091–1101, 2018

A. Entezari, K. Zhang, and E. Sakhaee. Sparse approximation for few view tomographic reconstruction. In Splines in Imaging, SIAM Imaging Science (IS). SIAM, 2018

A. Sinha. Experimental study of phase transitions of sparse recovery using l1 minimization techniques. Master’s thesis, University of Florida, 2019

K. Zhang and A. Entezari. A convolutional spline framework for forward and back-projection in fan-beam geometry. In 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI). IEEE, 2019

K. Zhang and A. Entezari. Box spline projection in non-parallel geometry. In 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI). IEEE, 2019

J. Sun, A. Entezari, and B. Vemuri. Exploiting structural redundancy in q-space for improved eap reconstruction from highly undersampled (k, q)-space in dmri. Medical image analysis, 2019

Last updated 27.jun.19