Direct Volume Rendering with Nonparametric Models of Uncertainty
T. M. Athawale, B. Ma, E. Sakhaee, C. R. Johnson, and A. Entezari
[Preprint, Supplementary material] [BibTex] [Preview video] [Presentation slides] [Presentation video (VIS 2020 uncertainty session, talk# 1)]
(IEEE Transactions on Visualization and Computer Graphics (TVCG), Special Issue on IEEE VIS Conf., vol. 27, no. 2, pp. 1797-1807, Feb. 2021.)
We present a nonparametric statistical framework for the quantification, analysis, and propagation of data uncertainty in direct volume rendering (DVR). The state-of-the-art statistical DVR framework allows for preserving the transfer function (TF) of the ground truth function when visualizing uncertain data; however, the existing framework is restricted to parametric models of uncertainty. In this paper, we address the limitations of the existing DVR framework by extending the DVR framework for nonparametric distributions. We exploit the quantile interpolation technique to derive probability distributions representing uncertainty in viewing-ray sample intensities in closed form, which allows for accurate and efficient computation. We evaluate our proposed nonparametric statistical models through qualitative and quantitative comparisons with the mean-field and parametric statistical models, such as uniform and Gaussian, as well as Gaussian mixtures. In addition, we present an extension of the state-of-the-art rendering parametric framework to 2D TFs for improved DVR classifications. We show the applicability of our uncertainty quantification framework to ensemble, downsampled, and bivariate versions of scalar field datasets.
Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps
T. M. Athawale, D. Maljovec, L. Yan, C. R. Johnson, V. Pascucci, and B. Wang
[Preprint, Supplementary material] [BibTex] [Preview video] [PubMed] [Presentation slides] [Presentation video (VIS 2020 uncertainty session, talk# 2)]
(IEEE Transactions on Visualization and Computer Graphics (TVCG), accepted, 2020)
Morse complexes are gradient-based topological descriptors with close connections to Morse theory. They are widely applicable in scientific visualization as they serve as important abstractions for gaining insights into the topology of scalar fields. Data uncertainty inherent to scalar fields due to randomness in their acquisition and processing, however, limits our understanding of Morse complexes as structural abstractions. We, therefore, explore uncertainty visualization of an ensemble of 2D Morse complexes that arises from scalar fields coupled with data uncertainty. We propose several statistical summary maps as new entities for quantifying structural variations and visualizing positional uncertainties of Morse complexes in ensembles. Specifically, we introduce three types of statistical summary maps -- the probabilistic map, the significance map, and the survival map -- to characterize the uncertain behaviors of gradient flows. We demonstrate the utility of our proposed approach using wind, flow, and ocean eddy simulation datasets.
Probabilistic Asymptotic Decider for Topological Ambiguity Resolution in Level-Set Extraction for Uncertain 2D Data
T. M. Athawale and C. R. Johnson
[Preprint,Appendix1,Appendix2] [BibTex] [PubMed] [Presentation slides] [Presentation video] [Source Code (MATLAB)]
(IEEE Transactions on Visualization and Computer Graphics (TVCG), Special Issue on IEEE VIS Conf, vol. 25, no. 1, pp. 1163-1172, Jan. 2019.)
We present a framework for the analysis of uncertainty in isocontour extraction. The marching squares (MS) algorithm for isocontour reconstruction generates a linear topology that is consistent with hyperbolic curves of a piecewise bilinear interpolation. The saddle points of the bilinear interpolant cause topological ambiguity in isocontour extraction. The midpoint decider and the asymptotic decider are well-known mathematical techniques for resolving topological ambiguities. The latter technique investigates the data values at the cell saddle points for ambiguity resolution. The uncertainty in data, however, leads to uncertainty in underlying bilinear interpolation functions for the MS algorithm, and hence, their saddle points. In our work, we study the behavior of the asymptotic decider when data at grid vertices is uncertain. First, we derive closed-form distributions characterizing variations in the saddle point values for uncertain bilinear interpolants. The derivation assumes uniform and nonparametric noise models, and it exploits the concept of ratio distribution for analytic formulations. Next, the probabilistic asymptotic decider is devised for ambiguity resolution in uncertain data using distributions of the saddle point values derived in the first step. Finally, the confidence in probabilistic topological decisions is visualized using a colormapping technique. We demonstrate the higher accuracy and stability of the probabilistic asymptotic decider in uncertain data with regard to existing decision frameworks, such as deciders in the mean field and the probabilistic midpoint decider, through the isocontour visualization of synthetic and real datasets.
A Statistical Framework for Quantification and Visualization of Positional Uncertainty in Deep Brain Stimulation Electrodes
T. M. Athawale, K. A. Johnson, C. R. Butson, and C. R. Johnson
[Preprint] [BibTex] [PubMed] [Poster] [Source code (MATLAB)]
(Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, vol. 7, no. 4, pp. 438-449, 2019.)
Deep brain stimulation (DBS) is an established therapy for treating patients with movement disorders such as Parkinson’s disease. Patient-specific computational modeling and visualization have been shown to play a key role in surgical and therapeutic decisions for DBS. The computational models use brain imaging, such as magnetic resonance (MR) and computed tomography (CT), to determine the DBS electrode positions within the patient’s head. The finite resolution of brain imaging, however, introduces uncertainty in electrode positions. The DBS stimulation settings for optimal patient response are sensitive to the relative positioning of DBS electrodes to a specific neural substrate (white/grey matter). In our contribution, we study positional uncertainty in the DBS electrodes for imaging with finite resolution. In a three-step approach, we first derive a closed-form mathematical model characterizing the geometry of the DBS electrodes. Second, we devise a statistical framework for quantifying the uncertainty in the positional attributes of the DBS electrodes, namely the direction of longitudinal axis and the contact-centre positions at subvoxel levels. The statistical framework leverages the analytical model derived in step one and a Bayesian probabilistic model for uncertainty quantification. Finally, the uncertainty in contact-centre positions is interactively visualized through volume rendering and isosurfacing techniques. We demonstrate the efficacy of our contribution through experiments on synthetic and real datasets. We show that the spatial variations in true electrode positions are significant for finite resolution imaging, and interactive visualization can be instrumental in exploring probabilistic positional variations in the DBS lead.
Isosurface Visualization of Data with Nonparametric Models for Uncertainty
T. M. Athawale, E. Sakhaee, and A. Entezari
[Preprint, Appendix] [BibTex] [PubMed] [Presentation slides] [Source code (C++, MATLAB)]
(IEEE Transactions on Visualization and Computer Graphics (TVCG), Special Issue on IEEE VIS Conf, vol. 22, no. 1, pp. 777-786, Jan. 2016.)
The problem of isosurface extraction in uncertain data is an important research problem and may be approached in two ways. One can extract statistics (e.g., mean) from uncertain data points and visualize the extracted field. Alternatively, data uncertainty, characterized by probability distributions, can be propagated through the isosurface extraction process. We analyze the impact of data uncertainty on topology and geometry extraction algorithms. A novel, edge-crossing probability based approach is proposed to predict underlying isosurface topology for uncertain data. We derive a probabilistic version of the midpoint decider that resolves ambiguities that arise in identifying topological configurations. Moreover, the probability density function characterizing positional uncertainty in isosurfaces is derived analytically for a broad class of nonparametric distributions. This analytic characterization can be used for efficient closed-form computation of the expected value and variation in geometry. Our experiments show the computational advantages of our analytic approach over Monte-Carlo sampling for characterizing positional uncertainty. We also show the advantage of modeling underlying error densities in a nonparametric statistical framework as opposed to a parametric statistical framework through our experiments on ensemble datasets and uncertain scalar fields.
Uncertainty Quantification in Linear Interpolation for Isosurface Extraction
T. M. Athawale and A. Entezari
[Preprint, Appendix] [Bibtex] [PubMed] [Presentation slides] [Source code (C++)]
(IEEE Transactions on Visualization and Computer Graphics (TVCG), Special Issue on IEEE VIS Conf, vol. 19, no. 12, pp. 2723-2732, Dec. 2013.)
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.
Quantification and Visualization of Spatial Uncertainty in Isosurfaces for Parametric and Nonparametric Noise Models
T. M. Athawale
[PhD dissertation preprint] [PhD defense presentation slides] [Proposal presentation slides]
(University of Florida, ProQuest Dissertations Publishing, 2015)
Defense committee: Dr. Alireza Entezari (Chair), Dr. Arunava Banerjee, Dr. Anand Rangarajan, Dr. Manuel Bermudez, and Dr. Haldun Aytug (External)
Isosurfaces are the key entities in visualization, and recent research has focused on the visualization of uncertain data. We study positional uncertainty in isosurfaces when data uncertainty is modeled with parametric and nonparametric density models. Noise in sampled data leads to topological and geometric variations in the extracted isosurface. We propose probabilistic techniques that reduce topological uncertainty in isosurfaces and resolve ambiguities in identifying topological configurations. We also characterize, in closed form, a random variable modeling the geometric uncertainty in isosurface extraction. Closed-form characterization of geometric uncertainty allows analytic computation of expected value of level-crossing location, as well as the variance. While former quantity can be used for constructing a stable isosurface for uncertain data, the latter can be used for visualizing the spatial uncertainty in the extracted isosurface. We show computational advantage of proposed analytic approach over expensive Monte-Carlo sampling approach for quantifying geometric uncertainty. The advantages of nonparametric statistical models over parametric noise models for uncertainty characterization are evident from our experiments on ensemble datasets and uncertain scalar fields. We also propose a novel approach that leverages nonlocal statistics for accurate characterization of underlying uncertainties in the nonparametric framework.