A Statistical Direct Volume Rendering Framework For Visualization Of .

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A Statistical Direct Volume Rendering Framework forVisualization of Uncertain DataElham Sakhaee, Alireza EntezariUniversity of Florida, Gainesville, FL1

Uncertainty visualization is important in final decision making. With no indication of uncertainty, a perception of accuracy is created.No indication ofdiscretization errorLow-resolution dataDiscretized datavisualized with uncertaintyOur Method2

Dataset courtesy of [Gröller et al., 2005]Propagating Uncertainty through the Rendering samplingQuantizationEnsemble ofSimulations Reconstruction/FilteringTransfer isition m2m423n1N 4 n2 5n3m7m8ShadingCompositinguncertaintyOur Method3

Contribution: Uncertainty sfer FunctionClassificationShadingCompositing A framework that:–––––propagates uncertainty throughout the rendering pipelineis independent of the source of uncertaintyallows real-time uncertainty visualizationcan be leveraged for different applicationscan be extended to non-parametric models [work in progress]4

Previous Work Visualizing uncertainty in ensembles [Sanyal et al., 2010; Whitaker et al., 2013] Uncertain iso-surface extraction [Grigoryan & Rheingans, 2004; Pöthkow & Hege, 2011, 2013] Uncertainty in data processing [Pang et al., 1997; Brodlie et al., 2012; Fout and Ma, 2012] Visualization of large-scale data [Schlegel et al., 2012]– interpolation of normally distributed data Rendering probable iso-surfaces [Thompson et al., 2011]– Hixels as a representation for bricks of large-scale data– Visualizing likelihood of presence of an iso-surface5

Interpolation of Probability ansfer FunctionClassificationShadingCompositingv4v2X Xv1w i Xiwith weightswi '(pv3vi )piv6v8 Assuming independent random variables:v5v7pdfX (x) pdfw1 X1 (x) pdfw2 X2 (x) · · · pdfwK XK (x)6

How to represent uncertainty? Box-splines are a suitable choice:––––The space of box-splines is closed under convolution.Convolution can be computed analytically (and efficiently).Box-splines can represent non-parametric distributions.Compact-support of box-splines avoid introducing additional uncertainty.7

Box-splines1: A Brief Overview Box-splines are[1] C.21– generalization of B-splinesns– projection of hyper-cubes in R ! Rn! Rsonto lower dimensionalRspaceR !R– defined by n direction vectorsin RsRn !M[x1 ,x2 ]de Boor, et al, “Box Splines”, 1993R3 ! R2M[x1 ,x2 ,x3 ]8

Box-splines: Statistical Viewpoint 1-D box-spline with one direction vector 1st order B-spline Uniform Distribution Example: Linear interpolation of 1D box-splines:pdfX (x) pdfw1 X1 (x) pdfw2 X2 (x) · · · pdfwK XK (x)M[0.5,0.5]M[1]M[1]0.5X1 0.5X2X1M[0.8,0.2]0.8X1 0.2X2X2M[0.3,0.7]0.3X1 0.7X29

Bilinear Interpolation of Histograms Histogram: superposition of (scaled) elementary box-splinesv2v1pv3v4 Higher-degree box-splines allow for modeling more generaldistributions, such as kernel density estimation10

Uncertain ringTransfer FunctionClassification Traditional post-classification (table-lookup): ( ) Z (t) (tShadingCompositingTransferFunc/onDistribu/on ofcertain data )dtintensityτ Expected optical properties: opacity, color, texture, etc.regardless of how pdf is computed:TransferE( ) ZFunc/onDistribu/on ofuncertain data (t)pdfX (t)dtintensity11

Shading Uncertain DataAcquisitionReconstruction/FilteringTransfer FunctionClassification Uncertain volumeX ShadingCompositinginterpolate distributions with interpolation weightsXw i Xiwith weightswi '(pvi )i Uncertain gradient fieldweights2323YxX i4 Yy 5 4 i 5 XiiYziinterpolate distributions with derivative filterswith weights24 iii35 (vvi )12

Sample Applications for Evaluation of the Proposed Framework Visualizing large datasets at reduced scaleIso-surface extraction in low-resolution volumesEnsemble visualizationVisualization of noisy volumes 13

Visualizing Large Datasets at Reduced Scaleb 8b 32b 16Mean field1 value/brick of size b329 : 1212 : 1215 : 1[Thompson et al., 2011]2b2 values/brick of size b329:27212:29215 : 211Proposed:2 values/brick of size b3(min, max)29 : 2212 : 214215:2

Visualizing large datasets at reduced scale Representing uncertainty with non-compactly-supported distributionsintroduces additional uncertainty due to modeling.Ground truthMean fieldGaussian distributedrandom fieldUniformly distributedrandom field15

Iso-surface Extraction in Low-resolution VolumesVisualization of a synthetic scalar field1: f (x) ( 1, 0, 0) High-resolution field, synthesized ona 255 255 255 gridx · (1, 0, 0)x at iso-value 1low-resolution field, synthesized ona 12 12 12 gridP (f 1) 50%Proposed statistical rendering onuniformly-distributed random field at resolution12 12 12[1] SchlegelGaussian Process Regression onnormally-distributed random field at resolution12 12 12 [1]et al., “On the interpola/on of data with normally distributed uncertainty for visualiza/on” , 201216

Interactive Uncertainty Exploration The interface allows for interactively changing the amount ofuncertainty at each grid pointData values 3/255Data values 4/255Data values 5/25517

Iso-surface extraction in low-resolution volumesLow-resolution fieldLow-resolution fieldwith uniform uncertainty18

Ensemble Visualization Propagating ensemble uncertainty through the rendering pipelineresults in a representative depiction of the underlying data.Original Fuel DatasetStatistical Rendering of an ensembleof 50 realizations of noise19

Visualizing Noisy VolumesOriginal Fuel DatasetOne sample of zero-meanuniform noise is added toeach data pointStatistical Rendering ofthe noisy volume20

Conclusion Box-splines provide analytical representation for interpolating(non-parametric) probability distributions Efficient computationReal-time uncertainty visualization Computing expected optical properties, by redefining the transferfunction classification, helps propagating uncertainty within thepipeline21

Conclusion: A Framework for Uncertainty Propagation Box-splines provide an analytical representation for interpolating (nonparametric) probability distributions. Efficient computations allow for interactive uncertainty visualization. Expected optical properties can be computed via the redefined postclassification.22

Future Research Directions Modeling uncertainty with correlations (box-splines with zonotopesupport) Rendering with multi-dimensional transfer functions Extension to uncertainty represented by multi-bin histograms23

Thank You!Questions?24

Ensemble Visualization (iso-contour extraction)Mean field visualization for an ensemble oftemperature field with 63 ensembles [1]Statistical rendering overlaid with mean fieldvisualization[1] T.25 2004.Palmer et al., “Development of a European mul/-model ensemble system for seasonal to inter-annual predic/on (demeter),”

Uncertain Post-Classification Traditional post-classification (table-lookup): Expected optical properties: opacity, color, texture, etc. regardless of how pdf is computed: intensity Transfer Funcon Distribu/on of uncertain data intensity Transfer Funcon τ Distribu/on of certain data Acquisition Reconstruction/ Filtering

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