A Statistical Direct Volume Rendering Framework for Visualization of Uncertain Data

Transcription

1 A Statistical Direct Volume Rendering Framework for Visualization of Uncertain Data Elham Sakhaee, Alireza Entezari University of Florida, Gainesville, FL 1

2 Uncertainty visualization is important in final decision making. With no indication of uncertainty, a perception of accuracy is created. No indication of discretization error Discretized data visualized with uncertainty Low-resolution data Our Method 2

3 Propagating Uncertainty through the Rendering Pipeline Dataset courtesy of [Gröller et al., 2005] m 2 m 4 m 1 m 3 m m 6 m 7 m 5 m 8 Transfer Func/on m m 1 m 2 m 3 m 4 N = 2 4 n 1 n 2 m 6 n 3 m 7 m 5 m Acquisition Reconstruction/ Filtering Transfer Function Classification Shading Compositing Noise Downsampling Quantization Ensemble of Simulations Transfer Func/on uncertainty Our Method 3

4 Contribution: Uncertainty Propagation Acquisition Reconstruction/ Filtering Transfer Function Classification Shading Compositing A framework that: propagates uncertainty throughout the rendering pipeline is independent of the source of uncertainty allows real-time uncertainty visualization can be leveraged for different applications can be extended to non-parametric models [work in progress] 4

5 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-surface 5

6 Interpolation of Probability Distributions Acquisition Reconstruction/ Filtering Transfer Function Classification Shading Compositing X = X i w i X i with weights w i = '(p v i ) v 1 v 2 v 3 v 4 p v 6 v 8 Assuming independent random variables: v 5 v 7 pdf X (x) =pdf w1 X 1 (x) pdf w2 X 2 (x) pdf wk X K (x) 6

7 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

8 Box-splines 1 : A Brief Overview Box-splines are generalization of B-splines projection of hyper-cubes in R n onto lower dimensional space R s R 2! R 1 R 3! R 2 defined by n direction vectors in R s M [x1,x 2 ] M [x1,x 2,x 3 ] [1] C. de Boor, et al, Box Splines,

9 Box-splines: Statistical Viewpoint 1-D box-spline with one direction vector = 1 st order B-spline = Uniform Distribution Example: Linear interpolation of 1D box-splines: pdf X (x) =pdf w1 X 1 (x) pdf w2 X 2 (x) M [0.5,0.5] M [1] M [1] X 1 0.5X X 2 X 2 M [0.8,0.2] M [0.3,0.7] 0.8X X 2 0.3X X 2 9

10 Bilinear Interpolation of Histograms Histogram: superposition of (scaled) elementary box-splines v 1 v 2 p v 3 v 4 Higher-degree box-splines allow for modeling more general distributions, such as kernel density estimation 10

11 Uncertain Post-Classification Acquisition Reconstruction/ Filtering Transfer Function Classification Shading Compositing Traditional post-classification (table-lookup): Z ( ) = (t) (t )dt Transfer Func/on Distribu/on of certain data intensity Expected optical properties: opacity, color, texture, etc. regardless of how pdf is computed: E( ) = Z (t)pdf X (t)dt Transfer Func/on τ Distribu/on of uncertain data intensity 11

12 Shading Uncertain Data Acquisition Reconstruction/ Filtering Transfer Function Classification Shading Compositing Uncertain volume X = X i interpolate distributions with interpolation weights w i X i with weights w i = '(p v i ) Uncertain gradient field weights interpolate distributions with derivative filters 2 Y x 3 4Y y 5 = X Y z i i i i 5 X i with weights i i i 5 = (v v i ) 12

13 Sample Applications for Evaluation of the Proposed Framework Visualizing large datasets at reduced scale Iso-surface extraction in low-resolution volumes Ensemble visualization Visualization of noisy volumes 13

14 Visualizing Large Datasets at Reduced Scale b=8 b = 32 b = 16 Mean field 1 value/brick of size b3 29 : : : 1 [Thompson et al., 2011] 2b2 values/brick of size b3 29 : : : 211 Proposed: 2 values/brick of size b3 (min, max) 29 : : :2

15 Visualizing large datasets at reduced scale Representing uncertainty with non-compactly-supported distributions introduces additional uncertainty due to modeling. Ground truth Mean field Gaussian distributed random field Uniformly distributed random field 15

16 Iso-surface Extraction in Low-resolution Volumes Visualization of a synthetic scalar field 1 : f(x) = ( 1, 0, 0) x (1, 0, 0) x at iso-value 1 High-resolution field, synthesized on a grid low-resolution field, synthesized on a grid P (f <1) = 50% Proposed statistical rendering on uniformly-distributed random field at resolution Gaussian Process Regression on normally-distributed random field at resolution [1] [1] Schlegel et al., On the interpola/on of data with normally distributed uncertainty for visualiza/on,

17 Interactive Uncertainty Exploration The interface allows for interactively changing the amount of uncertainty at each grid point Data values ± 3/255 Data values ± 4/255 Data values ± 5/255 17

18 Iso-surface extraction in low-resolution volumes Low-resolution field Low-resolution field with uniform uncertainty 18

19 Ensemble Visualization Propagating ensemble uncertainty through the rendering pipeline results in a representative depiction of the underlying data. Original Fuel Dataset Statistical Rendering of an ensemble of 50 realizations of noise 19

20 Visualizing Noisy Volumes Original Fuel Dataset One sample of zero-mean uniform noise is added to each data point Statistical Rendering of the noisy volume 20

21 Conclusion Box-splines provide analytical representation for interpolating (non-parametric) probability distributions Efficient computation Real-time uncertainty visualization Computing expected optical properties, by redefining the transfer function classification, helps propagating uncertainty within the pipeline 21

22 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

23 Future Research Directions Modeling uncertainty with correlations (box-splines with zonotope support) Rendering with multi-dimensional transfer functions Extension to uncertainty represented by multi-bin histograms 23

24 Thank You! Questions?? 24

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