1/60 Statistical Analysis for Uncertainty Quantification and Visualization of Ensemble/Large-Scale Data Tushar Athawale, Scientific Computing & Imaging (SCI) Institute, University of Utah Advisor: Dr. Chris R. Johnson !2 Outline • Need for uncertainty visualization Level-set Direct volume visualization rendering • Uncertainty analysis for scientific visualization techniques • Level sets • Direct volume rendering • Morse complexes Morse complex • Uncertainty visualization for visualization domain-specific data • Red Sea eddy simulations • Deep brain stimulation (DBS) imaging • Conclusion and future work Red Sea DBS simulations !3 Uncertainty Quantification: Example Attach confidence information or add error bounds denoting uncertainty to your answer Decision: Whether to carry umbrella or not? Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !4 Uncertainty Visualization: Example Can you identify a tumor boundary? Whole brain atlas: http://www.med.harvard.edu/AANLIB/home.html Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !5 Uncertainty Visualization: Example Red dotted lines: Likely tumor boundaries Blue lines: Error bounds Whole brain atlas: http://www.med.harvard.edu/AANLIB/home.html Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !6 Uncertainty Visualization [Johnson and Sanderson, A next step: Visualizing errors or uncertainty, 2004] [Potter, Rosen, and Johnson, From quantification to visualization: A taxonomy of uncertainty visualization approaches, 2011] Additional tool for domain scientists to reason regarding scientific decisions Measurement and quantization errors Algorithmic and + numerical View Data reduction Interpolation approximation transformation uncertainties errors errors errors Final Data Data Mapping Rendering Displayable Acquisition Enrichment Image The Visualization Pipeline [Brodlie, Osorio, and Lopes, A review of uncertainty in data visualization, 2012] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !7 Distributions for Uncertainty Modeling High-Resolution Data: Hixel representation/ in-situ statistical summaries for large-scale data [Thompson et al., Analysis of large-scale scalar data using hixels, 2011] Ensemble Data: Multiple simulations for PDE solutions (Store min/max, approximate distributions from samples) [Wang et al., Visualization and visual analysis of ensemble data: A survey, 2018] Distribution field Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !8 Distributions for Uncertainty Modeling Ground truth Mean Parametric Nonparametric High Uncertainty Low Memory Memory Memory Memory consumption = 100*X consumption = X consumption = 2*X consumption = b*X Brick size = 100 b = # representative samples per brick Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !9 Uncertainty Quantification (Abstract Statistical Methods) Monte Carlo (easy but expensive) vs. Analytical (difficult but fast) Measurement and quantization errors Algorithmic and numerical + View Unknown: Data reduction Interpolation approximation transformation uncertainties errors errors errors pdfY(y) Given: Y (Feature) pdfX(x) X (Input data) Final Data Data Mapping Rendering Displayable Ensemble/ Acquisition Enrichment High-Resolution Image The Visualization Pipeline Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !10 Level-Set Visualizations for Uncertain Data Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !11 Level-Set Visualization Deep Brain Stimulation (DBS) Temperature Field Bioelectric-field Simulation Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !12 Level-Set Extraction The inverse problem: The level-set S corresponding to the isovalue k is given by: n n S = x R f(x)=k , where f : R R { 2 | } ! k = 20 k = 30 Input: Discrete Scalar Field Output: Level-Sets Visualization Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !13 Level-Set Extraction in Uncertain Data Research Question: Analysis of topological and geometric variations in level sets for uncertain scalar fields k = 20 k = 30 Input: Noisy Scalar Field Output: Level-Sets Visualization May not represent the true level sets! Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !14 Uncertainty Visualization of Level Sets (Monte Carlo) Spaghetti plots Probabilistic marching cubes Contour/Surface box plots [Potter et al., 2009] [P¨o thkow et al., 2011] [Whitaker et al., 2013; Genton et al., 2014] MedianMedian Isocontour High uncertainty Isocontour OutlierOutlier Isocontour Isocontour Low uncertainty 0 0.5 1 Level-crossing Visualization software: The WeaVER probability [Quinan and Meyer, 2016] The visualization of uncertain temperature field For isovalue (k) = 60 ◦ F 1D box plot Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !15 Level-Set Extraction in Uncertain Data (Analytical Approach) • Marching squares/cubes algorithm in certain vs. uncertain data (our contribution!) • Closed-form formulation of PdfY(y) High Algorithmic Measurement and View and numerical Transformation quantization Interpolation approximation errors errors Errors errors Variance Low Final Data Data pdfY(y) Mapping Rendering Acquisition Enrichment Displayable pdfX(x) Image Y (Level set) X (Input data) Marching squares/cubes algorithm Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !16 Marching Squares/Cubes Algorithm for Level-Set Extraction [Lorenson and Cline, 1987] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !17 Marching Square Algorithm (MSA) Bilinear interpolation function: f: [0,1]x[0,1] → R For each cell (assume bilinear interpolation model): f(x,y) = ax + by + cxy + d, where a = d10 - d00 , b = d01 - d00 • Extract isocontour topology (Which edges are crossed?) c = d00 + d11 - d01 - d10, d = d00 • Compute geometry (Where on the edge?) Bilinear interpolation curve (k=30) d01 = 24 d11 = 29 d01 = 24 d11 = 29 d01 = 24 d11 = 29 Isovalue (k) - - - - = 30 - + - + d00 = 28 d10 = 33 d00 = 28 d10 = 33 d00 = 28 d10 = 33 Input: Discrete Scalar Field A grid cell Find vertex signs: Inverse linear interpolation: with vertex data dij k - d00 30 - 28 dxy > k : + α = = = 0.66 dxy < k : - d10 - d00 33 - 30 α ∈ [0, 1] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !18 Marching Cubes Algorithm (MCA): Topological Cases The Stag Beetle dataset is courtesy of Vienna University of Technology https://www.cg.tuwien.ac.at/research/vis/datasets/ Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !19 Marching Cubes Algorithm in Action! ¨o https://www.youtube.com/watch?v=LfttaAepYJ8 Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !20 MSA: Topological Uncertainty + : if data dxy > k - : if data dxy < k k = Isovalue d01 = 26 d11 = 28 Dxy = Uncertain data - - dxy = Sampled data k = 30 Realization 1 Isocontour D01 = 24 2 D11 = 29 2 ± ± + d00 =27 d10 = 35 - d01 = 22 d11 = 31 - + D00 = 28 1 D10 = 33 2 ± ± Realization k = 30 A grid cell 2 Isocontour with uncertain input data Dij + d00 = 27 d10 = 33 - Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !21 Uncertainty-Aware Topology Prediction k k pdfD11 k = Isovalue pdfD01 Dxy = Uncertain data pdfDxy = Probability distribution of Dxy D11 D01 + + Predict signs k pdfD00 k + - Input: Distribution Field A grid cell pdfD10 Isocontour D00 D10 [Athawale and Entezari, 2013; Pr(D00 > k / + ) > 0.5 Pr(D10 < k /- ) > 0.5 Athawale et al., 2016] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !22 Uncertainty-Aware Topology Prediction k k pdfD11 k = Isovalue pdfD01 M = Uncertain midpoint data Dxy = Uncertain data pdfDxy = Probability distribution of Dxy D11 D01 - + k Predict signs + M k pdfD00 k + - Input: Distribution Field A grid cell pdfD10 Isocontour (Solid/Dotted?) D00 D10 [Athawale and Johnson, 2018] Pr(D00 > k / + ) > 0.5 Pr(D10 < k /- ) > 0.5 Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !23 Geometric Uncertainty Quantification k k pdfD11 k = Isovalue pdfD01 Dxy = Uncertain data pdfDxy = Probability distribution of Dxy D11 D01 + + Predict signs P1 P2 k + pdfD00 k - Input: Distribution Field A grid cell pdfD10 Edge-crossing at P1/P2? D00 D10 k - D00 = Pr(D00 > k / + ) > 0.5 Pr(D10 < k / ) > 0.5 - α D10 - D00 [Athawale and Entezari, 2013; Inverse linear interpolation Athawale et al., 2016] random variable density: pdf( α )? Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !24 Isosurface Extraction in