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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 Uncertain Data

Tangle function (synthetic data)

Parametric Distribution Nonparametric Distribution Ground truth Mean Field Field Field

[Athawale and Entezari, 2013; Athawale et al., 2016] (Visualization software: The Geomview, Developer: The Geometry Center at the University of Minnesota) Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !25 Isosurface Extraction in Uncertain Data

Bonsai tree (real data)

Mean Nonparametric

[Athawale and Entezari, 2013; Athawale et al., 2016] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !26 Marching Squares Algorithm: Ambiguous Case Resolution in Uncertain Data Concentric circles (synthetic data)

+ - + -

Probabilistic Probabilistic Ground truth Asymptotic Midpoint Decider Decider - + - + (Our Method) MSA ambiguous case

Certain data: [Nielson and Hamann, 1991]

Uncertain data: [Athawale and Johnson, 2018] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !27 Isocontour Visualizations (K `a rm `a n Vortex Street)

Probabilistic Midpoint Decider

Probabilistic Asymptotic Decider (Our Method) The flow simulation dataset is courtesy of the Gerris project [Popinet, 2003] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !28

Direct Volume Rendering for Uncertain Data Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !29

Direct Volume Rendering [Hadwiger et al., 2006]

d2 d3 Samples (s) d4 d1 Color/opacity

s d6 d7 Intensity Imaging plane Data (d) d5 d8 s Transfer Volumetric Reconstruction Function data Classification

Compositing Shading

Final image

Imaging plane Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !30 Uncertainty-Aware Direct Volume Rendering

The teardrop function [Knoll et al., 2009] m2 m3 Color/opacity m1 m4

s Intensity m6 m7 s Mean m5 m8 Transfer High- Reduce Reconstruction Function Shading Compositing Resolution/ Classification ensemble Data Parametric Uncertainty Color/opacity in intensity (S)

S Intensity Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !31 DVR of Uncertain Data (Nonparametric)

(a) Ground truth (b) Mean (512x512x1559) (64x64x195)

(a) Ground (b) Mean (c) Parametric (d) Nonparametric truth [Sakhaee and Entezari, (Our contribution) 2017] [Athawale et al., 2020] (c) Parametric (d) Nonparametric The teardrop function (64x64x195) (64x64x195) (ensemble dataset) [Sakhaee and Entezari, (our contribution) 2017] [Athawale et al., 2020]

Visualization software: Voreen Osirix OBELIX dataset (http://medvis.org/datasets/) (http://voreen.uni-muenster.de) Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !32

Morse Complex Visualizations for Uncertain Data Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !33 Morse Complex Visualizations

Understanding structure of Segmenting molecular surfaces [Natarajan et al., 2006] turbulent mixing layers [Laney et al. 2006] [Shivashankar et al., 2012] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !34 Morse Complex Visualizations

Topological descriptors of gradient flows of a scalar field

Z

Y X The Ackley function Morse complex visualization [Ackley D. H., 1987] of the Ackley function Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !35 Effect of Noise on Morse Complexes

Ground truth Morse complex

Morse complex extraction

Ackley function [Ackley, 1987] Ensemble member 1 Ensemble member 2 Ensemble member 3

Mix noise and extract Morse complex Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !36 Uncertainty Visualization of Morse Complexes

Visualize agreement/certainty and disagreement/uncertainty among Morse complexes for ensembles

+

Ensemble of Agreement Regions Uncertainty Regions Morse Complexes

[Athawale et al., 2020] Expected boundaries Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !37 Interactive PDF Queries for Uncertain Regions

Selection 0 Selection 1 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 Probability 0.1 Probability 0.1 0 0 Cell Cell

Selection 2 Selection 3 0.5 0.4 0.4 0.3 0.3 0.2 [K. Potter, R. M. Kirby, D. Xiu, and C. R. 0.2 Probability Johnson; Interactive Visualization of Probability Probability 0.1 0.1 and Cumulative Density Functions; 2011] 0 0 Cell Cell Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !38 Wind Ensemble Dataset Expected

Certain (entropy=0) Uncertain (entropy>0)

Mean-field Morse complex (no indication of spatial uncertainty!)

Uncertainty Visualization

Dataset courtesy: http://iridl.ldeo.columbia.edu/SOURCES/.ECMWF/.S2S/ Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !39

Uncertainty Visualization of Red Sea Eddy Simulation Data Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !40 The Red Sea Eddy Simulations

IEEE SciVis Contest 2020:

• Kaust Supercomputing Core Lab

• Large-scale eddy simulations (~1.5 TB) • https://kaust-vislab.github.io/SciVis2020/ • We were selected as the contest finalist! Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !41 Statistical Volume Rendering: Interactive Exploration

Mean-field: Distribution per vertex Mean per-vertex Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !42 Quartile View: Uncertainty Visualization

(a) Lower quartile (b) Central 50%, IQR (c) Upper quartile

e3 e2

Confidence regarding the eddy e1 presence/position: e1: Low IQR e2: Moderate e3: High

1D box plot Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !43 Agreement Exploration Uncertain vortex

Probabilistic visualizations for Morse complexes

Vortex structures

60% 70% 80%

x%: Positions that have at least x% probability of flowing to the same local maximum Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !44

Uncertainty Visualization for Deep Brain Stimulation (DBS) Imaging Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !45 Deep Brain Stimulation (DBS)

Medtronic DBS electrode Mo. 3387

Voltage: 1-5 V Frequency: 120-185 Hz Pulse width: 60-200 µs Contacts: Cathode(-)/Anode(+)/.Off

Knowledge of precise electrode positions in the patient brain is essential in order to set optimal patient-specific stimulation pattern. Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !46 Patient Head Image with Implanted Electrodes

DBS lead Post-surgery MRI to capture schematic electrode positions

DBS lead DBS lead

Patient image illustration

c1 Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !47 Uncertainty Visualization of Implanted Electrodes

1 Isovalue=0.9 1mm Scale Standard Deviation = 0.71 mm PositionalLikelihood

Isovalue=0.95 [Athawale et al., 2019] Positional Likelihood Standard 0 Deviation = 0.48 mm

The volume The confidence visualization visualization

Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !48

Solar Farm Simulation Data Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !49 Temperature and Air Flow Visualizations

Eddy formation [Athawale, Staninslawski et al., 2021]

Visualization software: ParaView [Ahrens et al., 2005] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !50 Temperature Variation Analysis

Maximum

Q3

Mean

Q1

Minimum

(a) Minimum, (b) Maximum, (c) Mean, (d) Standard Deviation Quartile plot Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !51 Conclusions

• Uncertainty visualizations are important for avoiding misleading interpretations regarding the underlying data • Level sets • Direct volume rendering • Morse complex visualizations • Red Sea eddy simulations • Deep brain stimulation imaging • Statistical methods for uncertainty quantification • Monte Carlo vs. Analytical • Methods for uncertainty visualization • Color mapping proportional to the level of confidence/uncertainty • Color blending • Interactive probability distribution queries • Derive uncertainty volumes and visualize them using isosurfaces/direct volume rendering Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !52 Future Work: More Domain-Specific Data

Electrocardiogram Imaging Visualize uncertainty or inconsistency (ECGI) of ECGI inverse solutions ECG lead Topological Level sets analysis

Probability Confidence Potential (mV) Ensemble data representing the uncertainty in potentials on 6.54 epicardial (heart) surface 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 -39.4 -27.9 -16.4 -4.95

Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !53 Future Work: More Scientific Data Analysis Algorithms and High-Dimensional Data

Algorithms:

• Contour trees [Carr et al., 2003] • Fibre Surfaces [Carr et al., 2011] • Image segmentation and registration

High-dimensional data:

• Vector/tensor data • Extension of the published Feature level-sets Feature confidence uncertainty work for 2D domains to [Jankowai, 2018] level-sets 3D/higher dimensions [Sane, Athawale, and Johnson, 2021]

Visualization software: VisIt [Childs et al., 2011] Introduction Uncertainty Vis for SciVis Techniques Uncertainty Vis for Domain-Specific Data Conclusion & Future Work !54 Future Work: Value of Uncertainty Visualizations

• Extension of the IEEE VIS 2005 paper by J. J. van Wjik on the Value of Visualizations

• Understanding development costs relevant to uncertainty visualizations - What are implementation and hardware costs?

• Understanding exploration costs relevant to uncertainty visualizations - Are uncertainty visualizations informative or confusing for users? - Does decision-making quality vary for differernt uncertainty visualizations of same data? !55 Journal Publications

• T. M. Athawale, B. Ma, E. Sakhaee, C. R. Johnson, and A. Entezari; Direct Volume Rendering with Nonparametric Models of Uncertainty, IEEE Transactions on Visualization and Computer Graphics, Special Issue on IEEE VIS Conf, 2020.

• T. M. Athawale, D. Maljovec, L. Yan, C. R. Johnson, V. Pascucci, and B. Wang; Uncertainty Visualization of 2D Morse Complex Ensembles using Statistical Summary Maps, IEEE Transactions on Visualization and Computer Graphics (to appear), 2020.

• T. M. Athawale and C. R. Johnson; Probabilistic Asymptotic Decider for Topological Ambiguity Resolution in Level-Set Extraction for Uncertain 2D Data, IEEE Transactions on Visualization and Computer Graphics, Special Issue on IEEE VIS Conf, vol.25, no. 1, pp. 1163-1172, Jan. 2019.

• T. M. Athawale, K. A. Johnson, C. R. Butson, and C. R. Johnson; A Statistical Framework for Visualization of Positional Uncertainty in Deep Brain Stimulation Electrodes., Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, pp. 1-12, Oct. 2018.

• T. M. Athawale, E. Sakhaee, and, A. Entezari; Isosurface Visualization of Data with Nonparametric Models for Uncertainty, IEEE Transactions on Visualization and Computer Graphics, Special Issue on IEEE VIS Conf, vol.22, no.1, pp.777-786, Jan. 2016.

• T. M. Athawale and A. Entezari.; Uncertainty Quantification in Linear Interpolation for Isosurface Extraction, IEEE Transactions on Visualization and Computer Graphics, Special Issue on IEEE VIS Conf, vol.19, no.12, pp.2723-2732, Dec. 2013. !56 Conference Proceedings/ Workshops

• T. M. Athawale, B. Stanislawski, S. Sane and C. R. Johnson; Visualizing Interactions Between Solar Photovoltaic Farms and the Atmospheric Boundary Layer, to appear in EnergyVis '21, 2021, Torino, Italy (virtual).

• S. Sane, T. M. Athawale, and C. R. Johnson; Visualization of Uncertain Multivariate Data via Feature Confidence Level-Sets, to appear in EuroVis 21, 2021, Zurich, Switzerland (virtual).

• T. M. Athawale, A. Entezari, B. Wang, and C. R. Johnson; Statistical Rendering for Visualization of Red Sea Eddy Simulation Data, IEEE VIS 2020 SciVis Contest Finalist, Salt Lake City, USA.

• T. M. Athawale, K. A. Johnson, C. R. Butson, and C. R. Johnson; A Statistical Framework for Visualization of Positional Uncertainty in Deep Brain Stimulation Electrodes, 2019 IEEE Workshop on Visual Analytics in Healthcare (VAHC), Vancouver, Canada. !57 Thanks to Advisors and Co-authors!

Dr. Chris Johnson Dr. Alirea Entezari Dr. Chris Butson Dr. Bei Wang Dr. Valerio Pascucci Dr. Marc Calaf (Postdoctoral advisor) (PhD advisor) (DBS project) (Morse complex project) (Morse complex project) (Solar energy project)

Dr. Bo Ma Dr. Elham Sakhaee Dr. Kara Johnson Lin Yan Dr. Dan Maljovec Dr. Sudhanshu Sane Dennis Njeru Brooke Stanislawski (Direct volume (Direct volume (DBS project) (Morse complex (Morse complex (Multivariate (ECGI (Solar energy rendering project) rendering project) project) project) uncertainty analysis project) project) project) !58 Thank you!

This research is supported in part by by the NIH grants P41 GM103545-18 and R24 GM136986; the DOE grant DE-FE0031880; the Intel Graphics and Visualization Institutes of XeLLENCE; and the NSF grants IIS-1617101, IIS-1910733.

For any questions, please contact at: Email: [email protected] Personal website: http://tusharathawale.info