Quantification and Visualization of Spatial Uncertainty in - - PowerPoint PPT Presentation

1/115 Introduction Related work Problem Description and Approach Results and Conclusion

Quantification and Visualization of Spatial Uncertainty in Isosurfaces for Parametric and Nonparametric Noise Models

Tushar Athawale Department of Computer & Information Science & Engineering University of Florida April 3, 2015

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Outline

• Introduction
• 1. Need for uncertainty visualization
• 2. Problems arising in marching squares algorithm (MSA) for

uncertain data

• Related Work
• 1. Uncertainty visualization techniques
• 2. Uncertainty quantification techniques
• Problem Description and Approach
• 1. Topology prediction for marching cubes algorithm (MCA)
• 2. Uncertainty quantification in MCA geometry
• Results, Conclusion, and Future Work

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Data Visualization Paradigms

(a) Indirect Visualization (b) Direct Visualization

• Two fundamental paradigms for data visualization:
• Indirect Visualization
• Direct Visualization

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Isosurfaces: Biomedical Imaging

Figure: Blood Flow Visualization

(Image source: http://www.visitusers.org/images/c/c9/Bflow tutorial vel mag iso .png)

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Isosurfaces: Climate Studies

Figure: Isobar Visualization

(Image source: http://facweb.bhc.edu/academics/science/harwoodr/GEOL101/Labs/Wind/Images/IsobarMap3.gif)

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Sources of Uncertainty

• Errors in final visualization can lead to drawing wrong

conclusions and may impact sensitive applications.

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Sources of Uncertainty

• Errors in final visualization can lead to drawing wrong

conclusions and may impact sensitive applications.

• Uncertainty visualization, therefore, has become an important

research direction.

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Sources of Uncertainty

• Errors in final visualization can lead to drawing wrong

conclusions and may impact sensitive applications.

• Uncertainty visualization, therefore, has become an important

research direction.

• We study impact of uncertainty in data acquisition phase on

final visualization.

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Data Acquisition: Sampling/Discretization Errors

• Function of continuous variable is represented by finite

number of evaluations at grid locations.

• Sampling affects gradient values.

Figure: Continuous function sampled at finite space locations

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Data Acquisition: Quantization Errors

• Rounding off errors introduced due to limited instrument

precision

Figure: Quantization errors

(Image source:http://en.wikipedia.org/wiki/Quantization (signal processing))

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Data Acquisition: Simulation Data

• Physical systems characterized using state specified by space

and time

• Partial Differential Equations (PDE): Relate rate of change in

system state with respect to space and time (or other variables)

• Heat Equation:

∂u ∂t = h2 ∂2u ∂x2

• Wave Equation:

∂2u ∂t2 = h2 ∂2u ∂x2

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Data Acquisition: Ensemble Simulations

Consider PDE: ∂2u ∂x2 − ku = 0 When k = 1 then u = Aex + Be−x When k = 0 then u = Cx + D When k = −1 then u = Ecos(x) + Fsin(x)

• Uncertainty in coefficients of PDEs (k in this case) can lead

to diverse solutions.

• Diversity in solutions can be captured in ensemble dataset.
• Each ensemble member represents a solution corresponding to

sampled coefficient value.

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Marching Squares Algorithm (MSA)

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Contour Extraction Problem

• Isocontour represents a curve, where every point on the curve

attains an equal value. This value is called isovalue.

• Isocontour ’S’ corresponding to the isovalue c is given by

S = {x ∈ R2 | f (x) = c}, where f : R2 → R.

(a) Scalar Field (b) Isocontour Figure: Figure shows an example of isobars. Black solid lines represent points where pressure magnitude is one of 20, 30, 40, 50, and 60 units.

(Image source: http://www.econym.demon.co.uk/isotut/isobars.htm)

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Marching Squares Algorithm

For each cell of a scalar grid

• Determine isocontour topology.
• Determine isocontour geometry.

(a) Scalar Field (b) Isocontour Figure: Red lines mark a cell of the scalar grid. (Image source:

http://www.econym.demon.co.uk/isotut/isobars.htm)

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Research Problem

Analysis of the topological and geometric changes in the isocontour/isosurface for the given isovalue when sampled function values are uncertain.

(a) Scalar Field (b) Isocontour Figure: Red lines mark a cell of the scalar grid. (Image source:

http://www.econym.demon.co.uk/isotut/isobars.htm)

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Isocontour Topology

• f (x) > isovalue : Positive vertex (Marked with dark blue

circle)

• f (x) < isovalue : Negative vertex (Not marked)
• Identify cell edges crossed by isocontour (Highlighted in green)

Figure: Cell configuration for isovalue c = 30

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Isocontour Topology

Figure: Marked vertices represent positive vertices. Non-marked vertices indicate negative vertices. Topology of the isocontour for an isovalue c is determined by cutting off positive vertices from negative vertices.

24 possible cell configurations!

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MSA Cell Configurations

Figure: Basic cell configurations

• 16 configurations reduce to only 4 configurations using

symmetry.

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MCA Cell Configurations

Figure: Using symmetric properties, 28 possible configurations reduce to

• nly 15 basic configurations. However, number of basic configurations

grows to 88 when ambiguous configuration cases are considered. Figure shows configurations as published in [LC87].

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MCA Cell Configurations

Figure: The stag dataset: Isovalue c = 580

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Multilinear Interpolation

Figure: Left image: Bilinear interpolation is equivalent to linear interpolation in two directions. Right image: Piecewise linear approximation to the bilinear interpolation curve (dotted).

• Bilinear interpolation : F(x, y) = ax + by + cxy + d

(Equation of Hyperbola)

• Bilinear interpolation coincides with linear interpolation

(F(x) = ax + b) on the grid edges.

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MSA Cell Configurations

Figure: Basic cell configurations

• 16 configurations reduce to only 4 configurations using

symmetry.

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Asymptotic Decider

Figure: Asymptotic decider by Nielson and Hamann [NH91] to resolve the configuration

• ambiguities. Dotted straight lines

show asymptotes of the hyperbolic curves. Isovalue c = 30

.

• Value attained by the intersection of asymptotes of the

hyperbolic arcs, i.e., saddle point S, can be computed using the following formula: f (S) = x1x3 − x2x4 x1 + x3 − x2 − x4

• If saddle point is positive, cut off the negative cell vertices and

vice versa.

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Midpoint Decider

Figure: Isovalue c = 30

• Value attained at the cell midpoint, M, can be computed

using the following formula: f (M) = x1 + x2 + x3 + x4 4

• If midpoint is positive, cut off the negative cell vertices and

vice versa.

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Isocontour Geometry

Determine the isocontour-crossing locations on the edges using inverse linear interpolation.

Figure: Isocontour geometry for isovalue c = 30

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Marching Cubes Algorithm

For each cell of the scalar grid

• Determine the isosurface topology and isosurface geometry

that is consistent with trilinear interpolation.

Figure: Using symmetric properties, 28 possible configurations reduce to only 15 basic configurations. However, number of basic configurations grows to 88 when ambiguous configuration cases are

• considered. Figure shows configurations as published in [LC87].

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Marching Cubes in Action

. . . Video is courtesy of Koen Samyn.

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Marching Squares Algorithm (MSA) in Uncertain Data

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Topological Uncertainty

Isovalue c = 30

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Ambiguous Topology: Decider Uncertainty

Isovalue c = 30

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Geometric Uncertainty

Isovalue c = 30

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Uncertainty Visualization Techniques

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Color Mapping

Figure: Left image: Original isosurface. Right image: Isosurface with color-mapped uncertainties. Red regions indicate areas of high spatial

• uncertainties. Image is courtesy of [RLB+03]

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Primitive Displacement

Figure: The leftmost image: Original isosurface. The middle image: Color-mapped uncertainties. The rightmost image: Isosurface with points displaced in the surface normal direction proportional to the uncertainty. Image is courtesy of [GR04].

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Glyphs

Figure: Left image: Uncertainty in the wind direction visualized using uncertain arrow glyphs (image source: http://slvg.soe.ucsc.edu/

images.uglyph/uncertain.gif). Right image: Cylindrical glyphs to represent

local data uncertainty (image is courtesy of [NL04])

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Uncertainty Quantification Techniques

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Visualization of Correlation Structures

Figure: Multi-level clustering where each cluster stands for minimum positive correlation strength. Image is courtesy of [PW12].

Positively correlated regions imply low structural variability in the isosurface and vice versa [PW12].

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Isosurface Condition Analysis

Figure: For the noise amplitude of ǫ in f (x) = c, there is higher uncertainty in x2 than in x1.

Areas of high data gradient imply low spatial uncertainty in the isosurface and vice versa [PH11].

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Probabilistic Marching Cubes

Figure: Direct volume rendering of the cell-crossing probabili- ties [PWH11].

.

• Direct volume rendering of cell-crossing probabilities for

isosurface

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Isosurface Uncertainty: Direct vs. Indirect Visualization

• State of the art: Direct volume rendering of cell-crossing

probabilities for isosurface

• Our work: Quantification and visualization of isosurface

uncertainties while not shifting to direct visualization paradigm

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Problem Description and Approach

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Characterizing Data Uncertainty

• Field with independent random variables
• Characterization of uncertainty using probability density

function

• Propagation of data uncertainty into marching cubes

algorithm

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Topology Prediction for Isosurface in Uncertain Data

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Isosurface Topology Problem

• Classification of vertices (positive/negative) determine

isosurface topology.

Aim : Given access to each vertex pdf, design a scheme to recover vertex classification corresponding to underlying data?

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Scheme 1: Vertex-based Classification

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Vertex-based Classification

• Process each vertex independently
• If Pr(X > c) > Pr(X < c), classify vertex as positive and vice

versa.

Figure: Shaded areas show most probable vertex sign for isovalue c.

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Vertex-based Classification

• If Pr(X > c) > Pr(X < c), classify vertex as positive and vice

versa.

• Approach doesn’t consider signs of neighboring vertices!

Figure: Shaded areas show most probable vertex sign for isovalue c.

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Scheme 2: Edge-based Classification

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Edge-crossing Probability

• Edge-crossing probability for isosurface with isovalue c for

independent random variables X and Y :

1 − Pr(X > c) · Pr(Y > c) − Pr(X < c) · Pr(Y < c)

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Edge-based Classification

• When edge-crossing probability is relatively high, we want
• pposite signs and vice versa.

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Edge-based Classification

• What is a vertex classification (+1/-1) corresponding to

underlying data given edge-crossing probabilities?

Figure: Numbers on edges represent edge-crossing probabilities.

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Optimization Problem

s∗ = arg min

sn=±1

sTWs. W: weight matrix of edge-crossing probabilities s: sign vector sn: n’th entry of matrix s

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Optimization Problem

s∗ = arg min

sn=±1

sTWs. W: weight matrix of edge-crossing probabilities s: sign vector sn: n’th entry of matrix s

• Solution: Combinatorial approach (Not practical!)

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Relaxed Optimization Problem

s∗ = arg min

sn=±1

sTWs. w: weight matrix of edge-crossing probabilities s: sign vector sn: n’th entry of matrix s

• Solution: Eigenvector of W with largest (negative) eigenvalue
• Use signs of eigenvector entries for vertex classification
• Computationally expensive compared to scheme 1

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Ambiguous Configurations in Uncertain Data

Aim : Given access to each vertex pdf, design a scheme to recover topology corresponding to ambiguous configurations?

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Uncertain Midpoint Decider

• Random variable corresponding to 1-d cell midpoint:

M = X1+X2

2

• Sum of random variables corresponds to convolution of

densities.

Uniforms with equal bandwidths Uniforms with unequal bandwidths Multiple uniforms with unequal bandwidths

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Uncertain Midpoint Decider

Figure: Convolution of uniform kernels with unequal bandwidths

• Face midpoint random variable: M = X1+X2+X3+X4

4

• Face midpoint density (PdfM): Cubic univariate box-spline

with non-uniform knots

• Body (3-d cell) midpoint random variable: M = X1+··+X8

8

• Body (3-d cell) midpoint density (PdfM): Degree 7 univariate

box-spline with non-uniform knots

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Uncertain Midpoint Decider

• Random variable corresponding to cell midpoint:

M = X1+X2+X3+X4

4

• Sum of random variables corresponds to convolution of

densities.

• Vertex-based classification for M to make topological decision.

Figure: Shaded areas show most probable vertex sign for isovalue c.

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Uncertainty Quantification in Isosurface Geometry for Uncertain Data

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Uncertainty Quantification in Linear Interpolation

Figure: Left: Ratio density for uniform parametric model; Right: Ratio density for nonparametric model with uniform base kernel.

Aim : Closed-form characterization of the ratio random variable, Z =

c−X1 X2−X1 , assuming X1 and X2 have parametric or nonparametric

distributions.

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Ratio Density for Uniform Noise Model

63/115 Introduction Related work Problem Description and Approach Results and Conclusion

Uncertainty Quantification in Linear Interpolation

Figure: µi and δi represent mean and width, respectively, of a random variable Xi. c is the isovalue. v1 and v2 represent the grid vertices.

Aim : Closed-form characterization of the ratio random variable, Z =

c−X1 X2−X1 , when X1 and X2 are uniformly distributed.

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Joint Distribution

Find the joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1, where Z = Z1

Z2 = c−X1 X2−X1 .

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Joint Distribution

• Determine the range of

c − X1.

• X1 assumes values in the

range [µ1 − δ1, µ1 + δ1].

• Random variables Z1 and

Z2 are dependent. µi and δi represent mean and width, respectively, of a random variable Xi.

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Joint Distribution

• Determine the range of

X2 − X1.

• X2 assumes values in the

range [µ2 − δ2, µ2 + δ2].

• Random variables Z1 and

Z2 are dependent. µi and δi represent mean and width, respectively, of a random variable Xi.

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Joint Distribution

• Determine the range of

X2 − X1.

• X2 assumes values in the

range [µ2 − δ2, µ2 + δ2].

• Random variables Z1 and

Z2 are dependent. µi and δi represent mean and width, respectively, of a random variable Xi.

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Joint Distribution

• Parallelogram represents

the joint distribution of the dependent random variables Z1 = c − X1 and Z2 = X2 − X1.

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Joint Distribution

Shape and position of the joint distribution is impacted by relative configurations for X1 and X2 and the isovalue c.

(a) Non-overlapping (b) Overlapping (c) Contained

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Cumulative Density Function

What is Pr( Z1

Z2 ≤ m)?

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Cumulative Density Function

• What is Pr( Z1

Z2 ≤ m)?

• cdfZ(m) = Pr(−∞ ≤

Z1 Z2 ≤ m) (orange region).

cdfZ(m) represents cumulative density function of a random variable Z.

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Probability Density Function

• What is Pr( Z1

Z2 ≤ m)?

• cdfZ(m) = Pr(−∞ ≤

Z1 Z2 ≤ m) (orange region).

• Obtain pdfZ(m) by

differentiating cdfZ(m) with respect to m. pdfZ(m) represents probability density function of a random variable Z.

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Probability Density Function

• What is Pr( Z1

Z2 ≤ m)?

• cdfZ(m) = Pr(−∞ ≤

Z1 Z2 ≤ m) (orange region).

• Obtain pdfZ(m) by

differentiating cdfZ(m) with respect to m.

• A piecewise inverse

polynomial function.

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Probability Density Function

pdfZ(m) = (c−µ2)2+δ2

2

4δ1δ2(1−m)2

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Probability Density Function

pdfZ(m) = (c−µ2)2+δ2

2

4δ1δ2(1−m)2

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Probability Density Function

pdfZ(m) = (µ2+δ2−c)2m2+(µ1+δ1−c)2(1−m)2

8δ1δ2m2(1−m)2

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Probability Density Function

pdfZ(m) = (µ2+δ2−c)2m2+(µ1+δ1−c)2(1−m)2

8δ1δ2m2(1−m)2

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Probability Density Function

pdfZ(m) = (c−µ1)2+δ2

1

4δ1δ2m2

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Probability Density Function

pdfZ(m) = (c−µ1)2+δ2

1

4δ1δ2m2

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Probability Density Function

pdfZ(m) = (µ2+δ2−c)2m2+(µ1−δ1−c)2(1−m)2

8δ1δ2m2(1−m)2

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Probability Density Function

pdfZ(m) = (µ2+δ2−c)2m2+(µ1−δ1−c)2(1−m)2

8δ1δ2m2(1−m)2

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Probability Density Function

pdfZ(m) = (c−µ2)2+δ2

2

4δ1δ2(1−m)2

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Probability Density Function

We get a piecewise density function as follows, where each piece is an inverse polynomial: pdfZ(m) =                   

(c−µ2)2+δ2

2

4δ1δ2(1−m)2,

−∞ < m ≤ slope S.

(µ2+δ2−c)2m2+(µ1+δ1−c)2(1−m)2 8δ1δ2m2(1−m)2

, slope S < m ≤ slope Q.

(c−µ1)2+δ2

1

4δ1δ2m2

, slope Q < m ≤ slope P.

(µ2+δ2−c)2m2+(µ1−δ1−c)2(1−m)2 8δ1δ2m2(1−m)2

, slope P < m ≤ slope R.

(c−µ2)2+δ2

2

4δ1δ2(1−m)2,

slope R < m < ∞.

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Ratio Density for Triangle Kernel

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Uncertainty Quantification in Linear Interpolation

Figure: µi and δi represent mean and width, respectively, of a random variable Xi. c is the isovalue. v1 and v2 represent the grid vertices.

Aim : Closed-form characterization of the ratio random variable, Z =

c−X1 X2−X1 , when X1 and X2 have triangle distributions.

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Joint Distribution

Figure: P1, P2, P3, P4 represent quadratic polynomial functions of joint

• density. µi and δi represent mean and width, respectively, of a random

variable Xi. c is the isovalue.

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Cumulative Density Function

Figure: Cumulative density function can be obtained by integrating polynomials falling within red region. µi and δi represent mean and width, respectively, of a random variable Xi. c is the isovalue.

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Green’s Theorem

Figure: Integration of polynomial P1 over closed polygon ABC is equal to sum of line integrals of new polynomials L = ( −1

2 )

• P1 dZ1 and

M = 1

2

• P1 dZ2 along the edges AB, BC, and CA.

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Cumulative Density Function

Figure: Integrate polynomial P1 over orange region using Green’s theorm.

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Cumulative Density Function

Figure: Integrate polynomial P2 over orange region using Green’s theorm.

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Cumulative Density Function

Figure: Integrate polynomial P3 over orange region using Green’s theorm.

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Cumulative Density Function

Figure: Integrate polynomial P4 over orange region using Green’s theorm.

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Ratio Density for Nonparametric Noise Models

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Uncertainty Quantification in Linear Interpolation

Figure: Kδ(X − µX) represents a kernel with bandwidth δ centered at µX for random variable X. c is the isovalue. v1 and v2 represent the grid vertices.

Aim : Closed-form characterization of the ratio random variable, Z =

c−X1 X2−X1 , when X1 and X2 have nonparametric distributions.

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Cumulative Density Function

Figure: Joint density is superposition of joint densities for each pair of

• kernels. Cumulative density function can be computed by integrating

polynomials falling within orange region using Green’s theorem.

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Cumulative Density Function

• When kernel weights are not equal, each parallelogram

polynomial carries different weight.

Figure: Joint density is superposition of joint densities for each pair of

• kernels. Cumulative density function can be computed by integrating

polynomials falling within orange region using Green’s theorem.

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Results and Conclusion

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Noise Characterization: Parametric versus Nonparametric Densities

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Ensemble Dataset: Tangle Function (c = −0.59)

Tangle function: Commonly used dataset well-known for its complexity in isosurface reconstruction

(a) (b) (c) (d) Figure: Parametric versus nonparametric density. (a) Groundtruth, (b) uniform noise model, (c) nonparametric noise model with uniform base kernel, (d) color-mapped spatial uncertainties. Subfigures (b), (c), and (d) show expected isosurface with topology determined using edge-based classification.

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Ensemble Dataset: Teardrop Function (c = −0.002)

(a) (b) (c) (d) Figure: Parametric versus nonparametric density. (a) Groundtruth, (b) uniform noise model, (c) nonparametric noise model with uniform base kernel, (d) color-mapped spatial uncertainties. Subfigures (b), (c), and (d) show expected isosurface with topology corresponding to vertex-based classification.

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Ensemble Dataset: Temperature Field (c = 24 ◦C)

(a) Uniform noise model (b) Nonparametric noise model with Epanechnikov base kernel Figure: Boxes indicate places with different topology. Spatial uncertainties are colormapped.

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Uncertain Scalar Field: Fuel Dataset (c = 96.74)

(a) (b) (c) (d) (e) (f) Figure: (a) Mean field, (b) parametric density, (c) equally-weighed nonparametric density, (d) nonlocal means algorithm, (e) weighted nonparametric density (weights derived from nonlocal means technique [BCM05]), (e) colormapped uncertainties.

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Vertex-based Classification versus Edge-based Classification

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Gaussian Density Function (c = 0.008)

Green isocontour: Groundtruth Red isocontour: Vertex-based Classification(asteriskes) Blue isocontour: Edge-based Classification(circles)

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Ensemble Dataset: Temperature Field (c = 24 ◦C)

(a) Mean field (b) Vertex-based classification method (c) Edge-based classification method

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Conclusion

• We study impact of uncertainty on the marching cubes

algorithm for isosurface extraction.

• Isosurface topology is predicted using vertex-based and

edge-based classification techniques.

• Uncertain midpoint decider is derived analytically to resolve

topological ambiguties.

• Probability density function of inverse linear interpolation is

derived in closed-form.

• We quantify and visualize positional uncertainty in expected

isosurface using color-mapped uncertainties.

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Future Work

• Characterization of uncertain asymptotic decider for resolving

topological ambiguities.

• Study of isosurface extraction for dependent random variables.

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Publications and Submitted Work

• Athawale, T.; Entezari, A.;, Uncertainty Quantification in

Linear Interpolation for Isosurface Extraction, IEEE Transactions on Visualization and Computer Graphics (TVCG), Special Issue on IEEE Visualization, vol.19, no.12, pp.2723-2732

• Athawale, T.; Sakhaee E.; Entezari, A.;, Isosurface

Visualization of Data with Nonparametric Models for Uncertainty, [Submitted to IEEE Transactions on Visualization and Computer Graphics (TVCG)]

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References I

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William E. Lorensen and Harvey E. Cline, Marching cubes: A high resolution 3d surface construction algorithm, Computer Graphics (Proceedings of SIGGRAPH 87), vol. 21, July 1987,

• pp. 163–169.

Gregory M. Nielson and Bernd Hamann, The asymptotic decider: Removing the ambiguity in marching cubes, Visualization ’91, 1991, pp. 83–91.

110/115 Introduction Related work Problem Description and Approach Results and Conclusion

References II

Timothy S Newman and William Lee, On visualizing uncertainty in volumetric data: techniques and their evaluation, Journal of Visual Languages & Computing 15 (2004), no. 6, 463–491. Kai P¨

• thkow and H-C Hege, Positional uncertainty of

isocontours: Condition analysis and probabilistic measures, Visualization and Computer Graphics, IEEE Transactions on 17 (2011), no. 10, 1393–1406. Tobias Pfaffelmoser and R¨ udiger Westermann, Visualization of global correlation structures in uncertain 2d scalar fields, Computer Graphics Forum 31 (2012), no. 3, 1025–1034. Kai P¨

• thkow, Britta Weber, and Hans-Christian Hege,

Probabilistic marching cubes, Computer Graphics Forum,

• vol. 30, Wiley Online Library, 2011, pp. 931–940.

111/115 Introduction Related work Problem Description and Approach Results and Conclusion

References III

Philip J Rhodes, Robert S Laramee, R Daniel Bergeron, Ted M Sparr, et al., Uncertainty visualization methods in isosurface rendering, Eurographics, vol. 2003, 2003, pp. 83–88.

112/115 Introduction Related work Problem Description and Approach Results and Conclusion

Geometry Results: Linear Interpolation versus Expected Crossing

113/115 Introduction Related work Problem Description and Approach Results and Conclusion

Uncertain Scalar Field: Fuel Dataset (c = 56)

• Study of the impact of variation in the sample mean on the

linear interpolation.

Top: Linear interpolation, Middle: Expected crossing, Bottom: Colormapped distance between linear interpolation and the expected isosurface

114/115 Introduction Related work Problem Description and Approach Results and Conclusion

Future Work: Uncertain Asymptotic Decider

• Random variable corresponding to intersection of asymptotes:

S =

X1X3−X2X4 X1+X3−X2−X4

• Numerator: Convolution of piecewise logarithmic functions

Denominator: Piecewise cubic

• Support of joint distribution in four dimensions

Approximate solutions:

• Monte-Carlo sampling for approximating asymptotic decider

density

• Saddle point coordinates:(

X1−X2 X1+X4−X2−X3 , X1−X3 X1+X4−X2−X3 )

• Determine the density at expected saddle point, using

uncertain midpoint decider.

115/115 Introduction Related work Problem Description and Approach Results and Conclusion

Future Work: Correlated Random Fields

• Transform data to make it correlated (using technique like

PCA) and then extract isosurface.

• Determining joint density of a cell/edge requires marginalizing

density function over scalar field.

• Marginalization is a challenging task unless underlying density

is Gaussian or nonparametric density function with Gaussian base kernel.

• Support of joint density for ratio density is not a

parallelogram!