Introduction Related work Problem Description and Approach Results and Conclusion 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, 2 8 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]. 27/115
Introduction Related work Problem Description and Approach Results and Conclusion Marching Cubes in Action . . . Video is courtesy of Koen Samyn. 28/115
Introduction Related work Problem Description and Approach Results and Conclusion Marching Squares Algorithm (MSA) in Uncertain Data 29/115
Introduction Related work Problem Description and Approach Results and Conclusion Topological Uncertainty Isovalue c = 30 30/115
Introduction Related work Problem Description and Approach Results and Conclusion Ambiguous Topology: Decider Uncertainty Isovalue c = 30 31/115
Introduction Related work Problem Description and Approach Results and Conclusion Geometric Uncertainty Isovalue c = 30 32/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Visualization Techniques 33/115
Introduction Related work Problem Description and Approach Results and Conclusion 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] 34/115
Introduction Related work Problem Description and Approach Results and Conclusion 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]. 35/115
Introduction Related work Problem Description and Approach Results and Conclusion 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]) 36/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Quantification Techniques 37/115
Introduction Related work Problem Description and Approach Results and Conclusion 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]. 38/115
Introduction Related work Problem Description and Approach Results and Conclusion 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]. 39/115
Introduction Related work Problem Description and Approach Results and Conclusion Probabilistic Marching Cubes Figure: Direct volume rendering of . the cell-crossing probabili- ties [PWH11]. • Direct volume rendering of cell-crossing probabilities for isosurface 40/115
Introduction Related work Problem Description and Approach Results and Conclusion 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 41/115
Introduction Related work Problem Description and Approach Results and Conclusion Problem Description and Approach 42/115
Introduction Related work Problem Description and Approach Results and Conclusion Characterizing Data Uncertainty • Field with independent random variables • Characterization of uncertainty using probability density function • Propagation of data uncertainty into marching cubes algorithm 43/115
Introduction Related work Problem Description and Approach Results and Conclusion Topology Prediction for Isosurface in Uncertain Data 44/115
Introduction Related work Problem Description and Approach Results and Conclusion 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? 45/115
Introduction Related work Problem Description and Approach Results and Conclusion Scheme 1: Vertex-based Classification 46/115
Introduction Related work Problem Description and Approach Results and Conclusion 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 . 47/115
Introduction Related work Problem Description and Approach Results and Conclusion 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 . 48/115
Introduction Related work Problem Description and Approach Results and Conclusion Scheme 2: Edge-based Classification 49/115
Introduction Related work Problem Description and Approach Results and Conclusion 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 ) 50/115
Introduction Related work Problem Description and Approach Results and Conclusion Edge-based Classification • When edge-crossing probability is relatively high, we want opposite signs and vice versa. 51/115
Introduction Related work Problem Description and Approach Results and Conclusion 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. 52/115
Introduction Related work Problem Description and Approach Results and Conclusion Optimization Problem s ∗ = arg min s T Ws . s n = ± 1 W : weight matrix of edge-crossing probabilities s : sign vector s n : n’th entry of matrix s 53/115
Introduction Related work Problem Description and Approach Results and Conclusion Optimization Problem s ∗ = arg min s T Ws . s n = ± 1 W : weight matrix of edge-crossing probabilities s : sign vector s n : n’th entry of matrix s • Solution: Combinatorial approach (Not practical!) 54/115
Introduction Related work Problem Description and Approach Results and Conclusion Relaxed Optimization Problem s ∗ = arg min s T Ws . s n = ± 1 w : weight matrix of edge-crossing probabilities s : sign vector s n : 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 55/115
Introduction Related work Problem Description and Approach Results and Conclusion Ambiguous Configurations in Uncertain Data Aim : Given access to each vertex pdf, design a scheme to recover topology corresponding to ambiguous configurations? 56/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertain Midpoint Decider • Random variable corresponding to 1-d cell midpoint: M = X 1 + X 2 2 • Sum of random variables corresponds to convolution of densities. Uniforms with equal bandwidths Uniforms with unequal bandwidths Multiple uniforms with unequal bandwidths 57/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertain Midpoint Decider Figure: Convolution of uniform kernels with unequal bandwidths • Face midpoint random variable: M = X 1 + X 2 + X 3 + X 4 4 • Face midpoint density ( Pdf M ): Cubic univariate box-spline with non-uniform knots • Body (3-d cell) midpoint random variable: M = X 1 + ·· + X 8 8 • Body (3-d cell) midpoint density ( Pdf M ): Degree 7 univariate box-spline with non-uniform knots 58/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertain Midpoint Decider • Random variable corresponding to cell midpoint: M = X 1 + X 2 + X 3 + X 4 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 . 59/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Quantification in Isosurface Geometry for Uncertain Data 60/115
Introduction Related work Problem Description and Approach Results and Conclusion 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, c − X 1 Z = X 2 − X 1 , assuming X 1 and X 2 have parametric or nonparametric distributions. 61/115
Introduction Related work Problem Description and Approach Results and Conclusion Ratio Density for Uniform Noise Model 62/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Quantification in Linear Interpolation Figure: µ i and δ i represent mean and width, respectively, of a random variable X i . c is the isovalue. v 1 and v 2 represent the grid vertices. Aim : Closed-form characterization of the ratio random variable, c − X 1 Z = X 2 − X 1 , when X 1 and X 2 are uniformly distributed. 63/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution Find the joint distribution of the dependent random variables Z 1 = c − X 1 and Z 2 = X 2 − X 1 , where Z = Z 1 c − X 1 Z 2 = X 2 − X 1 . 64/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution • Determine the range of c − X 1 . • X 1 assumes values in the range [ µ 1 − δ 1 , µ 1 + δ 1 ]. • Random variables Z 1 and Z 2 are dependent. µ i and δ i represent mean and width, respectively, of a random variable X i . 65/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution • Determine the range of X 2 − X 1 . • X 2 assumes values in the range [ µ 2 − δ 2 , µ 2 + δ 2 ]. • Random variables Z 1 and Z 2 are dependent. µ i and δ i represent mean and width, respectively, of a random variable X i . 66/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution • Determine the range of X 2 − X 1 . • X 2 assumes values in the range [ µ 2 − δ 2 , µ 2 + δ 2 ]. • Random variables Z 1 and Z 2 are dependent. µ i and δ i represent mean and width, respectively, of a random variable X i . 67/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution • Parallelogram represents the joint distribution of the dependent random variables Z 1 = c − X 1 and Z 2 = X 2 − X 1 . 68/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution Shape and position of the joint distribution is impacted by relative configurations for X 1 and X 2 and the isovalue c. (a) Non-overlapping (b) Overlapping (c) Contained 69/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function What is Pr( Z 1 Z 2 ≤ m )? 70/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function • What is Pr( Z 1 Z 2 ≤ m )? • cdf Z ( m ) = Pr ( −∞ ≤ Z 1 Z 2 ≤ m ) (orange region). cdf Z ( m ) represents cumulative density function of a random variable Z . 71/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function • What is Pr( Z 1 Z 2 ≤ m )? • cdf Z ( m ) = Pr ( −∞ ≤ Z 1 Z 2 ≤ m ) (orange region). • Obtain pdf Z ( m ) by differentiating cdf Z ( m ) with respect to m. pdf Z ( m ) represents probability density function of a random variable Z . 72/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function • What is Pr( Z 1 Z 2 ≤ m )? • cdf Z ( m ) = Pr ( −∞ ≤ Z 1 Z 2 ≤ m ) (orange region). • Obtain pdf Z ( m ) by differentiating cdf Z ( m ) with respect to m. • A piecewise inverse polynomial function. 73/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( c − µ 2 ) 2 + δ 2 2 4 δ 1 δ 2 (1 − m ) 2 74/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( c − µ 2 ) 2 + δ 2 2 4 δ 1 δ 2 (1 − m ) 2 75/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 + δ 1 − c ) 2 (1 − m ) 2 8 δ 1 δ 2 m 2 (1 − m ) 2 76/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 + δ 1 − c ) 2 (1 − m ) 2 8 δ 1 δ 2 m 2 (1 − m ) 2 77/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( c − µ 1 ) 2 + δ 2 1 4 δ 1 δ 2 m 2 78/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( c − µ 1 ) 2 + δ 2 1 4 δ 1 δ 2 m 2 79/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 − δ 1 − c ) 2 (1 − m ) 2 8 δ 1 δ 2 m 2 (1 − m ) 2 80/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 − δ 1 − c ) 2 (1 − m ) 2 8 δ 1 δ 2 m 2 (1 − m ) 2 81/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function pdf Z ( m ) = ( c − µ 2 ) 2 + δ 2 2 4 δ 1 δ 2 (1 − m ) 2 82/115
Introduction Related work Problem Description and Approach Results and Conclusion Probability Density Function We get a piecewise density function as follows, where each piece is an inverse polynomial: pdf Z ( m ) = ( c − µ 2 ) 2 + δ 2 −∞ < m ≤ slope S . 4 δ 1 δ 2 (1 − m ) 2 , 2 ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 + δ 1 − c ) 2 (1 − m ) 2 , slope S < m ≤ slope Q . 8 δ 1 δ 2 m 2 (1 − m ) 2 ( c − µ 1 ) 2 + δ 2 slope Q < m ≤ slope P . 1 , 4 δ 1 δ 2 m 2 ( µ 2 + δ 2 − c ) 2 m 2 +( µ 1 − δ 1 − c ) 2 (1 − m ) 2 , slope P < m ≤ slope R . 8 δ 1 δ 2 m 2 (1 − m ) 2 ( c − µ 2 ) 2 + δ 2 4 δ 1 δ 2 (1 − m ) 2 , 2 slope R < m < ∞ . 83/115
Introduction Related work Problem Description and Approach Results and Conclusion Ratio Density for Triangle Kernel 84/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Quantification in Linear Interpolation Figure: µ i and δ i represent mean and width, respectively, of a random variable X i . c is the isovalue. v 1 and v 2 represent the grid vertices. Aim : Closed-form characterization of the ratio random variable, c − X 1 Z = X 2 − X 1 , when X 1 and X 2 have triangle distributions. 85/115
Introduction Related work Problem Description and Approach Results and Conclusion Joint Distribution Figure: P 1 , P 2 , P 3 , P 4 represent quadratic polynomial functions of joint density. µ i and δ i represent mean and width, respectively, of a random variable X i . c is the isovalue. 86/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function Figure: Cumulative density function can be obtained by integrating polynomials falling within red region. µ i and δ i represent mean and width, respectively, of a random variable X i . c is the isovalue. 87/115
Introduction Related work Problem Description and Approach Results and Conclusion 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 ) P 1 d Z 1 and M = 1 � P 1 d Z 2 along the edges AB , BC , and CA . 2 88/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function Figure: Integrate polynomial P 1 over orange region using Green’s theorm. 89/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function Figure: Integrate polynomial P 2 over orange region using Green’s theorm. 90/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function Figure: Integrate polynomial P 3 over orange region using Green’s theorm. 91/115
Introduction Related work Problem Description and Approach Results and Conclusion Cumulative Density Function Figure: Integrate polynomial P 4 over orange region using Green’s theorm. 92/115
Introduction Related work Problem Description and Approach Results and Conclusion Ratio Density for Nonparametric Noise Models 93/115
Introduction Related work Problem Description and Approach Results and Conclusion Uncertainty Quantification in Linear Interpolation Figure: K δ ( X − µ X ) represents a kernel with bandwidth δ centered at µ X for random variable X . c is the isovalue. v 1 and v 2 represent the grid vertices. Aim : Closed-form characterization of the ratio random variable, c − X 1 Z = X 2 − X 1 , when X 1 and X 2 have nonparametric distributions. 94/115
Introduction Related work Problem Description and Approach Results and Conclusion 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. 95/115
Introduction Related work Problem Description and Approach Results and Conclusion 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. 96/115
Introduction Related work Problem Description and Approach Results and Conclusion Results and Conclusion 97/115
Introduction Related work Problem Description and Approach Results and Conclusion Noise Characterization: Parametric versus Nonparametric Densities 98/115
Introduction Related work Problem Description and Approach Results and Conclusion 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. 99/115
Introduction Related work Problem Description and Approach Results and Conclusion 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. 100/115
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