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Visualizing the Variability of Gradients in Uncertain 2D Scalar Field
Authors: Tobais Pfaffelmoser, Mihaela Mihai and Rudiger Westermann(TU Munich)
Presented by: Subhashis Hazarika,The Ohio State University
Motivation
● Standard Deviation does not always give a rigorous analysis of uncertainty, specially when we want to study the differential quantities, like relative variability of data values at different points.
● To draw inference about the stability of geometric features in a scalar field like contour shapes.
Contribution
● Derivation of gradient based uncertainty parameters on discrete grid structures.
● Analytical expression of probability distribution describing gradient magnitude and orientation variation.
● A visualization technique using color diffusion to indicate the stability of the slope along gradient direction in 2D scalar fields.
● A family of colored glyphs to quantitatively depict the uncertainty in orientation of iso-contours in 2D scalar fields.
Gradient Uncertainty
● Applied on a discrete sampling of a 2D domain on a Cartesian grid structure with grid points.
● Y : is a multivariate RV modeling the data uncertainty at every point.
● Assumption: RVs follow a multivariate Gaussian Distribution
● Gradient at a point:
● Gradient follows a bivariate Gaussian Distribution whose mean and covariance is
● The bivariate PDF deltaY for a vector g is :
Uncertainty in Derivative● Choose the mean gradient direction as the direction along which to
determine the uncertainty in derivatives.
● The uncertainty of derivative in the mean gradient direction can be modeled by a scalar random variable
● RV D must also obey a Gaussian Distribution with mean and standard deviation:
Uncertainty in Orientation● Convert to polar coordinates:
● Integrating over r (0 to infinity)
● In order to analysis the stability of geometric features like iso-contours , the probability of occurrence of Theta should include theta+pi.
● Circular variance to determine the uncertainty in degree of orientation.
Visualization● To convey the basic shape of the iso-contours in the mean scalar
field they partition the range of mean values into a number of N equally spaced intervals.
● Use diffusion process to visually encode
● Diffusion at a point takes place along the normal curve, which is the curve passing through the point and oriented along the gradient direction.
● Diffusion value : fraction of initial black & white color
● Diffusion value 0.5 implies high degree of diffusion and 1.0 implies least diffusion.
● Generate a diffusion texture to lookup diffusion value.
● The parameter v(degree of diffusion) is controlled by the gradient uncertainty parameters.
● Now for a given point the texture coordinates u and v are calculated as:
● Use a final normalized diffusion value and compute the final color at each grid point by blending a diffusion color and a background color.
● Use different diffusion color to visualize
● The corresponding texture lookup for these 3 quantities are
● They are interested in the lower confidence interval so the final color is give by the following blending equation:
● The diffusion color encodes the relative position of
w.r.t the zero derivative.
●
● Four possible scenarios :
Orientation Visualization● Color of glyph is mapped to the circular variance [0,1] → [ green →
cyan → blue → magenta → red]
● To show individual distribution per glyph the transparency is controlled at all off-center vertices.
●
● Synthetic DataSet:
● Seismic Ensemble Dataset: