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: