1. 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

2. 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.

3. 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.

4. 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

5. ● Gradient at a point:
● Gradient follows a bivariate Gaussian Distribution whose mean and
covariance is
● The bivariate PDF deltaY for a vector g is :

6. 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:

7. Uncertainty in Orientation
● Convert to polar coordinates:
● Integrating over r (0 to infinity)

8. ● 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.

9. 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.

10. ● 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.

11. ● 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.

12. ● 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.
●

13. ● Four possible scenarios :

14. 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.
●

15. ● Synthetic DataSet:

16. ● Seismic Ensemble Dataset: