High Dimensional Function Visualization
An important goal of scientific data analysis is to understand the
behavior of a system or process based on a sample of the system. In many
instances it is possible to observe both input parameters and system
outputs, and characterize the system as a high-dimensional function. Such
data sets arise, for instance, in large numerical simulations, as energy
landscapes in optimization problems, or in the analysis of image data
relating to biological or medical parameters. This paper proposes an
approach to analyze and visualizing such data sets. The proposed method
combines topological and geometric techniques to provide interactive
visualizations of discretely sampled high-dimensional scalar fields. The
method relies on a segmentation of the parameter space using an
approximate Morse-Smale complex on the cloud of point samples. For each
crystal of the Morse-Smale complex, a regression of the system parameters
with respect to the output yields a curve in the parameter space. The
result is a simplified geometric representation of the Morse-Smale complex
in the high dimensional input domain. Finally, the geometric
representation is embedded in 2D, using dimension reduction, to provide a
visualization platform. The geometric properties of the regression curves
enable the visualization of additional information about each crystal such
as local and global shape, width, length, and sampling densities.
This provides succinct visual summary of the salient features of high dimensional
scalar functions.
Movie with method description and several examples here
Software is available here
Related publications
Samuel Gerber, Peer-Timo Bremer, Valerio Pascucci, Ross Whitaker,
"Visual Exploration of High Dimensional Scalar Functions",
IEEE Transactions on Visualization and Computer Graphics to appear,
Proceedings of VIS 2010
[pdf]