Multivariate Topology

Multivariate Topology

Since about 2012, one of my principal interests has been topological analysis of multiple functions, since real-world simulations in particular rarely generate only one function at a time for a given physical phenomenon.

Much of this work was carried out under the META grant (Multifield Extension of Topological Analysis), starting with a quantised approximation we called the Joint Contour Net or JCN (IEEE Visualization 2011 Honorable Mention Poster):

Joint Contour NetsJoint Contour Nets

This proved surprisingly effective at analysing a problem in nuclear physics, assisting scientists to better understanding of the process of scission during nuclear fission:

Nuclear Scission
Physics Review I
Physics Review II

As with scalar topology, simplification rapidly became important, and we explored how to achieve it:

Multivariate Simplification

And how it related to the existing field of Pareto analysis:

Pareto Analysis

We exploited the Joint Contour Net to build user interfaces to support mathematicians in studying fiber topology:

Fiber Visualisation

Along the way, we experimented with performing the same computation in functional programming languages:

Haskell JCN

Eventually, we were able to construct an accurate (non-quantised) algorithm for the volumetric bivariate case: (IEEE Visualization 2016 Best Paper Award, ACM Computing Reviews Best of Computing 2016, Also Presented at ACM SIGGRAPH 2017)

Jacobi Fiber Surfaces

A secondary result from this work was some consideration of pathological and test cases for computational topology, including some foldable models of Reeb Spaces:

Pathological Cases

Other Research Topics:

Contour Tree Computation
Scalar Topological Visualisation
Isosurface Acceleration
Isosurface Quality
Direct Volume Rendering
Histograms and Isosurfaces
Topological Comparisons
Fiber Surfaces
Aerial Urban LiDAR
Exascale Data Analysis

Authors from VCG

Hamish Carr