Supervisor: Torsten Möller, Daniel Pahr, Aleksandar Doknic
Visualizations of multi-dimensional surfaces often involve dimensionality reductions such as low-dimensional projection. Such projections lead to a loss of geometry information and create new axes that are difficult to interpret. Cartesian sampling would avoid this problem, but it becomes computationally expensive (exponential in the number of dimensions) as the number of dimensions increases. Additionally, visualizations become more difficult to interpret. The slicing method by Torsney-Weir, with adaptations by Doknic, could address those challenges. However, it is still unclear how interpretable and useful the resulting visualizations are.
Is it possible to make 4D and 5D spaces interpretable? If yes, can it be generalized for higher dimensions? We expect a detailed analysis of critical points ("Kurvendiskussion") for well-known functions such as the Rosenbrock Function. Here, the critical points are already known. The idea is to start with 1D and 2D functions that are easy to visualize and expand to 3D, 4D, and 5D.
The key part of this research is a user study that addresses the following questions:
- Is it possible to identify the critical points of a function through visual means?
- How does the slicing method [1] compare to low-dimensional projections and Schlegel diagrams [2] or topological spines [3]?
- Can Hasse diagrams [4] or Loewner-John ellipsoids [5] (or similar) help us understand multi-dimensional functions, for example, to support sampling? Literature with examples: Lectures on Polytopes by Günter Ziegler.
[1] Sliceplorer
[2] Schlegel diagrams
[3] C. Correa, P. Lindstrom, P.Bremer, Topological Spines: A Structure-preserving Visual Representation of Scalar Fields, see https://ieeexplore.ieee.org/document/6064947
[4] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order
[5] John ellipsoids, see https://arxiv.org/abs/2501.01801
Prerequisites: VIS, FDA
Contact: Torsten Möller, Daniel Pahr, Aleksandar Doknic