Visualizing nD Point Clouds as Topological Landscape Profiles to Guide Local Data Analysis

Abstract

Abstract-Analyzing high-dimensional point clouds is a classical challenge in visual analytics. Traditional techniques, such as projections or axis-based techniques, suffer from projection artifacts, occlusion, and visual complexity. We propose to split data analysis into two parts to address these shortcomings. First, a structural overview phase abstracts data by its density distribution. This phase performs topological analysis to support accurate and non-overlapping presentation of the high-dimensional cluster structure as a topological landscape profile. Utilizing a landscape metaphor, it presents clusters and their nesting as hills whose height, width, and shape reflect cluster coherence, size, and stability, respectively. A second local analysis phase utilizes this global structural knowledge to select individual clusters or point sets for further, localized data analysis. Focusing on structural entities significantly reduces visual clutter in established geometric visualizations and permits a clearer, more thorough data analysis. This analysis complements the global topological perspective and enables the user to study subspaces or geometric properties, such as shape.

Computing and Visualizing Time-Varying Merge Trees for High-Dimensional Data

“…We display the merge trees as occlusion-free 2-D topological landscape profiles [22], stacked in the third dimension according to their time stamp, and using orthographic projection to facilitate feature comparison. We relate features over time using visual links between the profiles, similar to standard isotracking graphs [25,5], but showing more feature properties and, most importantly, features for all thresholds in the landscape profiles.…”

“…Likewise, if features join, they first share a higher saddle and then one feature would die on the other (t 4 ). In t 3 , the first two hills change their order because the topological landscape profiles [22] sort subtrees of each merge tree saddle by persistence to put the highest hills to the left.…”

“…We represent time-varying merge trees as sequences of landscape profiles [22] and link hills visually to indicate structural changes. To reduce the clutter caused by showing many hills and links, we use topological simplification and represent preservation of topological feature groups by one visual link rather than many.…”

“…Repeatedly, the leaf superarc of lowest robustness is removed from the tree, potentially turning a saddle into a regular node, which, together with its two incident superarcs, is then replaced by a superarc. For each region mapped onto a superarc, we use three measures: persistence [12], the difference between the maximum and minimum function value of a region, size, the number of grid vertices of a region, and stability [22], a summed function value distribution of a region.…”

“…The merge tree can be represented as a 2-D landscape profile [22], which is a variation of 3-D topological landscapes [29,15]-a terrain metaphor for the more complex contour tree. A topological landscape has the same topology like its input tree and can represent structures of any dimension without structural occlusion.…”