Visual Signature of High-Dimensional Geometry in Parallel Coordinates

Although we can interactively rotate a 3D projected high-dimensional geometry and observe its dynamic changes, this traditional visualization method is limited and highly sensitive to the choice of viewing direction. Parallel-coordinates plots supplement this visualization scenario by providing statistical analysis of the geometry for distinct pairs of co-dimensions. Such analysis results in visual signatures that embed geometric structures such as symmetry, and thus allows us to overview the status of the missing dimensions while exploring the projected geometry. This paper presents a blue-noise sampling approach for efficient construction of continuous parallel-coordinates plots of high-dimensional geometric surfaces defined by mathematical equations. We employ the parallel-coordinates plots with the embedded visual signatures to assist the interactive exploration of high-dimensional geometries, typically for 2-manifold embedded in 4-space. While we interactively explore the 3D projected geometry, we can observe dynamic changes on its visual signature. Various geometric properties can also be identified and visualized. Moreover, we can interactively brush the plots, and see their counterparts in the 3D projection. Assorted geometric properties such as curvature can further be used to enhance the visual signature.