Generalized Hypercylinders: a Mechanism for Modeling and Visualizing N-D Objects

1. Generalized Hypercylinders: a Mechanism for Modeling and Visualizing N-D Objects Matt Ward Worcester Polytechnic Institute

2. Input data visualize Visual rep concept construct validate generate refine Model visualize Visual rep validate sample validate Outcome data Visual rep visualize Thinking with Models: courtesy of Penny Rheingans

3. Motivation • Abstract models useful for describing data in a compact manner - they can provide a language for encapsulating characteristics of data • Features that can be impossible to extract from the raw data can often be derived simply from the abstract model (e.g., critical points, holes) • Many modeling techniques developed for 2-D, 3-D, spatio-temporal data, but little for n-D data (we just have ‘points’) • Data mining and statistics have models for multivariate data features and relations – how about visualization?

4. Spatial enumeration Particles Primitive instancing Constructive Solid Geometry Hierarchical/subdivision (e.g., octrees) Boundary reps Implicit surfaces Sweep Representations (e.g., medial axis + radius – Thomas Wischgoll) Too much empty space Most common Sphere/Elliptic clusters Database queries with boolean operators Dimensional stacking (LeBlanc & Ward) Analytical surfaces (Yun Jang) H-BLOBS (Sprenger & Gross) None? 3-D vs. N-D Data Models 3-D N-D

5. Generalized Cylinders and Hypercylinders • Generalized cylinders (GC) are common modeling primitives in computer vision and graphics; can be used to represent the approximate shape of a rich variety of objects. • A GC consists of 2 3-D endpoints, a connecting spine (straight or curved), and a 2-D cross-section (fixed or variable). • A Generalized Hypercylinder (GHC) would thus consist of 2 N-D endpoints, a connecting N-D spine, and an N-1 dimensional cross-section that can vary along the spine.

6. Challenges with GHCs and Other N-D Models • Extraction • Endpoints: find points with maximum distance; alternately, centers of density • Spine: start with assumption of straight spine, break based on point distribution, fit parametric curve • Cross-section: start with assumption of constant radius, shrink-wrap based on points • Can build 1-D or 2-D at a time and form composite or with all dimensions at once • Problem: detecting branching and breaking conditions • Visualization • Can project to many 2-D projections, e.g., a scatterplot matrix • Endpoints and intermediate points on spine can be mapped using MDS or PCA • Conveying cross-section shape is challenging: • Could use same dimension reduction methods • Could use star-shape with N rays showing radius for each dimension • Interactions in data, display, model, and feature spaces

7. Summary • Much “uncharted land” in the area of N-D models for visual data exploration • Extending successful 2-D and 3-D models from graphics/visualization has potential • Features extractable from models could provide rich descriptors of data • Visualizing models and features can provide new insights • Interactions with models may be quite distinct from interacting with underlying data • GHCs are one possibility – too early to tell how useful they are (maybe someday in Leila’s shape workbench?)