1. Extraction and remeshing of ellipsoidal representations from mesh data Patricio Simari Karan Singh

2. Overview Input: surface data in mesh form. Output: ellipsoidal representation approximating input Ellipsoidal representation: surface defined piecewise by a set of ellipsoidal surfaces Ellipsoidal surface: ellipsoid plus boundaries Used ‘as is’ or remeshed if desired.

3. Motivation Efficient rendering and geometric querying Compact representation of large curved areas Can also be used to represent volumes Direct parameterization of each surface Objects perceptually segmented along concavities

4. Related work Bischoff et al., “Ellipsoid decomposition of 3D-models.” Hoppe et al., “Mesh optimization.” Cohen-Steiner et al., “Variational shape approximation.” Katz et al., “Hierarchical mesh decomposition using fuzzy clustering and cuts.”

5. Approximation error Total approximation error Mesh region (connected set of faces) Mesh face

6. Error metrics defined on vertices Radial Euclidean distance P vivi ∏P(vi)∏P(vi)

7. Error metrics defined on vertices Angular distance P nP(vi)nP(vi) nini

8. Error metrics defined on vertices Curvature distance P HP(vi)HP(vi) HiHi

9. Combining error metrics Combined vertex error Weights serve dual purpose: linearly scale metrics to comparable ranges Allow user to adjust for relative preference of one metric over another

10. Negative ellipsoids Ellipsoids have positive curvature so they would not capture surface concavities Negative ellipsoids remedy this

11. Ellipsoid segmentation algorithm Extension of Lloyd’s algorithm (k-means) Fitting step: compute P i that minimizes E(R i,P i ) Classification step: assign each face f j to a region R i that minimizes E(f j,P i ) Added constraint: regions must remain connected. Use flooding scheme (implies losing convergence guaranty.) Also include ‘teleportation’ to avoid local minima.

12. Remeshing ellipsoidal representations Parametric tessellation of surfaces unit sphere is sampled, cropped and tessellated Iterative vertex addition Boundary points are tessellated Faces are split at centre with highest error Edges are flipped

13. Error metric for ellipsoid volume Ellipsoids, being closed surfaces, can also be used to represent volume. Same algorithm can be used by adapting error metric Regions are approximated by an ellipsoid of similar volume.

14. Future work Segmentation boundaries: reduction or do away with explicit representation Initialization scheme that decides number of ellipsoids and gives a good initial placement

15. Using ellipsoidal boundaries Each primitive is a polygon which lies on an ellipsoidal surface Determine if a point is on the polygon Reduce to planar polygon using stereographic projection.

16. Smoothing segmentation boundaries

17. Impact of different metrics

18. Volume vs. surface fitting