Extraction and remeshing of ellipsoidal representations from mesh data

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 vi ∏P(vi) P

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

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

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 Pi that minimizes E(Ri,Pi) • Classification step: assign each face fj to a region Ri that minimizes E(fj,Pi) • 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