Least-squares Meshes

1. Least-squaresMeshes Olga Sorkine and Daniel Cohen-Or Tel-Aviv University SMI 2004

2. Our goal • Shape approximation • Connectivity data • Sparse geometric data • Exploit information in the mesh graph

3. Connectivity Shapes • Connectivity has geometric information in it • [Isenburg, Gumhold and Gotsman 01] showed how to get a shape from connectivity by assuming uniform edge length and smoothness • Non-linear optimization process to get shape from connectivity Images taken from the “Connectivity Shapes”, Isenburg et al., IEEE Visualization ‘01

4. Our approach • Enrich the connectivity by sparse set of control points with geometry • Solve a linear least-squares problem to reconstruct the geometry of all vertices (the approximated shape)

5. Geometry hidden in connectivity There is geometry in connectivity

6. The linear system • We have two types of constraints: • Fairness constraint L(vi) = 0 i • Geometric constraint vj = cjj  C • L is the mesh Laplacian operator:

7. Linear least-squares solution L

8. Connection to Tutte/Floater embedding • Similar system was used by [Tutte 63] and [Floater 97] for graph drawing and parameterization. • Constraints placed on boundary vertices along a convex 2D polygon

9. Connection to Tutte/Floater embedding • Tutte system has hard constraints • The smoothness condition is not satisfied at constrained vertices

10. Linear least-squares solution • We solve the system for 3D positions of the vertices • Arbitrary topology • The constraints are soft, not necessarily on a boundary L

11. Solving the least-squares problem • We need to solve an over-determined system: • We find the solution by solving the normal equations: • Very efficient solution by Cholesky factorization of ATA: • R is upper-triangular and sparse • Once R is computed, solving for x, y, z by back-substitution:

12. Basis functions • The geometry reconstructed by solving is in fact a combination of k basis functions:

13. Basis functions • The basis functions are defined on the entire mesh • Connectivity data – defined for arbitrary topology • Tagging of the control vertices • The bases satisfy (in LS sense): • Smooth everywhere : Lui = 0 • Large on the i-th control vertex (ui = 1)and vanish on all others 5 basis functions on a 2D mesh (simple chain)

14. Spectral basis vs. LS basis • Another basis for compact geometry representation on meshes was proposed by [Karni and Gotsman 2000]: • The basis functions are eigenvectors of L, sorted in increasing eigenvalue order Spectral Basis • The spectral basis does not take any geometric information into account • Requires eigendecomposition – impractical for today’s meshes LS basis • The LS basis tags specific vertices, which makes it “geometry-aware” • Require solving sparse linear least-squares problem – can be done efficiently

15. Selecting the control points • Random selection • Fast, but less effective approximation • Greedy approach • Place one-by-one at vertices with highest reconstruction error • Slow, but gives good approximation • Greedy selection combined with local error maxima • A reasonable compromise Random selection Greedy approach Combined approach 1000 control points

16. Some results – varying number of control points Original camel 39074 vertices 100 control points 600 control points 1200 control points 3600 control points

17. Some results – varying number of control points Original feline 49864 vertices 100 control points 500 control points 4000 control points 9000 control points

18. Running times • Pentium 2.4 GHz computer

19. Applications • Progressive geometry compression and streaming 100 control points 1000 control points 3000 control points 10000 control points 100,086 vertices, 8 seconds

20. Applications • Progressive geometry compression and streaming • Hole filling

21. Applications • Progressive geometry compression and streaming • Hole filling • Mesh editing

22. Conclusions • LS-mesh represents mesh geometry using the mesh connectivity and a sparse set of control points • Linear reconstruction • Arbitrary topology • For future work, we’d like to understand better how the mesh connectivity affects the shape or the reconstruction • governed by the shape of the basis vectors • the shape of the singular vectors of the LS system matrix A

23. Acknowledgments • Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) • Israeli Ministry of Science • German Israel Foundation (GIF) • Models courtesy of Max-Planck Institut für Informatik, Stanford University, Cyberware • Who created the beautiful Camel ??

24. Thank you!