1. Folding meshes: Hierarchical mesh segmentation based on planar symmetry Patricio Simari, Evangelos Kalogerakis, Karan Singh

2. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 2 Introduction and motivation Meshes may contain a high level of redundancy due to symmetry, either global or localized. We propose an algorithm for detecting approximate planar reflective symmetry globally and locally. Applications include: Compression Segmentation Repair Skeleton Extraction Mesh processing acceleration

3. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 3 Related work Perfect in polygons and polyhedra: Atallah ‘85, Wolter et al. ‘85, Highnam ‘86, Jiang & Bunke ‘96. Approximate in point sets: Alt et al. ‘88. 2D images/range images: Marola ‘89, Gofman & Kiryati ’96*, Shen et al. ‘99, Zabrodsky et al. ’95*. Global 3D: O’Mara & Owens ‘96, Sun & Sherrah ‘97, Sun & Si ‘99, Martinet et al. ‘05. Global as shape desc.: Kazhdan et al. ‘04. Local 3D: Thrun & Wegbreit ‘05, Podolak et al. ‘06, Mitra et al. ‘06.

4. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 4 Overview Property: A symmetric surface’s planes of symmetry are orthogonal to the eigenvectors of its covariance matrix and contain its centre of mass. Leverage this fact: iteratively re- weighted least squares (IRLS) approach with M-estimation to converge to a locally symmetric region.

5. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 5 Consider a candidate symmetry plane p and let d i be the distance of vertex v i to the reflected mesh wrt p. Each v i is associated a weight w i according to: Solving for plane of symmetry

6. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 6 Solving for plane of symmetry The plane of symmetry is estimated by the centre of mass m and the eigenvectors of the weighted covariance matrix C defined as: These eigenvectors and centre of mass determine three planes. One with smallest sum cost is chosen.

7. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 7 Support region: motivation

8. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 8 Controlling leverage

9. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 9 Controlling leverage

10. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 10 Controlling leverage

11. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 11 Controlling leverage

12. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 12 Controlling leverage

13. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 13 Finding support region Given the current ρ values we consider a face to be a support face if for all of its vertices d i ≤ 2σ. [Hampel et al. ‘86] We find the largest connected region of support faces, and set weights for all vertices outside this region to 0. The plane finding and region finding steps are iterated until convergence.

14. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 14 Initialization Initially, w i is defined to be the mesh area associated with vertex v i The initial support regions contains all faces. σ = 1.4826*median(d i ) [Forsyth and Ponce ‘02] during initial iterations and then is fixed to 2ε.

15. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 15 Convergence

16. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 16 Convergence

17. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 17 Convergence

18. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 18 Convergence

19. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 19 Convergence

20. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 20 Convergence

21. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 21 Convergence

22. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 22 Finding other local symmetries Converge to symmetric region Segment out locally symmetric region Apply recursively to one half of the symmetric region (nested symmetries) and to each remaining connected component.

23. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 23 Results: Local symmetry detection

24. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 24 Results: Local symmetry detection

25. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 25 Results: Local symmetry detection

26. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 26 Folding trees We introduce the folding tree data structure. Encodes the non redundant regions as well as the reflection planes. Created by recursive application of the detection method. Can then be unfolded to recover the original shape.

27. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 27 Folding tree example

28. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 28 Results: Folding trees

29. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 29 Results: Folding trees

30. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 30 Results: Folding trees

31. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 31 Results: Folding trees

32. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 32 Results: Folding trees

33. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 33 Conclusions We have presented a robust estimation approach to finding global as well as local planar symmetries. We have introduced a compact representation of meshes, called folding trees, and shown how they can be automatically constructed using the detection method.

34. P. Simari, E. Kalogerakis, K. Singh – University of Toronto Folding Meshes: Hierarchical mesh segmentation based on planar symmetry 34 Future work Investigation alternate initialization schemes Extension to translational and rotational symmetries Exploration of other applications Repair Robust skeleton extraction Shape description/retrieval