1. Scale Normalization for Isometric Shape Matching Yusuf Sahillioğlu and Yücel Yemez Computer Eng. Dept., Koç University, Istanbul, Turkey

2. Problem Definition & Scope 2 / 22 Scale problem inherent to isometric shape correspondence. Scale normalization tested by shape correspondence. One shape is an isometric part of the other up to an arbitrary scale. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

3. Motivation 3 / 22 Shape interpolation, animation. Shape registration. Shape recognition. Time-varying reconstruction. Statistical shape analysis. Mapping between two shapes initiates cool apps. Attribute transfer. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

4. Contributions A novel scale-invariant isometric distortion function. Embedding avoided. Correspondences are partial and dense at the same time. No restriction on topology and triangulation type. 4 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

5. Isometry Our method is purely isometric. Intrinsic global property. Similar shapes have similar metric structures. Metric: geodesic distance. 5 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

6. Scale Normalization Scale normalization to prepare geodesic distances for the upcoming isometric distortion computations. 6 / 22 Complete shapes (scale by max geodesic)Partial shapes (max geodesic based normalization fails) Partial shapes (scale by trusted matches)Partial shapes (scale by Euclidean embedding, e.g., Möbius) Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

7. Isometric Distortion Given, measure its isometric distortion: O(N 2 ) for a map of size N. 7 / 22 in the most general setting. : normalized geodesic distance b/w two vertices. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

8. Isometric Distortion 8 / 22 g g gg g g g g average for. in action: Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

9. Scale-invariant Isometric Distortion Given, measure its scale-inv. isometric distortion: This measure based on raw geodesics provides few trusted matches to be used in scale normalization. O(N 3 ) for a map of size N. 9 / 22 : raw geodesic distance b/w two vertices. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

10. Scale-invariant Isometric Distortion 10 / 22 in action: average for. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

11. Minimizing Isometric Distortion 11 / 22 Optimization for completely isometric shapes. Optimization for partially isometric shapes. Covers complete shape matching naturally. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

12. Shape Matching Algorithm Combinatorial optimization of for initial coarse correspondence. The most extreme M source vertices are matched w/ |T| target extremities in the guidance of scale-inv. isometric distortion measure. computational complexity where we set M=5. 12 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

13. Shape Matching Algorithm Two isometric distortion measures in action. Scale-invariant isometric distortion. Isometric distortion w/ normalized geodesics. 13 / 22 Winner Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

14. Shape Matching Algorithm Use initial coarse correspondence to bring the meshes to the same scale. Scale the target mesh by Dense sampling. 14 / 22 100 here Same radius Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

15. Shape Matching Algorithm Dense matching. Minimum-weight perfect matching on cost matrix C. c i,j = cost of matching s i to t j //generating is traversed by (s i, t j ). Symmetric flip caring: repeat above (scaling, sampling, matching) with K-1 more generating initial coarse correspondences that follow in sorted distortions list. 15 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

16. Computational Complexity 16 / 22 Initial coarse correspondence: Extension to dense map of size U : which also requires dense sampling. Initial coarse correspondence dominates all. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

17. Isometric part matching. Results 17 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

18. Also for complete matching and for pairs w/ incompatible max geodesics. Results 18 / 22 Comparison w/ Möbius Voting (MV). Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012. MV: bad extremity matching, triangulation-sensitive.

19. Different choices of M. Results 19 / 22 Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012. Too small. computation less reliable. Too large. High computational load. Inconsistent joint sampling.

20. 20 / 22 Limitation & Future Work Presence of uncommon parts may fail this framework which forces to match M=5 most extremes as a whole. Embedding into a more sophisticated framework should help as it handles arbitrary scaling of the similar parts to be matched. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

21. Conclusion 21 / 22 A novel scale normalization method that proves useful for isometric partial and complete correspondence. Scale-invariant isometric distortion minimized in the original 3D Euclidean space wherein isometry is defined. So does dense isometric matching. Correspondences are partial and dense at the same time. Care for symmetric flips. No restriction on topology and triangulation. Applicable in its current form to 3D part retrieval problem. Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

22. People Assoc. Prof. Yücel Yemez, supervisor Yusuf, PhD student 22 / 22