1. Robert Osada, Tom Funkhouser Bernard Chazelle, and David Dobkin Princeton University Matching 3D Models With Shape Distributions

2. Shape Similarity Determine similarity between 3D shapes Computer Graphics Computer Vision Computational Biology [Caltech][Insulin, PDB]

3. Previous Work in 2D Shape representations Fourier analysis[Arbter90] Turning function[Arkin91] Size function[Uras95] Metrics for comparing curves Hausdorff Fréchet Bottleneck etc.

4. Previous Work in 3D High-level representations Generalized cylinders [Binford71] Medial axis[Bardinet00] Skeletons[Bloomenthal99] Statistical Moments[Reeves45, Prokop92] Crease angle[Besl94] Shells decomposition around centroid [Ankerst99] Extended Gaussian Images [Horn84] etc.

5. Desired Properties Match global properties of shape Invariance Rotation, translation, scale, mirror Robustness Noise, cracks, insertions and deletions Practicality Concise representation Efficient comparison Working with degenerate models

6. Our Approach Shape distributions Concise shape descriptor Common parameterization Function of random points 3D Model Shape Distribution Parameterization Random sampling

7. Our Approach Similarity Measure Parameterization 3D Model Shape Distribution Shape Function

8. Issues Which shape function? How to compare shape distributions? Parameterization Similarity Measure

9. Issues Which shape function? How to compare shape distributions? Parameterization Similarity Measure

10. Which Shape Function? Computationally simple options (~ 1s) Based on random points Angles, distances, areas, volumes A3D1D2D3   [Ankerst99] D4

11. Shape Function – D2 Distance between two random points on surface Line SegmentCircle TriangleCube CylinderSphere Two adjacent spheres Two spheres moving apart

12. Which Shape Function? Sneak preview

13. Shape Function – Key Questions Invariant? Rotation, translation, mirror (not scale) Robust? Noise, cracks, insertions and deletions Descriptive?

14. Issues Which shape function? How to compare shape distributions? Parameterization Similarity Measure

15. Comparison 1. Normalize for scale 2. Compare shape distributions  Parameterization

16. Normalization for Scale max meansearch

17. Compare shape distributions Computationally simple options (~.1ms) L n norms of densities(PDF) or cumulative densities(CDF) More complex options Earth mover’s distance, Bhattacharyaa distance. PDFCDF

18. Experimental Results Goal is to address the following: Is the method robust? How well does it classify? Shape Function NormalizationComparison A3 D1 D2 D3 D4 Max Mean Search PDF L 1 L 2 L  CDF L 1 L 2 L 

19. Robustness Experiment 10 Models CarChairHumanMissileMug PhonePlaneSkateboardSubTable

20. Robustness Experiment 6 Transforms Rotate, scale, mirror, noise, delete, insert Total of 70 models 1% Noise5% Deletion

21. Robustness Results Resulting distributions stable 7 Mugs Distance Probability 7 Missiles

22. Classification Experiment 133 Models categorized into 25 Groups Large variety within a group among groups 4 Mugs 6 Cars 3 Boats

23. Classification Results 4 Balls5 Animals2 Belts3 Blimps3 Boats 6 Cars8 Chairs3 Claws4 Helicopters11 Humans 3 Lamps3 Lightnings6 Missiles4 Mugs4 Openbooks

24. Classification Results Distance Probability

25. 4 Balls5 Animals2 Belts3 Blimps3 Boats 6 Cars8 Chairs3 Claws4 Helicopters11 Humans 3 Lamps3 Lightnings6 Missiles4 Mugs4 Openbooks Classification Results Line SegmentCircle TriangleCube CylinderSphere Two adjacent spheres Two spheres moving apart

26. Nearest Neighbor 1 st Tier 2 nd Tier Classification Results Avoid bias due to varying group sizes Query … Results

27. Similarity matrix Nearest Neighbor 1 st Tier 2 nd Tier Blocks Tanks Mugs Humans Airplanes Boats Classification Results

28. Shape Function First Tier Second Tier Nearest Neighbor A338%54%55% D135%48%56% D249%66% D342%58% D432%42%47%

29. Comparison to Moments Method Align 1 st moments (translation) Align 2 nd moments (rotation and scale) Compare using remaining moments (L 2 ) Shape Function First Tier Second Tier Nearest Neighbor D249%66% M335%46%63% M441%52%64% M528%38%55% M634%44%54% M727%33%51%

30. Conclusion – Properties Match global properties of shape Invariance Rotation, translation, scale, mirror Robustness Noise, cracks, insertions and deletions Practicality Concise representation Efficient comparison Works for degenerate models

31. Conclusion – Key Ideas Sampling gives common parameterization Simplifies comparison Comparing distributions is fast and easy Avoids registration, correspondence, etc. Simple shape functions are discriminating Method suitable as preclassifier

32. Future Work Use a larger and more controlled database Combine shape distributions with other classifiers into a working shape-based retrieval system

33. Thank you Sloan Foundation NSF