1. Randomized Cuts for 3D Mesh Analysis
Aleksey Golovinskiy and Thomas Funkhouser

2. Motivation Input Mesh Segmentation[

3. Motivation

4. Motivation

5. Motivation

6. Key Idea
Partition Function

7. Key Idea

8. Applications
Visualization
Segmentation
Registration
Deformation

9. Outline
Related Work
Method
Results
Applications

10. Related Work – Shape Analysis
Local Shape Properties
Curvature
Global Shape Properties
Shape Diameter Function
[Rusinkiewicz 2004]
[Shapira et al. 2008]

11. Related Work – Mesh Segmentation
[Shapira et al. 2008]
Shape Diameter Function
Fuzzy clustering and min cuts
K-means
[Katz and Tal 2003]
[Shlafman et al. 2002]

12. Related Work – Mesh Segmentation
Partition function needs a segmentation method
Segmentation methods benefit from partition function:
Which is easier to segment?
Dihedral Angles
Partition Function

13. Related Work – Typical Cuts
[Gdalyahu et al. 2001]: image segmentation
Create many segmentations
Estimate likelihood of nodes in same segment
Extract connected components

14. Outline
Related Work
Method
Results
Applications

15. Method – Overview … Create randomized segmentations Output:
Partition function
Cut consistency
Say: could be 2 way or multi-way .5 .3
.01 …

16. Method – Randomization
[Shapira et al. 2008]
Vary algorithms
Vary parameters
Jitter mesh
Algorithm-specific choices
[Katz and Tal 2003]
α= .1
β= 500
γ= 20
α= .05
β= 700
γ= 18
α= .07
β= 650
γ= 11
α= .12
β= 400
γ= 26

17. Method – K-Means Initialize K segment seeds, iterate:
Assign faces to closest seed
Move seed to cluster center
Randomization: random initial seeds

18. Method – Hierarchical Clustering
Initialize with a segment per face
Iteratively merge segments
Randomization: choose merge randomly

19. Method – Min Cut Initialize with source + sink seed
Find min-cut (weighted towards middle)
Randomization: random source + sink

20. Outline
Related Work
Method
Results
Applications

21. Results – Examples

22. Results – Articulation

23. Results – Intra-Class Variation

24. Results – Noise

25. Results – Tessellation

26. Results – Comparison to Alternatives

27. Results – Timing 4K models: 4 min per model Not a problem:
4K models capture salient parts
Computed once in model lifetime
Method-specific optimizations possible
Future work: recursive

28. Outline
Related Work
Method
Results
Applications

29. Applications – Visualization
Shaded Surface
Dihedral Angles
Partition Function

30. Applications – Segmentation
Compute cut consistency
Split among most consistent cut, recurse .5 .3
.01 …

31. Applications – Segmentation

32. Partition Function Sampling
Applications – Surface Correspondence X
Uniform Sampling
Partition Function Sampling
End: intuition is that high partition function values are stable, global features that should align across instances in the set

33. Applications – Deformation
Input Mesh
Partition Function
Uniform Deformation
Partition Function Deformation

34. Conclusion Randomized Segmentations Discrete Segmentation
Partition Function

35. Future work Other randomization methods
Other applications: saliency analysis, feature-preserving smoothing, skeleton embedding, feature detection, …

36. Future work
Multi-dimensional partition function
Scale

37. Acknowledgements Suggestions, code, feedback:
Adam Finkelstein, Szymon Rusinkiewicz, Philip Shilane, Yaron Lipman, Olga Sorkine and others
Models:
Stanford, Cyberware, Lior Shapira, Marco Attene, Daniela Giorgi, Ayellet Tal and others
Grants:
NSF (CNFS , IIS , and CCF ) and Google

38. Related Work – Shape Analysis
Local Shape Properties
Shape Diameter Function
Diffusion Distance
[Rusinkiewicz 2004]
[de Goes et al. 2008]
[Shapira et al. 2008]

39. Related Work – Random Cuts
[Karger and Stein 1996]
Randomized algorithm for finding min cut of a graph …

40. Related Work – Random Cuts
Our method vs Typical Cuts:
3D domain
Goal is partition function
Different segmentation algorithm

41. Method – Dual Graph Graph Nodes represent faces
Graph Arcs between adjacent faces
Lower cut cost at concave edges
Input Model
Graph Weights

42. Method – Min Cuts Initialize with source + sink seed Find min-cut
Often trivial
Increase weight close to source + sink
Discourage cuts at relative distance < s
Randomization: random source + sink
Scale: s

43. Results – Noise

44. Results – Tessellation
Reorder images

45. Applications – Deformation
Uniform
Partition Function

46. Method – Scale
Multi-scale features?