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

Motivation Input Mesh Segmentation [http://www.aimatshape.net/research]

Motivation

Motivation

Motivation

Key Idea Partition Function

Key Idea

Applications Visualization Segmentation Registration Deformation

Outline Related Work Method Results Applications

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

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]

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

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

Outline Related Work Method Results Applications

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

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

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

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

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

Outline Related Work Method Results Applications

Results – Examples

Results – Articulation

Results – Intra-Class Variation

Results – Noise

Results – Tessellation

Results – Comparison to Alternatives

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

Outline Related Work Method Results Applications

Applications – Visualization Shaded Surface Dihedral Angles Partition Function

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

Applications – Segmentation

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

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

Conclusion Randomized Segmentations Discrete Segmentation Partition Function

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

Future work Multi-dimensional partition function Scale

Acknowledgements Suggestions, code, feedback: Adam Finkelstein, Szymon Rusinkiewicz, Philip Shilane, Yaron Lipman, Olga Sorkine and others Models: Aim@Shape, Stanford, Cyberware, Lior Shapira, Marco Attene, Daniela Giorgi, Ayellet Tal and others Grants: NSF (CNFS-0406415, IIS-0612231, and CCF-0702672) and Google

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

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

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

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

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

Results – Noise

Results – Tessellation Reorder images

Applications – Deformation Uniform Partition Function

Method – Scale Multi-scale features?