Randomized Algorithms for Comparing and Understanding 3D Geometry

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Randomized Algorithms for Comparing and
Understanding 3D Geometry
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Need for Digital 3D Models
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Creating Geometry 3D Modelers
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Capturing Geometry 3D Scanners
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Shape Acquisition
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Shape Acquisition
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Shape Acquisition

• Partial similarity between shapes

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Shape Acquisition

• Partial similarity between shapes

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Shape Acquisition

• Partial similarity between shapes
• Efficient shape retrieval for partial queries

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Geometry Processing
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Model Organization and Retrieval
Kazdhan et al. 04
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Partial Shape Similarity
partial similarity
Self-similarity of an object ? symmetry
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Total vs Partial Matching

• Total matching is easy

PCA (Principal Component Analysis) Axes
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Total vs Partial Matching

• Partial matching is difficult
• Which region matches which other region(s)?
• Space of rigid transforms ? rotation
  translation
• Brute force approach ? not feasible
• Instead of exhaustive searching, use local
  geometry to guide where to search
• Easy to verify a transform

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Contributions

• Algorithms to
• Identify and extract similar (symmetric) patches
  of different size
• Estimate partial shape similarity between models
  without explicitly aligning them
• Properties
• Scalable and parallel
• Theoretical error bounds
• Output sensitive ? depends on complexity of
  solution and not on the complexity of model(s)

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Outline

• Introduction
• Related Work
• Symmetry Detection
• Probabilistic Fingerprints
• Conclusion and Future Work

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Related Work Global Alignment

• Feature based alignment
• Combinatorial search, need multiple objects
  together

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Related Work Desc. Based Align.

• Feature based alignment
• Combinatorial search, need multiple objects
  together
• Descriptor based alignment
• Fails for partial similarity

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Related Work Geometric Hashing

• Feature based alignment
• Combinatorial search, need multiple objects
  together
• Descriptor based alignment
• Fails for partial similarity
• Geometric hashing
• Tradeoff memory for time

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Related Work Symmetry Detection
brute force O(n6)
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Outline

• Introduction
• Related Work
• Symmetry Detection
• Probabilistic Fingerprints
• Conclusion and Future Work

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Symmetry in Nature
Symmetry is a complexity-reducing concept ...
seek it everywhere. - Alan J. Perlis
"Females of several species, including
humans, prefer symmetrical males." -
Chris Evan
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Partial Symmetry Detection (SIGGRAPH 2006)

• Given

Object/shape (represented as point cloud, mesh,
... )
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Partial Symmetry

• Transform Types
• Reflection
• Rotation Translation
• Uniform Scaling

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Reflective Symmetry
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Reflective Symmetry A Pair Votes
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Reflective Symmetry Voting Continues
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Reflective Symmetry Voting Continues
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Reflective Symmetry Largest Cluster

• Height of cluster ! size of patch
• Spread of cluster ! approximation level

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Pipeline
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Pruning Local Signatures

• Local signature ? invariant under transforms
• Signatures disagree ? points dont correspond

Use (?1, ?2) for curvature based pruning
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Reflection Normal-based Pruning
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Point Pair Pruning
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Transformations

• Reflection ? point-pairs
• Rigid transform ? more information

Robust estimation of principal curvature frames
Cohen-Steiner et al. 03
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Mean-Shift Clustering

• Kernel
• Type ? radially symmetric hat
• Radius

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Verification

• Clustering gives a good guess
• Verify ? build symmetric patches
• Locally refine solution using ICP algorithm Besl
  and McKay 92

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Random Sampling

• Height of clusters related to symmetric region
  size
• Random samples ! larger regions likely to be
  detected earlier
• Output sensitive

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Analysis

• Assumptions
• Smooth surface ?-sampled
• No noise
• Relates number(n) of random samples to
• Size of symmetric patch (p)
• Confidence (1-?)
• Sampling spacing, kernel radius, continuity of
  signature

• Tools for Analysis
• Signature continuity
• Chernoff bound

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Compression Chambord
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Compression Chambord
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Compression Chambord
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Approximate Symmetry Dragon
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Limitations
Castro et al. 06

• Cannot differentiate between small sized
  symmetries and comparable noise

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Articulated Motion Horses
registration ? symmetry detection between two
objects
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Outline

• Introduction
• Related Work
• Symmetry Detection
• Probabilistic Fingerprints
• Conclusion and Future Work

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Partial Shape Similarity (SGP 2006)

• Are two shapes similar in parts?
• Efficient tests require compact signatures
• database query
• online setting

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Probabilistic Fingerprints
compact
independent
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Insight

• Partial matching ! difficult problem
• Total matching ! easy problem
• Reduce partial matching ! many small total
  matching problems
• Results in few false positives ! quick to verify
  and discard

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Input Shapes
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Sample Points
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Shingles Overlapping Patches
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Shingles Overlapping Patches
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Bag of Patches Ordering Discarded
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Pipeline
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Pipeline Uniform Sampling

• Uniform spacing ? use Turk92
• Sample spacing ? ?

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Pipeline Shingle Generation

• Shingles overlapping unordered patches
• Shingle radius ?

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Pipeline Signatures

• Stable signatures
• Invariant to rigid transforms
• Spin-images
• Shape ? unordered high-dimensional point set
  with rigid transform factored out

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Pipeline Resemblance

• Similarity/resemblance
• Defined wrt. signatures
• Compare two bag of points in high-dim space
• No alignment required
• Brute force evaluation impractical

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How to Compare Point Sets

• Compare two point sets ? no need to align
• Dont have red and blue points together

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Reduce Sample Size

• Randomly sample red points
• Randomly sample blue points
• still need to solve for correspondence

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Min-hashing Broder97
2
3

• Each of m random experts
• Has an ordering of space-boxes
• Selects the point that lies in lowest ordered
  box

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Min-hashing Broder97

• Each of m random experts
• Has an ordering of space-box
• Selects the point that lies in lowest ordered
  box

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Pipeline Min-hashing Broder97

• Feature selection by random experts
• Features only useful for correspondence
• Need not have any visual importance
• Reduces set comparison to element-wise comparison

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Applications Adaptive Features
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Applications Adaptive Features
merged scan
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Applications Shape Space

• Partial similarity
• Articulated motion

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Applications Database Retrieval
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Statistics

• Pre-processing time in seconds
• Query time ? 15 msec/model
• Fingerprint size ? 10kb

model vertices uniform sampling spin image Rabin hash min-hash
skull 54k 0.8 7.5 0.05 4.5
Caesar 65k 1.4 7.3 0.08 10.3
bunny 121k 1.8 13.8 0.04 2.9
horse 8k 0.7 5.7 0.05 7.3
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Limitations

• Fails if resemblance is small
• How to handle uniform scaling?
• Stability of spin-images

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Outline

• Introduction
• Related Work
• Symmetry Detection
• Probabilistic Fingerprints
• Conclusion and Future Work

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Conclusion

• Simple probabilistic framework
• Local evidence ? global reasoning
• Geometric information for guidance
• Complexity of problem, not complexity of model
• Symmetry information ? High level model
  understanding
• Possible to compare two shapes using very compact
  fingerprints without aligning the models
• Local reasoning ? possible false positives ?
  verification

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Future Works

• Continuous scanning, assembly, hole filing
• Extension to deformable, time varying models
• Understanding of high dimensional data
• Online transmission, authentication, and security

Data courtesy Prof. B. Chen
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Collaborators
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Acknowledgements

• Gunnar Carlson
• Leonidas Guibas
• Jean-Claude Latombe
• Marc Levoy
• Mark Pauly

Pierre Alliez, Mario Botsch, Pat Hanrahan,
Michael Hoffer, Rajiv Motwani, Richard Keiser,
Doo Young Kwon, Bob Sumner, Martin Wicke
Manuela Cavegn, Heather Gentner, John Gerth, Ada
Glucksman, Hoa Nguyen
Joseph W. and Hon Mai Goodman Stanford Graduate
Fellowship Cargo, Darpa, ITR, NIH, and NSF funding
Emilio Antunez, Qing Fang, Natasha Gelfand, Olaf
A. Hall-Holt, Kyle Heath, Rachel Kolodny, Nikola
Milosavljevic, An Nguyen, Steve Oudot, Maksim
Ovsjanikov, Daniel Russel, Aneesh Sharma, Jaewon
Shin, Primoz Skraba, Michael Wand, Yusu Wang,
Danny Yang, Afra Zomorodian
Mike Cammarano, Billy Chen, Milton Chen , Kayvon
Fatahalian, Gaurav Garg, Eran Guendelman, Daniel
Horn, Mike Houston, Jeff Klingner, David Koller,
Manu Kumar, Ren Ng, John Owens, Doantam Phan,
Marie Ringel, Pradeep Sen, Eino-Ville Talvala,
Vaibhav Vaish, Ron Yeh
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Acknowledgements
Kamran Ahsan, Abhishek Bapna, Akanksha Bapna,
Indrahit Bhattacharya, Gaurav Chandra, Anirban
Dasgupta, Anupam Datta, Amal Ekbal, Gaurav Garg,
Mahesh Hardikar, Sara Kalantari, Uma Kelkar, Neha
Kumar, Subhasish Mitra, Shoubhik Mukhopadhyay,
Subha Nabar, Anindya Pathak, Inam Ur-Rehman,
Mitul, Saha, Debasis Sahoo, Sriram
Sankaranarayanan, Arjun Singh, Padma Sundaram,
Vaibhav Vaish, gsb-SIE fellows, climbing buddies,
cricket club folks, many I missed
Stanford Cricket Club Stanford Outing
Club Stanford Climbing Wall Stanford Alpine Club
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Acknowledgements

• Parents
• Brother
• Devasree

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Acknowledgements
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thank you!