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Symmetry in Shapes – Theory and Practice
Niloy Mitra Maksim Ovsjanikov Mark Pauly Michael Wand Duygu Ceylan
Symmetry in Shapes – Theory and Practice
Michael Wand
Saarland University / MPI Informatik
Symmetry in Shapes Theory and Practice Niloy Mitra Maksim - - PowerPoint PPT Presentation
Symmetry in Shapes Theory and Practice Niloy Mitra Maksim - - PowerPoint PPT Presentation
Symmetry in Shapes Theory and Practice Niloy Mitra Maksim Ovsjanikov Mark Pauly Michael Wand Duygu Ceylan Symmetry in Shapes Theory and Practice Representations & Applications Michael Wand Saarland University /
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Representations
& Applications
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Toy Example
How many building blocks are these?
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Toy Example
How many building blocks are these?
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What is Symmetry?
Set of operations 𝑔 that leave object 𝑌 intact
- 𝑔 𝑌 = 𝑌
Operations 𝐻 = 𝑔 𝑔 𝑌 = 𝑌 form a group 𝐻 encodes absent information
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Pairwise Correspondences
Derived Properties
Pairwise matches T
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Pairwise Correspondences Permutation Groups
Derived Properties
Pairwise matches Exchangeable building blocks T
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Pairwise Correspondences Permutation Groups Transformation Groups
Derived Properties
Pairwise matches Exchangeable building blocks Regular transformations 𝐔𝑗|𝑗 ∈ ℤ T T T T T
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Pairwise Matches
T
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Input Data (Point Cloud)
[data set: C. Brenner, IKG Univ. Hannover]
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Feature Representation
[data set: C. Brenner, IKG Univ. Hannover]
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[data set: C. Brenner, IKG Univ. Hannover]
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Result
[data set: C. Brenner, IKG Univ. Hannover]
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Symmetry Detection
Partial Symmetry Detection
- Yields pairwise partial correspondences
- No symmetry groups (yet)
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Applications
Pairwise correspondences
- Non-local denoising
- Symmetrization
- Constrained editing
Techniques
- Correspondences transport information
- Simplification of pairwise relations
- Pairwise constraints as invariants
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Non-Local Denoising
data MLS non-local [Gal et al. 2007]
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Non-Local Denoising
16 parts
[data set: C. Brenner, University Hannover]
MLS non-local [Bokeloh et al. 2009]
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Non-Local Denoising
data non-local denoising [Zheng et al. 2010]
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Symmetrization
[Mitra et al. 2007]
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Symmetry Preserving Editing
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iWires
[Gal et al. 2009]
Symmetry-based propagation of edits: additional references [Wang et al. 2011], [Zheng et al. 2011]
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Permutation & Building Blocks
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Example Scene
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Pairwise Correspondences
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Cutting at the Boundaries
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Microtiles
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3D Result
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Properties
General framework
- Need point-wise equivalent relations
Canonical, unique decomposition Every point of every piece is unique
- Microtiles cannot have partial correspondences
Microtiles reveal permutation groups
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Symmetry Factored Embedding
Related Concept
- Points that map together in once piece
- Consistent orbits
- Ignores transformation, point-wise orbits
[Lipman et al. 2010]
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Inverse Procedural Modeling
Rules from example geometry
- Example model
- Compute rules describing
a class of similar models
Input Output
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Inverse Procedural Modeling
r-Similarity
- Local neighborhoods match exemplar
- utput
radius r radius r radius r input
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Inverse Procedural Modeling
[data set: G. Wolf]
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Theoretical Results
All 𝑠-similar objects are made out of (𝑠 − 𝜗)-microtiles
- Unique construction
- Connectivity same as in the example
Implications
- Canonical representation
- Synthesis
= solving jigsaw puzzles
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Shape Grammar
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Practice: Context Free Grammar
A a1 a2 B C D d1 d2 c1 c2 b1 b2 b3 Grammar: A a1 B C | a2 D B b1 | b2| b3 C c1 | c2 D d1 | d2
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Practical Results
[data sets: G. Wolf, Dosch 3D]
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Fast Pairwise Matches
T
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Quadratic Complexity?
[data set: C. Brenner, IKG Univ. Hannover]
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Cliques / Equivalence Classes
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Scalable Symmetry Detection
Hannover scans: 128M points / 14GB detection: 23 min preproc.: 43 min
[data set: C. Brenner, IKG Univ. Hannover] [data set: C. Brenner, IKG Univ. Hannover]
[Kerber et al. 2013]
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Regular Transformations
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Applications
Symmetry: regularity (transformations)
- Inverse procedural modeling
- Regularity preserving editing
- Shape recognition
- Shape understanding
Techniques
- Transformation groups characterize shapes
- Transformation group structure as invariants
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Inverse Procedural Modeling
[Pauly et al. 2008] [Mitra et al. 2008]
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Regularity Aware Deformation
[Bokeloh et al. 2011]
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Algebraic Shape Editing
[Bokeloh et al. 2012]
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Shape Recognition
[Kazhdan et al. 2004] [Podolak et al. 2006] [Thrun et al. 2005]
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Shape Understanding
[Mehra et al. 2009] [Mitra et al. 2010]
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Conclusions
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Symmetry
Principle
- Absence of information
- Invariance under operations
Structure
- Global Symmetry: transformation groups
- Partial Symmetry: permutations of building blocks
Detection
- Pairwise matching (efficient pruning, segmentation)
- Regular transformations: estimate generators
- Intrinsic formulations
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Applications
Different structural insights
- Correspondence
- Equivalence
- Pairwise relations
- Permutations
- Building blocks
- Shape grammar
- Hierarchical encoding
- Regularity
- Structural invariant
- Regularity relations
Different Applications
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Open Problems
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Open Problems
Future Work & Open Problems
- Detection
- Scalability
- Partial intrinsic symmetry detection
- Approximate (deformable) symmetry
- Modeling
- More general, semantic symmetry
- Equivalence of chairs, cars, houses?
Avoid overfitting?
- Theoretical framework
- Approximate group theory?