Compu&ng Correspondences in Geometric Datasets. 4.2 Symmetry & Symmetriza/on
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1 Compu&ng Correspondences in Geometric Datasets 4.2 Symmetry & Symmetriza/on
2 Symmetry Invariance under a class of transformations Reflection Translation Rotation Reflection + Translation + global vs. partial exact vs. approximative Rotation + Scaling 2
3 Symmetry Example: Reflection in 2D y x 3
4 Symmetry Example: Reflection in 2D y x 4
5 Symmetry Example: Reflection in 2D y x 5
6 Symmetry Example: Reflection in 2D y x 6
7 Symmetry Example: Reflection in 2D y φ d φ spatial domain x transformation space d 7
8 Symmetry Example: Reflection in 2D y φ φ d spatial domain x d φ transformation space d 8
9 Symmetry Local analysis: Symmetry as a pair relation y spatial domain x 9
10 Symmetry Local analysis: Symmetry as a pair relation y φ spatial domain x transformation space d 10
11 Symmetry Local analysis: Symmetry as a pair relation y φ spatial domain x transformation space d 11
12 Symmetry Local analysis: Symmetry as a pair relation y φ spatial domain x transformation space d 12
13 Symmetry Local analysis: Symmetry as a pair relation y φ local evidence for symmetry plane spatial domain x transformation space d 13
14 Symmetry Accumulation of local evidence y φ local evidence for symmetry plane spatial domain x transformation space d 14
15 Symmetry Accumulation of local evidence y φ local evidence for symmetry plane φ d x d clustering to extract symmetry transformation verification to extract symmetric patches 15
16 Symmetry Accumulation of local evidence y φ local evidence for symmetry plane φ d x d stochastic Algorithm with provable guarantees E(n, µ, σ) < 1 2 log α/np np/2 d 16
17 Sydney Opera House 17
18 Sydney Opera House Eurographics 2011 Course Compu/ng Correspondences in Geometric Data Sets 18
19 Articulated Shapes Random samples on two poses Correspondences between points are not known 19
20 Articulated Shapes Correspondence candidates 20
21 Articulated Shapes segmentation transform plot Eurographics 2011 Course Compu/ng Correspondences in Geometric Data Sets 21
22 Symmetrization Goal: Symmetrize 3D geometry Applications reverse engineering recognition, retrieval compression symmetric meshing, etc. Approach Minimally deform the model by optimizing the distribution in transformation space 22
23 Optimal Displacements Find minimal displacements that make two points symmetric with respect to a given transformation p i p i + p j 2 p j p i p j + p i 2 p j 23
24 Optimal Displacements Find minimal displacements that make two points symmetric with respect to a given transformation d i p i minimize d i 2 + d j 2 d j p j 24
25 Optimal Displacements Find optimal transformation and minimal displacements for a set of corresponding points 25
26 Optimal Displacements Find optimal transformation and minimal displacements for a set of corresponding points 26
27 Optimal Displacements Find optimal transformation and minimal displacements for a set of corresponding points 27
28 Optimal Displacements Find optimal transformation and minimal displacements for a set of corresponding points closed form solution exists! 28
29 Optimization Embedded deformation 29
30 Optimization Embedded deformation 30
31 Optimization Embedded deformation 31
32 Optimization Embedded deformation 32
33 Optimization Embedded deformation 33
34 Symmetrization 2D Example original transformation space 34
35 Symmetrization 2D Example original transformation space 35
36 Symmetrization 2D Example original transformation space 36
37 Symmetrization 2D Example transformation space 37
38 Symmetrization 2D Example transformation space 38
39 Symmetrization 2D Example transformation space 39
40 Symmetrization 2D Example transformation space 40
41 Symmetrization 2D Example transformation space 41
42 Dragon 42
43 Dragon 43
44 Dragon 44
45 Symmetrizing the Dragon 45
46 Shape Matching 46
47 References Mitra, Guibas, Pauly: Partial and Approximate Symmetry Detection for 3D Geometry, ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2006 Mitra, Guibas, Pauly: Symmetrization, ACM Transactions on Graphics (Proceedings of SIGGRAPH),
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