Compu&ng Correspondences in Geometric Datasets. 4.2 Symmetry & Symmetriza/on

Transcription

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),