Symmetry Transforms 1

Transcription

1 Symmetry Transforms 1

2 Motivation Symmetry is everywhere 2

3 Motivation Symmetry is everywhere Perfect Symmetry [Blum 64, 67] [Wolter 85] [Minovic 97] [Martinet 05] 3

4 Motivation Symmetry is everywhere Local Symmetry [Blum 78] [Thrun 05] [Simari 06] 4

5 Motivation Symmetry is everywhere Partial Symmetry [Zabrodsky 95] [Kazhdan 03] 5

6 Goal A computational representation that describes all planar symmetries of a shape? 6

7 Symmetry Transform A computational representation that describes all planar symmetries of a shape? 7

8 Symmetry Transform A computational representation that describes all planar symmetries of a shape? Perfect Symmetry Symmetry = 1.0 8

9 Symmetry Transform A computational representation that describes all planar symmetries of a shape? Local Symmetry Symmetry = 0.3 9

10 Symmetry Transform A computational representation that describes all planar symmetries of a shape? Partial Symmetry Symmetry =

11 Symmetry Measure Symmetry of a shape is measured by correlation with its reflection 11

12 Symmetry Measure Symmetry of a shape is measured by correlation with its reflection D( f, γ ) = f γ ( f ) D( f, γ ) = f γ ( f ) Symmetry =

13 Symmetry Measure Symmetry of a shape is measured by correlation with its reflection D( f, γ ) = f γ ( f ) D( f, γ ) = f γ ( f ) Symmetry =

14 Symmetry Measure Symmetry of a shape is measured by correlation with its reflection D( f, γ ) = f γ ( f ) D( f, γ ) = f γ ( f ) 14

15 Symmetry Measure Symmetry of a shape is measured by correlation with its reflection D( f, γ ) = f γ ( f ) Symmetry =

16 Previous Work Zabrodsky 95 Kazhdan 03 Thrun 05 Martinet 05 16

17 Symmetry Distance Define the Symmetry Distance of a function f with respect to any transformation γ as the L 2 distance between f and the nearest function invariant to γ SD( f, γ ) = min f g g γ ( g ) = g Can show that Symmetry Measure D( f, γ ) = f γ ( f ) is related to symmetry distance by D ( f, γ ) = 2SD + f

18 Previous Work Zabrodsky 95 Kazhdan 03 Thrun 05 Martinet 05 18

19 Previous Work Zabrodsky 95 Kazhdan 03 Thrun 05 Martinet 05 19

20 Previous Work Zabrodsky 95 Kazhdan 03 Thrun 05 Martinet 05 20

21 Computing Discrete Transform Brute Force O(n 6 ) Convolution Monte-Carlo O(n 3 ) planes X O(n 3 ) dot product 21

22 Computing Discrete Transform Brute Force O(n 6 ) Convolution O(n 5 Log n) Monte-Carlo O(n 2 ) normal directions X O(n 3 log n) per direction 22

23 Computing Discrete Transform Brute Force O(n 6 ) Convolution O(n 5 Log n) Monte-Carlo O(n 4 ) For 3D meshes Most of the dot product contains zeros. Use Monte-Carlo Importance Sampling. 23

24 Monte Carlo Offset Angle 24

25 Monte Carlo Monte Carlo Sample for single plane Offset Angle 25

26 Monte Carlo Offset Angle 26

27 Monte Carlo Offset Angle 27

28 Monte Carlo Offset Angle 28

29 Monte Carlo Offset Angle 29

30 Monte Carlo Offset Angle 30

31 Weighting Samples Need to weight sample pairs by the inverse of the distance between them P 2 d P 1 31

32 Weighting Samples Need to weight sample pairs by the inverse of the distance between them Two planes of (equal) perfect symmetry 32

33 Weighting Samples Need to weight sample pairs by the inverse of the distance between them Vertical votes concentrated 33

34 Weighting Samples Need to weight sample pairs by the inverse of the distance between them Horizontal votes diffused 34

35 Application: Alignment Motivation: Composition of range scans Morphing PCA Alignment 35

36 Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system. 36

37 Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system. 37

38 Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system. 38

39 Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system. 39

40 Application: Alignment Results: PCA Alignment Symmetry Alignment 40

41 Application: Matching Motivation: Database searching = Query Database Result 41

42 Application: Matching Observation: All chairs display similar principal symmetries 42

43 Application: Matching Approach: Use Symmetry transform as shape descriptor = Query Transform Database Result 43

44 Application: Matching Results: Symmetry provides orthogonal information about models and can therefore be combined with other descriptors 44

45 Summary Planar-Reflective Symmetry Transform Captures degree of reflectional symmetry about all planes Monte Carlo computation Applications: alignment, search, completion, segmentation, canonical viewpoints, 45