Symmetry Transforms 1 1 Motivation Symmetry is everywhere 2 - PowerPoint PPT Presentation

Symmetry Transforms 1 1

Motivation Symmetry is everywhere 2

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

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

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

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

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

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

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

Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Partial Symmetry Symmetry = 0.2 10

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

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection γ = ⋅ γ ( , ) ( ) D f f f γ = ⋅ γ ( , ) ( ) D f f f Symmetry = 0.7 12

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection γ = ⋅ γ ( , ) ( ) D f f f γ = ⋅ γ ( , ) ( ) D f f f Symmetry = 0.3 13

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

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection γ = ⋅ γ ( , ) ( ) D f f f Symmetry = 0.1 15

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

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 γ = − ( , ) min SD f f g γ = γ = ⋅ γ | ( ) g g g Can show that Symmetry Measure ( , ) ( ) D f f f is related to symmetry distance by 2 γ = − + 2 ( , ) 2 D f SD f 17

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

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

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

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

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

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

Monte Carlo Offset Angle 24

Monte Carlo Monte Carlo Sample for single plane Offset Angle 25

Monte Carlo Offset Angle 26

Monte Carlo Offset Angle 27

Monte Carlo Offset Angle 28

Monte Carlo Offset Angle 29

Monte Carlo Offset Angle 30

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

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

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

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

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

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

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

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

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

Application: Alignment Results: PCA Alignment Symmetry Alignment 40

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

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

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

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

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