symmetry transforms - max planck...
TRANSCRIPT
![Page 1: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/1.jpg)
1
Symmetry Transforms
1
![Page 2: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/2.jpg)
2
Motivation
Symmetry is everywhere
![Page 3: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/3.jpg)
3
Motivation
Symmetry is everywhere
Perfect Symmetry[Blum ’64, ’67]
[Wolter ’85][Minovic ’97] [Martinet ’05]
![Page 4: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/4.jpg)
4
Motivation
Symmetry is everywhere
Local Symmetry[Blum ’78][Thrun ‘05] [Simari ’06]
![Page 5: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/5.jpg)
5
Motivation
Symmetry is everywhere
Partial Symmetry[Zabrodsky ’95][Kazhdan ’03]
![Page 6: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/6.jpg)
6
Goal
A computational representation that describes all planar symmetries of a shape
?
![Page 7: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/7.jpg)
7
Symmetry Transform
A computational representation that describesall planar symmetries of a shape
?
![Page 8: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/8.jpg)
8
Symmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 1.0Perfect Symmetry
![Page 9: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/9.jpg)
9
Symmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 0.3Local Symmetry
![Page 10: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/10.jpg)
10
Symmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 0.2Partial Symmetry
![Page 11: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/11.jpg)
11
Symmetry Measure
Symmetry of a shape is measured by correlation with its reflection
![Page 12: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/12.jpg)
12
Symmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD γγ ⋅= Symmetry = 0.7)(),( fffD γγ ⋅=
![Page 13: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/13.jpg)
13
Symmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD γγ ⋅= Symmetry = 0.3)(),( fffD γγ ⋅=
![Page 14: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/14.jpg)
14
Symmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD γγ ⋅=
)(),( fffD γγ ⋅=
![Page 15: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/15.jpg)
15
Symmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD γγ ⋅= Symmetry = 0.1
![Page 16: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/16.jpg)
16
Previous Work
Zabrodsky ‘95
Kazhdan ‘03
Thrun ‘05
Martinet ‘05
![Page 17: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/17.jpg)
17
Symmetry Distance
Define the Symmetry Distance of a function f with respect to any transformation γ as the L2
distance between f and the nearest function invariant to γ
Can show that Symmetry Measureis related to symmetry distance by
ggggffSD
=
−=)(|
min),(γ
γ)(),( fffD γγ ⋅=
222),( fSDfD +−=γ
![Page 18: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/18.jpg)
18
Previous Work
Zabrodsky ‘95
Kazhdan ‘03
Thrun ‘05
Martinet ‘05
![Page 19: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/19.jpg)
19
Previous Work
Zabrodsky ‘95
Kazhdan ‘03
Thrun ‘05
Martinet ‘05
![Page 20: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/20.jpg)
20
Previous Work
Zabrodsky ‘95
Kazhdan ‘03
Thrun ‘05
Martinet ‘05
![Page 21: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/21.jpg)
21
Computing Discrete Transform
Brute Force O(n6)ConvolutionMonte-Carlo
O(n3) planes
X
O(n3) dot product
![Page 22: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/22.jpg)
22
Computing Discrete Transform
Brute Force O(n6)Convolution O(n5Log n)Monte-Carlo
O(n2) normal directions
X
O(n3 log n) per direction
![Page 23: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/23.jpg)
23
Computing Discrete Transform
Brute Force O(n6)Convolution O(n5Log n)Monte-Carlo O(n4) For 3D meshes
• Most of the dot product contains zeros.
• Use Monte-Carlo Importance Sampling.
![Page 24: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/24.jpg)
24
Monte Carlo
Offs
et
Angle
![Page 25: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/25.jpg)
25
Monte Carlo
Monte Carlo Sample for single plane
Offs
et
Angle
![Page 26: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/26.jpg)
26
Monte Carlo
Offs
et
Angle
![Page 27: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/27.jpg)
27
Monte Carlo
Offs
et
Angle
![Page 28: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/28.jpg)
28
Monte Carlo
Offs
et
Angle
![Page 29: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/29.jpg)
29
Monte Carlo
Offs
et
Angle
![Page 30: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/30.jpg)
30
Monte Carlo
Offs
et
Angle
![Page 31: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/31.jpg)
31
Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
P1
P2
d
![Page 32: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/32.jpg)
32
Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
Two planes of (equal) perfect
symmetry
![Page 33: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/33.jpg)
33
Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
Vertical votesconcentrated…
![Page 34: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/34.jpg)
34
Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
Horizontal votesdiffused…
![Page 35: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/35.jpg)
35
Application: Alignment
Motivation:Composition of range scansMorphing
PCA Alignment
![Page 36: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/36.jpg)
36
Application: Alignment
Approach:Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
![Page 37: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/37.jpg)
37
Application: Alignment
Approach:Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
![Page 38: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/38.jpg)
38
Application: Alignment
Approach:Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
![Page 39: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/39.jpg)
39
Application: Alignment
Approach:Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
![Page 40: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/40.jpg)
40
Application: Alignment
Symmetry Alignment
PCA Alignment
Results:
![Page 41: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/41.jpg)
41
Application: Matching
Motivation:Database searching
Database ResultQuery
=
![Page 42: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/42.jpg)
42
Application: Matching
Observation:All chairs display similar principal symmetries
![Page 43: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/43.jpg)
43
Application: Matching
Approach:Use Symmetry transform as shape descriptor
Database ResultQuery
=
Transform
![Page 44: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/44.jpg)
44
Application: Matching
Results: Symmetry provides orthogonal informationabout models and can therefore becombined with other descriptors
![Page 45: Symmetry Transforms - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011_Tutori… · 9 Symmetry Transform. A computational representation that describes all](https://reader033.vdocuments.us/reader033/viewer/2022060507/5f22313d9d088c58a0701242/html5/thumbnails/45.jpg)
45
Summary
Planar-Reflective Symmetry Transform
Captures degree of reflectional symmetry about all planes
Monte Carlo computation
Applications: alignment, search, completion,segmentation, canonical viewpoints, …