Top Banner
Symmetrization Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007
39

Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

Dec 28, 2015

Download

Documents

Aldous Hensley
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

SymmetrizationNiloy J. Mitra Leonidas J. Guibas Mark Pauly

TU Vienna Stanford University ETH Zurich

SIGGRAPH 2007

Page 2: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

2

Invariance under a class of transforma-tions

Types of Symmetry

Page 3: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

3

Goal: Symmetrize 3D geometry

Approach: Minimally deform the model in the spatial domain by optimizing the distri-bution in transformation space

Symmetrization

Page 4: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

4

Given an explicit point‐pairing, a closed form solution for symmetrizing the point set

A symmetrization algorithm that uses trans-form domain reasoning to guide shape de-formation in object domain

Applications:◦ Extend the types of detected symmetries◦ Symmetric remeshing◦ Automatic correspondence for articulated bodies

Contributions

Page 5: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

5

Mitra, Guibas, Pauly: Partial and Approx-imate Symmetry Detection for 3d Ge-ometry. ACM Trans. Graph. 25, 3, 2006

Prior Work: Symmetry Detec-tion

Page 6: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

6

Initial pairs are sampled randomly Pruning based on curvature and normal

Prior Work: Pair pruning

Page 7: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

7

Use mean-shift algorithm◦ Non-Parametric Density Estimation

Prior Work: Clustering

Tessellate the space with windows

Run the procedure in parallel

The blue data points were traversed by the windows towards the mode

Page 8: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

8

Goal : Extracting the connected compo-nents of the model from cluster

Starting with a random point of cluster◦ Corresponds to a pair (pi, pj) of points on the

model surface Look at the one-ring neighbors pi and apply

T Check distances of the transformed points

to the surface around pj

Prior Work: Verification

Page 9: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

9

2D Example: Symmetry Detec-tion

Transformation space

d

Page 10: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

10

2D Example: Another point‐pair votes

Page 11: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

11

Pairs of sample points define reflective symmetry transform

2D Example: Voting Con-tinues

Page 12: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

12

Density plot → accumulation of symmetry evidence

2D Example: Density Plot

Page 13: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

13

Density cluster → reflective symmetry

2D Example: Density Peaks

Page 14: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

14

2D Example: Symmetry Detec-tion

Page 15: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

15

2D Example: Symmetry Detec-tion

A set of potential corresponding point pairs extracted

Page 16: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

16

2D Example: Local Symmetriza-tion

Page 17: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

17

2D Example: Local Symmetriza-tion

Cluster contraction

Local symmetrization

Cluster contraction in transform space

Constrained deformation in object space

Page 18: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

18

Object space point pairs → points in transform space

Cluster in transform space corresponds to approximate symmetry

Cluster contraction in transform space corresponds to constrained in deforma-tion in object space that enhances object symmetry

Recap

Page 19: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

19

2D Example: Global Symmetrization

Cluster merging → global symmetrization

Page 20: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

20

2D Example: Global Symmetrization

Cluster merging/contraction → Global symmetrization

Page 21: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

21

Local Symmetrization◦ Cluster contraction How to deform in the spatial domain ? Where to move in transform space ?

Global Symmetrization◦ Cluster merging

Sub‐problems

Page 22: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

22

Goal: Minimally displace two points to make them symmetric with respect to a given transformation

Optimal Displacements

[Zabrodsky et al. 1997]

2

)('

2

)(' 1 qpd

pqd

Tand

Tqp

Page 23: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

23

Goal: Find optimal transformation and minimal displacements for a set of point‐pairs

Optimal Transformation

Page 24: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

24

Reflection◦ Minimize energy

◦ Reduced to eigenvalue problem

Rigid Transform◦ Minimize energy

◦ SVD problem

Optimal Transformation

2/)(21

2

1

2

1

22

m

iii

m

ip

m

ippT TE

ijipqddd

m

iFqpiiR ii

CRCRE1

22

),( qtpt

Page 25: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

25

Initial random sampling does not respect symmetries.

The correspondences estimated during the symmetry detection stage are potentially inaccurate and incomplete

Optimization

Page 26: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

26

Every sample p shifted in the direction of displacement dp (white circle)

Project them onto the surface (colored square)

The procedure is iterated until the variance of the cluster is no longer reduced.

Optimizing Sample Posi-tions

Page 27: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

27

Local Symmetrization◦ Cluster contraction Where to move in transform space ? How to deform in the spatial domain ? Optimal transformation

Global Symmetrization◦ Cluster merging

Sub‐problems

Page 28: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

28

Using existing shape deformation method◦ Symmetrizing displacements positional constraints◦ 2D : As-rigid-as-possible shape manipulation method[Igarashi

et al.2005]◦ 3D : Non-linear PriMo deformation model [Botsch et al. 2006]

Symmetrizing Deformation

As-Rigid-As-Possible Shape Manipulation[Igarashi 2005] PriMo: Coupled Prisms for Intuitive Surface Modeling

[Botsch 2006]

Page 29: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

29

Find sample pairs Optimize sample positions on surface Compute the optimal transformation Update pi : p are used as deformation constraints Re-compute the optimal transformation Find new sample pairs every 5 time step

Contracting Clusters

iii tdpp

Page 30: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

30

Sort clusters by height Select the most pronounced cluster for

symmetrization Apply the symmetrizing deformation Repeat the process with next biggest cluster Finally, Merge clusters based on distance

greedily

Merging Clusters

Page 31: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

31

User controls the deformation by modifying the stiffness of the shape’s material

Soft materials allow for better symmetriza-tion

Stiffer materials more strongly resist the symmetrizing deformation

System allow spatially varying stiffness User controls the symmetrization by inter-

actively selecting clusters for contraction or merging

Control

Page 32: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

32

Results

Page 33: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

33

Results

Page 34: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

34

Results - Symmetric Mesh-ing

Symmetry Based Remeshing [Podolak al SGP 2007]

Page 35: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

35

Results

Page 36: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

36

Results –Computation Time

Page 37: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

37

Results- Articulated Bodies

Page 38: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

38

Some case, method is fails to process the entire model◦ The front feet of the bunny and the right foot of

the male character Small-scale features are sometimes ignored Insufficient local matching

The deformation model does not respect the semantics of the shape.

Limitations

Page 39: Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007.

39

Symmetrization algorithm◦ Robust and efficient, requires minimal user inter-

vention ◦ Handle both local and global symmetries

Future Work◦ Symmetry respecting geometry processing◦ Hierarchical shape semantics◦ Perception, art, design◦ Other data, e.g. motion data, derived spaces

Conclusion