6/26/2006CGI'06, Hangzhou China1 Sub-sampling for Efficient Spectral Mesh Processing Rong Liu, Varun Jain and Hao Zhang GrUVi lab, Simon Fraser University,

6/26/2006 CGI'06, Hangzhou China 1

Sub-sampling for Efficient Spectral Mesh Processing

Rong Liu, Varun Jain and Hao ZhangGrUVi lab, Simon Fraser University,

Burnaby, Canada


Roadmap

Background Nyström Method Kernel PCA (KPCA) Measuring Nyström Quality using KPCA Sampling Schemes Applications Conclusion and Future Work


Roadmap



Spectral Applicationsspectral clustering

[Ng et. al., 02]

spectral mesh compression

[Karni and Gotsman, 00]

watermarking[Ohbuchi et. al., 01]

spectral mesh segmentation [Liu and Zhang, 04]

face recognitionin eigenspace

[Turk, 01]

spectral meshcorrespondence[Jain and Zhang, 06]

“affinity matrix” W,its eigen-decomposition

texture mapping using MDS

[Zigelman et. al., 02]


Spectral Embedding

W = 0.56

j

i

i

j

W = EΛET

n points, dimension 2

1 nE =

1e ne…

embedding space, dimension n

row i

i

j

22/ eWijjiD


Bottlenecks

Computation of W, O(n2) .

Apply sub-sampling to compute partial W.

Eigenvalue decomposition of W, O(n3).

Apply Nyström method to approximate the eigenve

ctors of W.

How to sample to make Nyström work better ?


Roadmap



Sub-sampling

Compute partial affinities

n points

O (n2)

O (l . n)complexity:Z = X U Y

l sample points

W =

affinities between X and Yaffinities within

X


Nyström Method [Williams and Seeger, 2001]

Approximate Eigenvectors

W =

A

BT

B

C, A = UΛUT

O (n3)

O (l2 . n)

complexity:

U =

U

BTUΛ-1

approximate eigenvectors


Schur Complement

U

BTUΛ-1

ΛU

BTUΛ-1

T

=

A

BTA-1B

B

BT

W = UΛUT =

W =

A

C

B

BT

Schur Complement = C - BTA-1BF F

Practically, SC is not useful to measure the quality of a sample set.

SC =


Roadmap



PCA and KPCA [Schölkopf et al, 1998]

covariance matrix CX

dimension 2

covariance matrix Cφ(X)

X

)(

feature space, high dimension (infinite)

)( X

is implicitly defined by a kernel matrix K, where Kij= ji ,


Training Set for KPCA

)(

K =

L

MT

M

N

L = EΛET

E =

E

MTEΛ-1

˙Λ-1/2


Nyström Method and KPCA

W =

A

BT

B

C

A = UΛUT

U =

U

BTUΛ-1

Nyström

KPCA w/ training

set

K =

L

MT

M

N

L = EΛET

E =

E

MTEΛ-1

˙Λ-1/2


Roadmap



When Nyström Works Well ? When the training set of KPCA works well ?

1 2

3

4

5

]||||[ 54321 P22 ||)(||||)(|| j

T

jj yPPy

j

jTraining set should minimize:

subspace spanned by training points

)( jy


Objective Function

)(tr 1BABT

j

jminimize:

maximize:

W =

A

BT

B

C

)( 32 lmlO evaluation:


Compare Γ and SCGiven two sampling sets

S1 and S2

,21 12 SCSC

1. Test data are generated using Gaussian distribution;

2. Test is repeated for 100 times;

3. 4% inconsistency.


Roadmap



How to sample: Greedy Scheme

Maximize:

Greedy Sampling Scheme:

)(tr 1BABT W =

A

BT

B

C

A B

Best candidate sampling scheme:To find the best 1% with probability 95%, we only need to search for the best one from a random subset of size 90 (log(0.01)/log(0.95)) regardless of the problem size.


Properties of Γ

)(

)(tr

)(tr

)(tr

1

1

1

1

11

11

A

A

BBA

BAB

T

T

T

T

TTBB 11

(0, m), m is the column size of B

maximize 1T(A-11)1. A is symmetric.

2. Diagonals of A are 1.

3. Off-diagonals of A are in (0, 1).

It can be shown that when A’s columns are

canonical basis of the Euclidean space, the

maxima is obtained.


How to Sample: Farthest Point Scheme

A = 1

1

1

…

In order for A’s columns to be close to canonical basis, the off-diagonals should be close to zero.

This means the distances between each pair of samples should be as large as possible, namely

22/ jiD

ij eA

Samples are mutually farthest away.


Farthest Sampling Scheme


Roadmap



Mesh Correspondence

M(1) D(1) W(1) EΛ-1/2M(1)

M(2) D(2) W(2) EΛ-1/2 M(2)


without sampling

farthest point sampling

random sampling

(vertices sampled: 10,

total vertices: 250)


(vertices sampled: 10

total vertices: 2000)


correspondence error against mesh size

• correspond a series a slimmed mesh with the original mesh

• a correspondence error at a certain vertex is defined as the geodesic distance between the matched point and the ground-truth matching point.


Mesh Segmentation

M D W EΛ-1/2


• (b, d) obtained using farthest point sampling

• (a, c) obtained using random sampling

• faces sampled: 10

• number in brackets: value of Γ


w/o sampling, it takes 30s to handle a mesh with 4000 faces.

2.2 GHz Processor

1GB RAM


Roadmap



Conclusion

Nyström approximation can be considered as using training data in Kernel PCA.

Objective function Γ effectively quantifies the quality of a sample set.

Γ leads to two sampling schemes: greedy scheme and farthest point scheme.

Farthest point sampling scheme outperforms random sampling.


Future Work

Study the influence of kernel functions to Nyström method.

Further improve the sampling scheme.


Thank you !

Questions ?


Mesh Correspondence

1. Given any two models, M(1) and M(2), build the geodesic distance matrices D(1) and D(2). Dij encodes the geodesic distance between vertices i and j;

2. D(1) W(1) , D(2) W(2) , using Gaussian kernel.

3. Compute the eigenvalue decomposition of W(1) and W(2), and use the corresponding eigenvectors to define the spectral-embedded models M(1) and M(2).

handle bending, uniform scaling and rigid body transformation.

4. Compute the correspondence between M(1) and M(2).


Mesh Segmentation

1. Given a model M, somehow define the distances between each pair of faces; the distances are stored in matrix D;

2. D W ;3. Compute the eigenvalue decomposition of W, and use the

eigenvectors to spectral-embed the faces.4. Cluster (K-means) the embedded faces. Each cluster corre

sponds to a segment of the original model.


Maximize:

Given any two sampling sets S1 and S2 , S1 is superior to S2 iff

Efficient to compute.

Minimize: (schur complement)

S1 is superior to S2 iff

Very expensive to compute.

Γ and Schur Complement

)(tr 1BABT

SC = C - BTA-1B

21 SS

21SCSC SS

Documents

6/26/2006CGI'06, Hangzhou China1 Sub-sampling for Efficient Spectral Mesh Processing Rong Liu, Varun Jain and Hao Zhang GrUVi lab, Simon Fraser University,