3D Face Recognition Using M

Summation Invarian

Wei-Yang Lin, Kin-Chung Wong, Nigel Bos

Dept. of Electrical & Computer Eng

University of Wisconsin-Madis

This research is partially supported by the NSF under Grant 
Agenda

1. Introduction Introduction

– Focus of FR research

2. Invariant Theory3. MSI Algorithm and Result4. Conclusion
Focus of FR Research1. Currently focus on 3D FR algo2. A novel family of geometrical 

based on the method of movi

3. Extract summation invariants 

profiles on multi-regions

4. LDA-based fusion method
Agenda1. Introduction2. Invariant Theory Invariant Theory

– Method of Moving Frame– Summation Invariants– Feature Extraction on 3D surfac

3. MSI Algorithm and Results4. Conclusion
Method of Moving Frame• A power tool for finding inv

under group actions (É. Ca

• Definition: a moving fram

moving fram

equivariant mapping 

ρ : M 

ρ

o

=

ρ xgxg )()(
Summation Invariants• Procedures

1. Given a transformation over points of a 2. Define jet space.

3. Solve the moving frame from the norm

equations.

4. Invariants can be derived by applying 

space.

• A systematical way to derive g

invariants for pattern recognit
ExampleSTEP 1 : A point (x,y) under rotaSTEP 2 : Define jet space as STEP 3 : Normalization equation STEP 4 :Apply moving frame to j

0cossin 

a

yxy 

θθθ =→=+=

22 sincos θθ +=−= yxyxx

yx 

,(
Example : Euclidean SumInvariants for CurvesSTEP 1 : Given a curve (x[n], y[n]) un

transformation

STEP 2 : Jet space is defined as

⎢⎣⎡

=⎥

⎦⎤

⎢⎣⎡

+⎥

⎦⎤ ⎢

⎣⎡

⎥⎦⎤ ⎢

⎣⎡

−

][

][

][

][

cossin

sincos

ny

nx

b

a

ny

nx

θθ

θθ

),,],[],1[],1[( PPNyyx 1,00,1

K 

, 

P 

ji

=
Example : Euclidean SumInvariants of CurvesSTEP 3 : We can find a moving frame moving fram

normalization equations

STEP 4 : Invariants can be obtained b

moving framewhere and 

][][

1

, nynx j

N

n

i

ji

∑ ⋅= =

η

o xx −

1

=

ρ

o 

y −

1

=

ρ

Nyyx

= )0,0,0(])[],1[],1[(
Example : Euclidean SumInvariants of Curves• The first-order summation inva

explicitly shown below

where 

()()(

)()()(

101,0010,11,0 

001

011,0010,10,1 100 

yxyxNxxPyyP

xxNxyyPxxP

−+−+−=

=

−

+

−

+

−

+

ηη

][][

][],1[],[],1[

1

,

0 1 0 1

nynxP

NyyyyNxxxx

j N

n

i

ji

∑= ⋅=

=

=

=

=
Feature extraction on 3D 

Local profile
Feature extraction on 3D 
Agenda1. Introduction2. Invariant Theory3. MSI Algorithm and Res

MSI Algorithm and Res

4. Conclusion
Block diagram

Image Preproces

Subsimilarity Gene

Similarity Normaliz

Analysis

Modified BioBo
Single Region Algorithm 

SI PCA 

Lin et. al.,“Fusion of Summation Invariants in 3D Human Face Recognition”, accepted to appear in CVPR’06

on cropped regibaseline on whobaseline on crop
Multi-Region Summation InvariaAlgorithm



Optimal Fusion with LDA

Similarity matrices

from 10 regions

+

w

1

w

n

W = 

Final simi
# of regions #1 #2 #3

Weight on each region

#4 #5 #6 #7 #8 #9 

9 regions .3586 .3222 .2665 .3661 .4849 .4008 .2396 .2749 .18

8 regions .3630 .3162 .2675 .3784 .4958 .4067 .2564 .2763 

7 regions .3708 .3125 .2709 .3759 .6225 .2881 .2819 6 regions .3922 .2400 .3906 .6289 .2994 .3044 5 regions .3980 .3736 .3923 .6679 .3192 4 regions .4052 .3754 .3950 .7339 3 regions .4692 .4374 .7671 2 regions .5367 .8437 1 regions 1.0 

LDA Fusion Results and Feat

Conclusion and Future Wo• Summation invariants 

– A systematical way to derive geo

for pattern recognition

– extract useful shape information

• Fusion of multiple regions

– LDA can improve performance

• Future work

– Apply SI on non-normalized shap

Lin et. al.,“Fusion of Summation InvarianFace Recognition”, accepted to appear in