Face Recognition Using

Laplacianfaces Xiaofei He, Shuicheng Yan, Yuxiao u,Partha Niyogi,

Hong-Jiang Zhang

IEEE Trans. on PATTERN ANALYSIS AND MACHINE INTELLIGENCE, MARCH 2005

Presented By

Sreekanth Raja

M Tech Computational Science

SERC

Indian Institute of Science, Bangalore

10/19/2011 1

Outline

• Face Recognition – Introduction

• Motivation and Current Research

• Laplacian Faces

• Results and Conclusions

10/19/2011 2

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

3

• Given a face image that belongs to a person in

a database, tell whose image it is.

• Applications – Access control, biometrics, HMI

• Face Recognition – Feature based, Appearance

Based

• Feature based – Local Feature Analysis(LFA),

Gabor wavelets etc

• Appearance Based – PCA, ICA,LDA etc..

• Recent studies reveal that face images reside in a nonlinear sub manifold

• Nonlinear techniques to discover nonlinear structure of manifolds – Isomaps, Local Linear Embedding(LLE) , Laplacian Eigen maps etc

• Kernel based methods for dimensionality reduction also discover non linear structure of face images

• All these are computationally expensive

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

4

10/19/2011

• The Laplacianfaces method is proposed

against this background.

• Laplacianfaces method preserves the local

structure of the image space.

• Laplacianfaces method is linear and is

computationally efficient compared to other

nonlinear techniques

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

5

• Appearance-based approach to human face

representation and recognition

• Uses Locality Preserving Projection(LPP)

• It creates a face subspace which explicitly

considers the face manifold structure

• Better discriminating power than PCA

• Reduces the dimension of the face image

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

6

10/19/2011

• Appearance-based approach to human face

representation and recognition

• Face manifold structure modeled by nearest

neighbor graph which preserves the local

structure of image space

• PCA, LDA and LPP can be derived from

different graph models

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

7

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

• Eigenface(PCA) – preserves global structure of

image space

• Fischerface(LDA) – preserves discriminating

information

• Laplacianface(LPP) – preserves local structure

of image space

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10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

• Appearance based recognition – face image

modeled as d – dimensional vector

• Consider n d – dimensional zero mean face

vectors

• Generalized approach : can we find a linear

map w, and a vector such that

9

• Different objective function will give different algorithms

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

PCA

Solution:

Set of Principal Eigenvectors

LDA

M - total sample mean

m(i) - average of ith class

SW - within-class scatter matrix

SB - between-class scatter

matrix. 10

• Locality Preserving Projection(LPP)

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Objective Function

Where S is a similarity matrix defined as :

OR

11

• The matrix S defines the “locality” of images

• Minimizing this objective function ensures

that and are close.

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Some Notations:

(Column/Row sum of S)

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10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Matrix form of objective function

The matrix D provides a natural

Measure on data points

gives a measure importance of

the ith image (hence )

So a constraint can be imposed:-

Thus the optimization problem is:-

The solution is the solution to the

Generalized Eigenvalue problem :

The solutions

Are the so called Laplacianfaces

13

• Connection of LPP to PCA

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Suppose

i.e, we are not considering “local structure ” – all images equally close

Then

The Laplacian Matrix and let

Covariance Matrix of Dataset

14

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

• Connection of LPP to LDA

Recall LDA objective function :

Equivalently, solve the generalized Eigenvalue problem :

15

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

is the Covariance matrix of the ith class

For further simplification, define

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10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Thus the generalized Eigen vector

problem of LDA can be written as

Thus optimal projections correspond to

Eigen vectors corresponding to the

Smallest eigenvalues

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• LPP - Algorithm

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Locality Preserving Projections

• PCA , LDA and LPP

• Laplacianfaces for recognition

Step1 : Construct Adjacency graph

and are “close”

Step2 : Choosing the weights

Step3 : Eigen map: generalized eigenvector

problem

with

locality preserving face subspace

is spanned by

Step 4 : Recognition

A new face is projected into the

face space by

To determine which face class

find the minimum value of

where is the vector representing

the k th face class

In case is singular, use PCA to reduce

Dimensionality so that the resulting

Is non singular. In this case, the embedding

Is :-

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10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

2- D embedding of Laplacianfaces 19

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

Distribution of 10 test samples 20

• Performance Evaluated on 3 datasets:-

– Yale Database

– PIE(Pose Illumination and Expression) database

– MSRA(Microsoft Research Asia) database

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• Comparison with Eigenface and Fischerface

• NN metric used to define neighborhood and

Euclidean metric used as the distance measure

• Each image is 32x32 pixels(1024 dim vector)

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• Yale Database

– 165 gray scale image of 15 individuals

– A random subset of 6 image/person taken for training and rest for

testing

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

22

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• PIE Database

– 68 subjects, total of 41,368 face images

– 13 synchronized cameras with 21 flashes

– 170 images used for each individual - 85 for training and 85 for testing

Some images

of PIE database

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10/19/2011 10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

• MSRA Database

– 12 subjects, captured in 2 sessions with different BG and illumination

– All images used – 1st session for training and 2nd for testing

Some images

of MSRA database

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• A few observations and Discussions:-

– All three methods performed better in their optimal subspaces than full image space

– In all three, Laplacianfaces performed better

– Laplacian faces takes advantage of more training samples

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

25

• Laplacian faces outperforms Eigen faces and Fischer faces approaches

• It’s a linear transform that optimally preserves the nonlinear local manifold structure

• Possible Extension of work:-

– use of sophisticated and better distance metrics like variance normalized distance may improve the recognition performance

– The present work is that of face analysis. Possibly this can be extended to unlabeled classes

10/19/2011

Face Recognition – Introduction

Motivation and Current Research

Laplacian Faces

Results and Conclusions

26

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