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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
8
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)
12
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
16
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
17
• 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 :-
18
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)
21
• 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
23
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
24
• 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
10/19/2011 27
10/19/2011 28
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