Principal Component Analysis
and Applications in Image Analysis
Presented by Judy Jun Feng
Image Computing Group, Department of Computer Science
City University of Hong Kong
Image Computing Group. All rights reserved 2002.
Outline
•
Introduction and History
•
Bivariate or two variables case
•
Mathematical derivation in multivariate case
•
Graphical representation of PCA
•
Application in Face Recognition (FR)
•
Application in Active Shape Models (ASM)
•
A useful deriving for computation in practice
•
Conclusion
Image Computing Group. All rights reserved 2002.
Basic idea
One
of
the
multivariate
techniques
for
data
analysis
Main
purpose
is
to
reduce
the
dimension
of
the
data
set
and
retain
information
as
much
as
possible,
especially
retain
information
which
characterizes
the
most
important
variation
of
the
data
.
Transforming
the
original
data
to
a
new
set
of
uncorrelated
variables
(PCs)
History
1901: Karl Pearson

First proposed
1933: Harold Hotelling

General procedures
1963: Anderson

Theoretical developments of variance of sample PCs
1963: Rao

New ideas about interpretation and extension
1966: Gower

The links between PCA and other statistic techniques and
geometric insights
1970’s

Popularly used in the variety of areas
1989: Esbensen

Multivariate Image Analysis
Introduction and History
Image Computing Group. All rights reserved 2002.
Data Set:
Length and width of the similar shape objects in images: (
X
iL
X
iW
) (
i=1..n
)
Objective:
Find 2 uncorrelated components which are the normalized linear combinations of
X
L,
X
W
with maximum variance .
Procedures:
(1)
Transform the data set to the center:
(2)
Find two components:
with the constraint:
That is
PCA in Bivariate Case
LW
W
L
W
L
W
W
L
L
R
S
S
b
b
S
b
S
b
U
S
1
1
2
2
1
2
2
1
1
2
2
)
(
1
2
1
2
1
W
L
b
b
90
1
2
)
,
(
W
L
X
X
)
cos(
1
1
L
b
)
sin(
1
1
W
b
W
W
L
L
X
b
X
b
U
1
1
1
W
W
L
L
X
b
X
b
U
2
2
2
)
cos(
2
2
L
b
)
sin(
2
2
W
b
Image Computing Group. All rights reserved 2002.
Statistic and Geometric meaning
Original Axes
Principal Components
Components
X
L
X
W
U
1
=

0.59
X
W
+0.81
X
L
U
2
=0.81
X
W
+0.59
X
L
Standard Deviation
0.81
0.63
0.95
0.39
Correlation Coefficient

0.69
0
Variance Proportion
62%
38%
86%
14%
)
/(
2
2
2
W
L
x
x
S
S
S
p
L
W
X
X
U
81
.
0
59
.
0
1
L
W
X
X
U
59
.
0
81
.
0
2
Transform each pair of (
X
L,
X
W
) to a pair (
U
1
,
U
2
), by:
Transform each pair of (U1, U2) to a pair (XL, XW), by:
2
1
59
.
0
81
.
0
U
U
X
L
2
1
81
.
0
59
.
0
U
U
X
W
Image Computing Group. All rights reserved 2002.
Principal Components in the
multivariate case (1)
Data Set:
**
X=(X
1
, X
2
, X
3
…X
p
)
T
are vectors of p

dimensional random variables with
covariance matrix
C.
**
X
have been standardized to the same scales
Statistic background:
The symmetric matrix
C
is positive definite with eigenvalues .
Matrix formed by the eigenvectors is an orthogonal matrix
From the relations the covariance matrix
C
can be diagonalized to
And
0
...
2
1
p
]
[
i
e
T
I
T
T
T
p
T
CT
T
0
0
0
0
...
0
0
0
0
0
0
0
0
2
1
i
i
e
Ce
T
T
T
C
Image Computing Group. All rights reserved 2002.
Principal Components in the
multivariate case (2)
If principal components are chosen to be :
Then
That means the covariance matrix of
U
has been diagonalized to
with the largest variance
So
U
i
is known as the
i

th principal component of the original variables and
U
i
can be
calculated from
And X can be reconstructed by the principle components:
X
T
U
or
X
e
U
T
T
i
i
j
i
for
j
i
for
Ce
e
x
e
x
e
Cov
U
U
Cov
i
j
T
i
T
j
T
i
j
i
0
)
,
(
)
,
(
CT
T
U
U
Cov
U
Cov
T
j
i
)
,
(
)
(
1
1
1
1
)
,
(
)
(
U
U
Cov
U
Var
X
e
U
T
i
i
U
e
X
i
Image Computing Group. All rights reserved 2002.
Principal Components in the
multivariate case (3)
Total variance
Proportion variance
The cumulative proportion
is often calculated to find if
k
components have reproduced the data sufficiently well
Reduce the number of variables or dimension from p to k ( k<<p ).
p
i
p
i
i
i
total
S
S
1
1
2
2
2
/
total
i
S
2
1
/
)
(
total
k
i
i
S
Image Computing Group. All rights reserved 2002.
Graphical representation of PCA
2
2
)
(
)
(
w
w
L
L
Euclid
x
x
x
x
D
2
2
2
2
2
2
2
2
)
(
)
)(
(
2
)
(
xy
y
x
x
xy
y
M
R
S
S
S
y
y
R
y
y
x
x
S
x
x
D
2
2
2
2
2
1
1
1
2
u
u
M
S
u
u
S
u
u
D
2
2
2
2
2
2
1
1
1
2
...
p
p
p
M
u
u
u
u
u
u
D
E=2.56
M=2.8
M=6.4
Mahalanobis Distance
Euclidean Distance
**The
ith
principal
component
lies
on
the
ith
longest
axes
of
ellipsoids
.
**All
of
the
points
on
the
contour
of
one
ellipse
or
on
the
surface
of
one
ellipsoid
with
the
orthogonal
axes
of
the
principal
components,
have
the
same
Mahalanobis
distance
.
Image Computing Group. All rights reserved 2002.
Using PCA in the Face Recognition
(FR) (1)
Raw Faces
Eigen Faces
Faces: complex natural objects
with high dimension
.
Very difficult
to
develop a
computational model
Feature images decomposed
from face images
Speed, simplicity,
insensitivity to small or
gradual changes of the faces.
Image Computing Group. All rights reserved 2002.
Using PCA in the Face Recognition
(FR) (2)
Collect
a
number
of
face
images
as
the
training
set,
a
few
for
each
person
.
X

N
丬
ⴭⴭ
a
ow
晡捥
v散瑯e
潦
摩d敮獩en
†
N
2
1
.
The
eigenvectors
and
eigenvalues
can
be
found
from
the
covariance
matrix
C
from
the
face
vectors
of
those
images
.
Calculate
the
principal
components
from
linear
combinations
:
Use
principal
components
to
reconstruct
the
raw
image
We
call
these
eigenvectors
as
"eigenfaces"
.
The
reconstruction
can
also
be
viewed
as
the
combinations
of
eigenfaces
weighted
by
corresponding
values
of
PCs
.
In
practice,
each
individual
face
is
actually
approximated
by
only
a
few
k
eigenfaces
and
PCs
with
the
largest
eigenvalues
.
256
㈵2
†
8
扩b
†
㈰2
業ag敳
†
㐰
敩e敮晡捥e
†††
2
2
N
N
X
e
U
T
i
i
U
e
X
i
Image Computing Group. All rights reserved 2002.
Using PCA in the Face Recognition
(FR) (3)
Define a face class using a group of images of one person
Get a group of
k

d PC vectors
Defined the face class centered at the mean of these PC vectors.
Choose a distance threshold to define the maximum allowable distance .
Define a face space using all of the images in the training set
Get the PC vectors of all the face images in the training set
Find the mean and allowable distance to determine if an input image is a face
images
Face Recognition
A
new
face
is
:
(
1
)
A
known
people
:
if
it
belongs
to
a
face
class
.
(
2
)
An
unknown
people
:
add
to
training
set
.
(
3
)
Not
a
face
at
all
:
far
away
from
the
face
space
.
]
,...,
,
[
2
1
k
T
u
u
u
U
X
e
U
T
i
i
i
i
u
e
X
Image Computing Group. All rights reserved 2002.
Applying PCA in the Active Shape
Models (1)
•
Flexible models, or deformable templates, can be used to allow for some
degree of variability in the shape of the image objects
•
PCA is often used to find some most important "mode of variation", and
helps to obtain a more compressed statistical description of the shapes and
their variation
Metacarpal shapes in X

ray images
Image Computing Group. All rights reserved 2002.
Applying PCA in the Active Shape
Models (2)
•
Labeling and Aligning the shapes training set
Each shape has been described by a position vector( A points in 2p

d space )
Mean shape:
•
Get the most important modes of variation
Find the eigenvectors from the 2p
2瀠捯va物湣攠ma瑲楸
The position vectors can be reproduced by the eigenvectors weighted by the values of
PCs, and every eigenvectors accounts for a kind of mode of variation .
K largest eigenvalues and their eigenvectors, the most important modes of variations.
All of the PC vectors in the training set can consist of the "Allowable Shape Domain" .
A possible shape similar to the class of shapes can be expressed by those modes with
different weights in the “Allowable Shape Domain” .
)
,
,...
,
,
(
2
2
,
1
1
ip
ip
i
i
i
i
i
y
x
y
x
y
x
x
N
i
i
x
N
x
1
1
Image Computing Group. All rights reserved 2002.
A useful deriving for computation in
practice
•
High computer load to find the eigenvectors in image analysis
Face Recognition,
256
256⁎
2
=65,536 And C = 65,536
65ⰵ,6
•
There are only M meaningful eigenvectors
( M is the number of the images in the training data)
Suppose
V
is the eigen vectors of
is the eigen vectors of
i
i
i
T
V
AV
A
i
i
i
T
AV
AV
AA
i
AV
T
M
i
T
i
i
AA
M
C
1
1
)
(
2
2
N
N
A
A
T
)
(
M
M
T
AA
C
I
]
,...,
,
[
2
1
M
A
)
1
(
2
N
)
(
2
M
N
M
k
k
ik
i
i
V
AV
U
1
Image Computing Group. All rights reserved 2002.
Conclusion
PCA
has
been
proved
an
effective
approach
for
image
recognition,
segmentation
and
other
kind
of
analysis
.
•
Advantages:
(1)Extract uncorrelated basic feature sets to describe the properties of a data set.
(2)Reduce the dimensionality of the original image space.
•
Disadvantages:
(1) Provide little quantitative information and visualization implications
(2)
No way to discriminate between variance due to object of interest and variance
due to noise or background.
•
Researchers proposed different schemes and make many improvements for
better applying PCA in image analysis.
Image Computing Group. All rights reserved 2002.
The End
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