# Principal Components Analysis & Eigenpictures - ElderLAB

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Nov 17, 2013 (4 years and 5 months ago)

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Principal Components
Analysis

Vida Movahedi

December 2006

Outline

What is PCA?

PCA for images

Eigenfaces

Recognition

Training Set

Test Set

Summary

Principal Component Analysis

Eigen Vectors show the direction of axes of a
fitted ellipsoid

Eigen Values show the significance of the
corresponding axis

The larger the Eigen value, the more separation
between mapped data

For high dimensional data,

only few of Eigen values

are significant

What is PCA?

Finding Eigen Values and Eigen Vectors

Deciding on which are significant

Forming a new coordinate system defined
by the significant Eigen vectors

(

lower dimensions for new coordinates)

Mapping data to the new space

Compressed Data

How is PCA used in Recognition?

A training set is used for
LEARNING

phase

Applying PCA to training data to form a new
coordinate system defined by significant Eigen
vectors

Representing each data in PCA coordinate system
(weights of Eigen vectors)

A test set is used for
TESTING

phase

Same PCA coordinate system is used

Each new data is represented in PCA coordinates

New data is recognized as the closest training data
(Euclidean distance)

PCA for images

Each image is represented as a 1
-
D data

i

Finding Eigen values/vectors is expensive

Turk/Pentland Trick:

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T
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T
T
M
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AA
Av
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v
Av
Av
AA
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M
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A
N
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AA
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C
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C

of
r
eigenvecto
:

of
r
eigenvecto
:
:
but

:
:
Trick
:

:
image

:
Matrix

Covariance

:
picture

average
2
2
2
2
1
1
1
1

What are Eigenfaces?

Turk and Pentland used PCA method for
face images

All faces are about the same size

Each face image is a data vector.

Each Eigen vector is actually an image
called an Eigenface.

Average image

Eigenfaces

Training set
-

before preprocessing

Training Set

Eigen Pictures

Significant Components

1
2
3
4
5
6
7
8
9
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

m
i
i
m
g
1
)
(

Recognition of Training set

L: No of
eigenvectors

Recognition Rate

10

10 of 10

9

10 of 10

8

10 of 10

7

9 of 10

6

8 of 10

5

8 of 10

Test set: Noisy images

P
n
=0.1

P
n
=0.2

P
n
=0.3

Recognition of Noisy images

P
n
: Probability of Noise

Recognition Rate

0.10

10 of 10

0.20

9 of 10

0.30

3 of 10

Summary

PCA gives a high compression rate

Performance is good when noise is
present

Performance is very bad if scale of
image is changed

References

1) Smith, L.I. (2002), “A tutorial on Principal Components Analysis”,
http://csnet.otago.ac.nz/cosc453/student_tutorials/principal_compon
ents.pdf.

2) Zhao, W., Chellappa, R., Rosenfeld, A., Phillips, P.J. (2000), “Face
Recognition: A literature survey”, UMD CfAR Technical Report CAR
-
TR
-
948, http://citeseer.ist.psu.edu/zhao00face.html.

3) Turk, M. and Pentland, A. (1991), “Eigenfaces for recognition”,
Journal of Cognitive Neuroscience, vol. 3, no. 1, p.71
-
86.

4) ‘Principal components analysis’,
http://en.wikipedia.org/wiki/Principal_ component_analysis.

5) “Eigenface”, http://en.wikipedia.org/wiki/Eigenface

6) Dailey, M. (2005), “Matt’s Matlab Tutorial Source Code Page”,
http://ai.ucsd.edu/ Tutorial/matlab.html.