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```2010
Scientific Computing
Face Recognition
Image database:
Test image:
A:
B:
Who is this guy?
C:
D:
E:
F:
Characteristics of FR:
• A mode of biometric
identification
• Easy for human, hard for
machine
G:
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Scientific Computing
Biometric Identification
Identification of people from their physical
characteristics, such as
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faces
voices
fingerprints
palm prints
hand vein distributions
hand shapes and sizes
retinal scans
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2010
Scientific Computing
FR via PCA
First paper:
• M. Turk and A. Pentland, "Eigenfaces for
Recognition", Journal of Cognitive Neuroscience,
vol. 3, no. 1, pp. 71-86, 1991
Characteristics
• Efficient computation
• Proven mathematics
• Applicable to face detection
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Scientific Computing
Problem Definition
Input
• A dataset of face images of n person
• An unknown person’s face image
Output:
• Determine the identity of the unknown person
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Scientific Computing
ATT Face Dataset
Origin
• Olivetti Research
Laboratory,
1992~1994
Stats:
• 40 subjects, each
with 10 images
Characteristics
6
• Same-size photos
of black and white
• Centered faces of
different poses
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Scientific Computing
Face Recognition via PCA
Compute
Mean Face
Select 6
Principal
Eigenfaces
Facial Signatures
Compute
Eigenvectors
(Eigenfaces)
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f   wi ui
Subtract
i 1
400
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400
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Scientific Computing
Steps of Feature Extraction via PCA
3 simple steps:
1. Data preprocessing
- Each sample image is rearranged into a column
vector of length 112*92=10304. All images are
put into a matrix F of size 10304x400.
- Mean face is subtracted from each column.
2. PCA
- Find the eigenvectors of F*F’.
3. Projection
- Select top k eigenvectors with k largest
eigenvalues  k eigenfaces!
- Do projection along these eigenfaces to find new
features for classification
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Scientific Computing
Details for Step 2: PCA
Problem: FF T is large,10304x10304! (849MB!)
How to compute the eigenvectors of FF T ?
Observation:
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If u is the eigenvector of F’F, then Fu is the
eigenvector of FF’.
If l is the eigenvalue of F’F, then l is also the
eigenvalue of FF’.
F T Fu  lu  FF T Fu  lFu  FF T  Fu   l  Fu 
Note that:
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FF’ has 10304 eigenvalues.
F’F has 400 eigenvalues, corresponding to the
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400 largest eigenvalues of FF’.
2010
Scientific Computing
Details for Step 3: Projection (1/2)
Each face (minus the mean) in the training set
can be represented as a linear combination
of the best k eigenvectors:
k
f  f mean   wi ui
i 1
Typical eigenfaces when k=4:
u1
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u2
u3
u4
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Scientific Computing
Details for Step 3: Projection (2/2)
Since u1, u2 , u3 , u4  is an orthonormal basis, any
face (after mean subtraction) can be
represented by this basis:
The feature vector of the face is then the new
coordinates obtained by:
u1T 
 T
u2 
T
T

U f  T f   0.9571, 0.1945, 0.0461, 0.0586
u3 
 T
u4 
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Scientific Computing
Classification
Once the features for images are extracted, we
can then apply any classification methods to
obtain the final recognition results, including
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Minimum distance classifier
Support vector machines
Neural networks
Gaussian mixture models
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Scientific Computing
Face Detection Using Eigenfaces
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Scientific Computing
Distance from Face Space
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