biometrics and security tutorial 3. 1 (a) what is the scatter matrix (p4: 21)? understand what about...
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Biometrics and Security
Tutorial 3
• 1 (a) What is the scatter matrix (P4: 21)? Understand what about eigenvector and eigenvalue as well as their functions? (P4: 22)
Step 1
11
12
21
12
120 100
200 110
255 0
255 50
T
T
T
T
X
X
X
X
Step 2
11
12
21
12
1 1 1 11 1 1 2 2
2 2 2 22 1 1 2 2
1 21 2
120 100
200 110
255 0
255 50
1 1120 100 200 110 160 105
2 21 1
255 0 255 50 255 252 2
1 1160 105 255 25 207.5 65
2 2
T
T
T
T
T T T
T T T
T T T
X
X
X
X
m p X p X
m p X p X
m p m p m
Step 3
1 1 1 1 1 1 1 2 2 2 2 2 2 21 1 1 2 2 2 1 1 1 2 2 2
87.5 7.5 47.5 47.5187.5 35 7.5 45 47.5 -65 47.5 -15
35 45 65 154 1
4075 2400
2400 2566.7
T T T T
tS p p X m X m p X m X m p p X m X m p X m X m
1 1 2 11 2 1 2120 100 200 110 255 0 255 50
207.5 65
T T T T
T
X X X X
m
Step 4
1 2
11
21
1 00, (unit matrix)
0 1
0 4075 2400 4075 24000 0 4075 2566.7 2400* 2400 0
0 2400 2566.7 2400 2566.7
5836.5, 805.1
4075 24005836.5
2400 2
t t
i i t i
S I S I
S
11 12 12
21 22 22
4075 2400 805.1
566.7 2400 2566.7
(Hint: To solve that euqations manually, first assume that one of the variables is equal to 1,then find the other
and
11 11 21
11 21
21 11 1112 2
2111 21
one,
and finally normalize the vector to unit-length.)
5836.5 4075 2400
1, 0.7340
0.734 0.8062 0.8062
0.5917 0.59171
Assume Then
Normalization
2
0.5917,
0.8062Similarly
Step 5
1 2
11
1 2
1 22
1 2
5836.5, 805.1
100% 87.86%
100% 100%
r
r
0102030405060708090100
0 1 2 3
Step 6
1 2
1
1 1 2 11 2 1 2
11
12
21
'
'
'
0.8062 0.5917
0.5917 0.8062
'
120 100 200 110 255 0 255 50
0.8062120 100 37.57
0.5917
0.8062200 110 96.15
0.5917
0.8062255 0
0.5917
T
T T T T
W
X X W
X X X X
X
X
X
22'
205.58
0.8062255 50 175.99
0.5917X
0 2 5 5
**
o
o
1 1
2 2
2 2
1 1
11X
22X
21X
12X
11'X
12'X
22'X
21'X
P C A P ro jec tio n
• 1.(b) Understand the PCA application to facial recognition: Eigenface. (P4: 26-30)
12X
22X
21X
11X
T ra ining S e t:
E ige nv e c to rs :1 2
• 1. (c) Compare two StatPR techniques, PCA and LDA (P4: 13) and point out their main difference (P4: 42-45)
• 1. (d) Linear discrimination analysis (LDA) is introduced in P5:37-40. Please understand the two steps in P5:37 and compare within-class scatter matrix with between-class scatter matrix.
1 1 2 21 2 1 2
1 2
1 1 1 1 1 1 1 2 2 2 2 2 2 21 1 1 2 2 2 1 1 1 2 2 2
2
1
1 1 1 1 1 1 11 1 1 1 1 2 2 1 2
120 100 200 110 255 0 255 50
160 105 255 25 207.5 65
T T T T
T T T
T T T T
t
iW i
i
T
X X X X
m m m
S p p X m X m p X m X m p p X m X m p X m X m
S p S
p p X m X m p X m X
2 2 2 2 2 2 21 1 1 2 1 2 2 2 2 2 2
1 21 1 2 2
3200 400 0 01 1
400 50 0 12502 2
1600 200
200 650
47.5 47.51 147.5 40 47.5 40
40 402 2
T T T
T T
B
m p p X m X m p X m X m
S p m m m m p m m m m
2256.3 1900
1900 1600
Step 1
1
1
1 2
1600 200 2256.3 1900
200 650 1900 1600
0.0006 -0.0002
-0.0002 0.0016
1.8466 1.5550
3.4913 2.9400
1.8466 1.55500 0
3.4913 2.9400
5.1634 0
5.1
W B
W
F W B
i i F i
F F
S S
S
S S S
S
S I S
11 11
21 21
11 21
11 21 1
1
'1 '1 '1 2 1
1.8466 1.5550634
3.4913 2.9400
1 2.1330
0.4675 0.4675 0.8840
0.8840
Pr W= '
32.3 3.74
T
Assume
Normalization
ojection and X X W
X X X
2 '22119.21 75.01X
Step 2
0 2 5 5
**
o
o
1 1
11X
22X
21X
12X
11'X
12'X
22'X
21'X
L D A P ro jec tio n
1S
2S
BS
0 2 5 5
**
o
o
1 1
2 2
2 2
1 1
11X
22X
21X
12X
11'X
12'X
22'X
21'X
P C A P ro jec tio n
1 1 2 21 2 1 2
2 2 1 11 2 1 2
2 2 1 11 2 1 2
' ' ' '
' ' ' '
' ' ' '
After PCA projection
37.57 96.15 205.58 175.99
1 1123.925 ("Scatter between classes")
2 2
88.17 ("Scatter within class")
123.925/88.17=1.4055 (" / ")B W
X X X X
X X X X
X X X X
S S
'1 '1 '2 '21 2 1 2
2 2 1 11 2 1 2
2 2 1 11 2 1 2
' ' ' '
' ' ' '
205.28-37.37=167.88 ("Total scatter")
After LDA projection
32.3 3.74 119.21 75.01
1 1115.13 ("Scatter between classes")
2 2
72.77 ("Scatter within clas
X X X X
X X X X
X X X X
s")
115.13/ 72.77 1.5821 (" / ")
119.21 ( 32.3) 151.51 ("Total scatter")B WS S
2. There are three PR approaches: StatPR, SyatPR and NN. What difference between them (P4:4-10)? Based on your knowledge, can you give a simple application for each approach?
3. Please check the example of PR approaches in P4:10. Try to analysis the character “H” by statistical PR approach and structural PR approach. (StatPR approach - the feature set: (intersections, -, |, holes) and x=[2,1,4,0]; SyntPR approach - primitives and relations: ++++)
4. From P4:19-25, PCA method given by using image data is defined, which projects an image space with 644 dimension into 6 dimension eigenvector space. Please understand each step.
5. According to the figure in P4: 24, how to find its minimum λ if we hope to get rλ > 60? (λ = 3)