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2
I: M A
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• R: • R
• A
• S D.S A .
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O
S
R
SA P
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−
100
0cossin
0sincos
θ θ
θ θ
100
10
01
y
x
d
d
100
00
00
y
x
s
s
100
010
01 xe
100
01
001
ye
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-
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O
S R
O R S
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L R
L R
L CR
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• I:
• O: C 1
6
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• I: • O: C 1 6
F ,
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• I: • O: C 1 6
F ,
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• M ...
• C , ,
( )
∑=
−+++−++=
n
j j j j j j j
ya ya xa xa ya xa E 1
2'
654
2'
321 )()()(a
=
∑
∑
∑
∑
∑∑
∑∑∑
∑∑∑
∑∑∑
∑∑∑
∑∑∑∑∑∑
j
j
j
j j
j
j j
j
j
j
j j
j
j j
j j
j
j
j
j
j
j
j
j
j j
j
j
j
j j
j
j
j j
j
j
j
j
j
j
j
j
j j
j
j
j
j j
j
j
y
y y
y x
x
x y
x x
a
a
a
a
a
a
y x
y y y x
x y x x
y x
y y y x
x y x x
'
'
'
'
'
'
6
5
4
3
2
1
2
2
2
2
1000
000
000
0001
000
000
ia E
∂∂
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F
M
(
=
'
''
'
'
'
'
'
1000
00011000
0001
1000
0001
1000
0001
4
4
3
3
2
2
1
1
6
5
4
3
2
1
44
44
33
33
22
22
11
11
y
x y
x
y
x
y
x
aa
a
a
a
a
y x
y x y x
y x
y x
y x
y x
y x
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• I:
• I
• A A = 1 2 1 3 4 2
• O:
• G = A
• O P:
• F • A
• A ,
• P?
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•
• B !
• R
• F • A
• P?
• , • S: B 4
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• I 1D
• =+
• :
• :
• = 1, = 2
• S =4.3, => = 2.3
4
2
5
3
4.3 ?
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• F ( x, y)
( ) y x y x y x y x
,,,,,,,
1
1
)int(
)int(
+=
+=
=
=
y y
x x
y y
x x
where
y x,
( ) y x,
y x,
y x,
y x,
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y y
x x
y y
x x
y x f y x f y x f y x f y x f
y
x
y
x
y x y x y x y x
−=
−=
−=
−=
+++=
ε
ε
ε
ε
ε ε ε ε ε ε ε ε
),(),(),(),(),('
y x,
y x,
y x,
y x,
xε
xε
yε yε
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( (
( (
( (
( (
0, 1
• 0,1
• C ()
• F (,) I2
• A ()
• E..
• (0,0) 0, 1 ::::::
• (0,1) 0, 0 :::::: (0,0)
I1 I2
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( (
( (
( (
( (
0, 0.7
0, 0.7
• 0,0.7• C (), 0, 0.7
• F (,) I2
• A ()
• E..
• (0,0)
0, 0.7 :::::: • (0,1) 0, 0 .3 :::::
(
• B I
I1 I2
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• S G S I
• S C
• M 2D
• I RGB
• A
• B I
• A
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• P 3D (1 1 1)
• ( )
2 = 1 +
2 = 1 +
2 = 1 +
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=
11000
100
010001
1
1
1
1
2
2
2
Z
Y X
dz
dydx
Z
Y X
• I
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=
11000
000
000
000
1
1
1
1
2
2
2
Z
Y
X
Sz
Sy
Sx
Z
Y
X
• I
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=
1000000
000
000
Sz
Sy
Sx
S
=
−
1000
01
00
001
0
0001
1
Sz
Sy
Sx
S
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• ?
1000
0100
0010
001 e
1000
0100
0010
001 e
1000
0100
0010
01 ed
1000
0100
001
001
e
d
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• R
• • =
• I , 2D
( 1 1 1)
( 2 2 2)
θ
−
=
110000100
00cossin
00sincos
1'
'
'
Z
Y
X
Z
Y
X
θ θ
θ θ
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• P
( 1 1 1)
( 2 2 2)
θ
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• R
• :
• P
•
• +
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Z Z
Y X Y
Y X X
=
+=
−=
'
cossin'
sincos'
θ θ
θ θ
X X
Z Y Z
Z Y Y
=
+=
−=
'
cossin'
sincos'
θ θ
θ θ
Y Y
X Z X
X Z Z
=
+=
−=
'
cossin'
sincos'
θ θ
θ θ
R
R
R
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Z Z
Y X Y
Y X X
=
+=
−=
'
cossin'
sincos'
θ θ θ θ
X X
Z Y Z
Z Y Y
=
+=
−=
'
cossin'
sincos'
θ θ θ θ
Y Y
X Z X
X Z Z
=
+=
−=
'
cossin'
sincos'
θ θ θ θ
−
1000
0100
00cossin
00sincos
θ θ
θ θ
−
1000
0cossin0
0sincos0
0001
γ γ
γ γ
−
1000
0cos0sin
0010
0sin0cos
β β
β β
R R R
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• R γ
β α
X Y Z R R R R
γ β α =
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• R M
+1
.. = =
•
• 3D
+
=
z
y
x
T
T
T
Z
Y
X
r r r
r r r
r r r
Z
Y
X
333231
232221
131211
'
'
'
=
110001
'
'
'
333231
232221
131211
Z
Y
X
T r r r
T r r r
T r r r
Z
Y
X
Z
y
X
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• 9 , 6
• H 3D 9
6 = 3
=
333231
232221
131211
r r r
r r r
r r r
R
∑ ∑= =
=
≠==
3
1
3
1 1
0
j j
jk jikjijk i
k ir r r r
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• A
.. P?
• Q ,
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39
0.793560.30367-0.5273-
0.46367-0.86294-0.20084-0.39404-0.403880.8256-
H I ?
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• A
=
0
0
1
13
12
11
333231
232221
131211
r
r
r
r r r
r r r
r r r
=
0
1
0
23
22
21
333231
232221
131211
r
r
r
r r r
r r r
r r r
=
1
0
0
33
32
31
333231
232221
131211
r
r
r
r r r
r r r
r r r
?
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• A
θ
• , 1, 2, 3
θ
• S 3
12
3
2
2
2
1 =++ nnn
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• θ
1. S H?
2. R θ
3. 1
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• Q: G 3D ,
θ
?
0.793560.30367-0.5273-
0.46367-0.86294-0.20084-
0.39404-0.403880.8256-Given R on the left, how
can we tell n and θ ?
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• 3D 1, θ + θ
θ θ
• ?
•
•
•
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• O C
• G ,
• E:
• Q: H ?
• I ?
=
100
0
0
100
10
01
100
43
21
2
1
243
121
aa
aa
b
b
baa
baa
43
21
aa
aa
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• S C:
• E
• I ΦΦΦΦ=ΦΛΦΛΦΛΦΛ
=ΦΛΦΦΛΦΦΛΦΦΛΦ (I?)
• A
= ( ,
• S ,
• L A = MM
• A = (S)(S)
• A = S2 = (S)1M
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• L
.
=
:
• •
•
• E
://..//S
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• I: ,
• 2D E: D A
M = 0.95 0.49 0.46
0.23 0.89 0.02
0 0 1
>> , S, = (M(1:2, 1:2))
=
0.78156 0.62384
0.62384 0.78156
S =
1.2904 0
0 0.56789
=
0.68658 0.72705
0.72705 0.68658
>> * S * '
=
0.95 0.49
0.23 0.89
I
?
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• I: E →→
•
F P S: F C G
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