linear algebra review - university of maryland · 2015. 9. 10. · linear algebra review. 09-sep-15...
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![Page 1: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/1.jpg)
Linear Algebra Review
![Page 2: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/2.jpg)
09-Sep-15 Octavia I. Camps 2
Why do we need Linear Algebra?
• We will associate coordinates to– 3D points in the scene– 2D points in the CCD array– 2D points in the image
• Coordinates will be used to– Perform geometrical transformations– Associate 3D with 2D points
• Images are matrices of numbers– We will find properties of these numbers
![Page 3: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/3.jpg)
09-Sep-15 Octavia I. Camps 3
Matrices
nmnn
m
m
m
mn
aaa
aaa
aaa
aaa
A
21
33231
22221
11211
mnmnmn BAC Sum:
ijijij bac
64
78
51
26
13
52Example:
A and B must have the same dimensions
![Page 4: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/4.jpg)
09-Sep-15 Octavia I. Camps 4
Matrices
pmmnpn BAC Product:
m
k
kjikij bac1
1119
2917
51
26.
13
52
Examples:
1017
3218
13
52.
51
26
nnnnnnnn ABBA
A and B must have compatible dimensions
![Page 5: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/5.jpg)
09-Sep-15 Octavia I. Camps 5
Matrices
mnT
nm AC
Transpose:
jiij ac TTT ABAB )(
TTT BABA )(
If AAT A is symmetric
Examples:
52
16
51
26T
852
316
83
51
26T
![Page 6: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/6.jpg)
09-Sep-15 Octavia I. Camps 6
Matrices
Determinant:
1315213
52det
Example:
A must be square
3231
2221
13
3331
2321
12
3332
2322
11
333231
232221
131211
detaa
aaa
aa
aaa
aa
aaa
aaa
aaa
aaa
12212211
2221
1211
2221
1211det aaaa
aa
aa
aa
aa
![Page 7: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/7.jpg)
09-Sep-15 Octavia I. Camps 7
Matrices
IAAAA nnnnnnnn
11
Inverse: A must be square
1121
1222
12212211
1
2221
1211 1
aa
aa
aaaaaa
aa
Example:
61
25
28
1
51
261
10
01
280
028
28
1
51
26.
61
25
28
1
51
26.
51
261
![Page 8: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/8.jpg)
09-Sep-15 Octavia I. Camps 8
2D Vector),( 21 xxv
P
x1
x2
v
Magnitude: 2
2
2
1|||| xx v
Orientation:
1
21tanx
x
||||,
||||||||
21
vvv
v xxIs a unit vector
If 1|||| v , v Is a UNIT vector
![Page 9: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/9.jpg)
09-Sep-15 Octavia I. Camps 9
Vector Addition
),(),(),( 22112121 yxyxyyxx wv
vw
V+w
![Page 10: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/10.jpg)
09-Sep-15 Octavia I. Camps 10
Vector Subtraction
),(),(),( 22112121 yxyxyyxx wv
vw
V-w
![Page 11: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/11.jpg)
09-Sep-15 Octavia I. Camps 11
Scalar Product
),(),( 2121 axaxxxaa v
v
av
![Page 12: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/12.jpg)
09-Sep-15 Octavia I. Camps 12
Inner (dot) Product
v
w
22112121 .),).(,(. yxyxyyxxwv
The inner product is a SCALAR!
cos||||||||),).(,(. 2121 wvyyxxwv
wvwv 0.
![Page 13: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/13.jpg)
09-Sep-15 Octavia I. Camps 13
Orthonormal Basis
),( 21 xxv
1||||
1||||
j
i
jiv .. 21 xx
P
x1
x2
v
ij )1,0(
)0,1(
j
i0 ji
12121 0.1.)...(. xxxxx ijiiv
22121 1.0.)...(. xxxxx jjijv
![Page 14: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/14.jpg)
09-Sep-15 Octavia I. Camps 14
Vector (cross) Product
wvu
The cross product is a VECTOR!
w
v
u
0)(
0)(
wwvwuwu
vwvvuvuOrientation:
sin|||||||||.|| ||u|| wvwv Magnitude:
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09-Sep-15 Octavia I. Camps 15
Vector Product Computation
),,(),,( 321321 yyyxxx wvu
1||||
1||||
1||||
k
j
i
kji
kji
u
)()()( 122131132332
321
321
yxyxyxyxyxyx
yyy
xxx
)1,0,0(
)0,1,0(
)0,0,1(
k
j
i
0,0,0 kjkiji
w
v
u
![Page 16: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/16.jpg)
2D Geometrical Transformations
![Page 17: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/17.jpg)
09-Sep-15 Octavia I. Camps 17
2D Translation
t
P
P’
![Page 18: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/18.jpg)
09-Sep-15 Octavia I. Camps 18
2D Translation Equation
P
x
y
tx
ty
P’t
tPP ),(' yx tytx
),(
),(
yx tt
yx
t
P
![Page 19: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/19.jpg)
09-Sep-15 Octavia I. Camps 19
2D Translation using Matrices
P
x
y
tx
ty
P’t
),(
),(
yx tt
yx
t
P
11
0
0
1' y
x
t
t
ty
tx
y
x
y
xP
t P
![Page 20: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/20.jpg)
09-Sep-15 Octavia I. Camps 20
Homogeneous Coordinates
• Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example,
0 ),,,(),,(
0 ),,(),(
wwwzwywxzyx
zzzyzxyx
• NOTE: If the scalar is 1, there is no need for the multiplication!
![Page 21: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/21.jpg)
09-Sep-15 Octavia I. Camps 21
Back to Cartesian Coordinates:
• Divide by the last coordinate and eliminate it. For example,
)/,/,/(0 ),,,(
)/,/(0 ),,(
wzwywxwwzyx
zyzxzzyx
![Page 22: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/22.jpg)
09-Sep-15 Octavia I. Camps 22
2D Translation using Homogeneous Coordinates
P
x
y
tx
ty
P’t
1100
10
01
1
' y
x
t
t
ty
tx
y
x
y
x
P
)1,,(),(
)1,,(),(
yxyx tttt
yxyx
t
P t
P
PTP 'T
![Page 23: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/23.jpg)
09-Sep-15 Octavia I. Camps 23
Scaling
P
P’
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09-Sep-15 Octavia I. Camps 24
Scaling Equation
P
x
y
Sx.x
P’Sy.y
1100
00
00
1
' y
x
s
s
ys
xs
y
x
y
x
P
)1,,(),('
)1,,(),(
ysxsysxs
yxyx
yxyx
P
P
SPSP '
![Page 25: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/25.jpg)
09-Sep-15 Octavia I. Camps 25
Scaling & Translating
P
P’=S.P
P’’=T.P’
P’’=T.P’=T.(S.P)=(T.S).P
S
T
![Page 26: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/26.jpg)
09-Sep-15 Octavia I. Camps 26
Scaling & TranslatingP’’=T.P’=T.(S.P)=(T.S).P
11100
0
0
1100
00
00
100
10
01
''
yy
xx
yy
xx
y
x
y
x
tys
txs
y
x
ts
ts
y
x
s
s
t
t
PSTP
![Page 27: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/27.jpg)
09-Sep-15 Octavia I. Camps 27
Translating & Scaling Scaling & Translating
P’’=S.P’=S.(T.P)=(S.T).P
11100
0
0
1100
10
01
100
00
00
''
yyy
xxx
yyy
xxx
y
x
y
x
tsys
tsxs
y
x
tss
tss
y
x
t
t
s
s
PTSP
![Page 28: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/28.jpg)
09-Sep-15 Octavia I. Camps 28
Rotation
P
P’
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09-Sep-15 Octavia I. Camps 29
Rotation Equations
Counter-clockwise rotation by an angle
y
x
y
x
cossin
sincos
'
'
P
x
Y’P’
X’
y R.PP'
![Page 30: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/30.jpg)
09-Sep-15 Octavia I. Camps 30
Degrees of Freedom
R is 2x2 4 elements
BUT! There is only 1 degree of freedom:
1)det(
R
IRRRR TT
The 4 elements must satisfy the following constraints:
y
x
y
x
cossin
sincos
'
'
![Page 31: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/31.jpg)
09-Sep-15 Octavia I. Camps 31
Scaling, Translating & Rotating
Order matters!
P’ = S.PP’’=T.P’=(T.S).PP’’’=R.P”=R.(T.S).P=(R.T.S).P
R.T.S R.S.T T.S.R …
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09-Sep-15 Octavia I. Camps 32
3D Rotation of PointsRotation around the coordinate axes, counter-clockwise:
100
0cossin
0sincos
)(
cos0sin
010
sin0cos
)(
cossin0
sincos0
001
)(
z
y
x
R
R
R
P
x
Y’P’
X’
y
z
![Page 33: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/33.jpg)
09-Sep-15 Octavia I. Camps 33
3D Rotation (axis & angle)
0
0
0
sin)cos1(cos
angle ,
12
13
23
2
33231
32
2
221
3121
2
1
321
nn
nn
nn
nnnnn
nnnnn
nnnnn
nnnT
IIR
n
![Page 34: Linear Algebra Review - University Of Maryland · 2015. 9. 10. · Linear Algebra Review. 09-Sep-15 Octavia I. Camps 2 Why do we need Linear Algebra? • We will associate coordinates](https://reader034.vdocument.in/reader034/viewer/2022051903/5ff31d63b221fd7ddf10b36a/html5/thumbnails/34.jpg)
09-Sep-15 Octavia I. Camps 34
3D Translation of Points
Translate by a vector t=(tx,ty,tx)T:
1000
100
010
001
z
y
x
t
t
t
T
PxY’
P’
x’
yz
z’
t