Linear Algebra Review. 6/26/2015Octavia I. Camps2 Why do we need Linear Algebra? We will associate coordinates to –3D points in the scene –2D points in.

Linear Algebra Review

04/19/23 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


Matrices

nmnn

m

m

m

mn

aaa

aaa

aaa

aaa

A

21

33231

22221

11211

mnmnmn BAC Sum:Sum:

ijijij bac

64

78

51

26

13

52Example:Example:

A and B must have the A and B must have the same dimensionssame dimensions


Matrices

pmmnpn BAC Product:Product:

m

kkjikij bac

1

1119

2917

51

26.

13

52

Examples:Examples:

1017

3218

13

52.

51

26

nnnnnnnn ABBA

A and B must have A and B must have compatible dimensionscompatible dimensions


Matrices

mnT

nm AC Transpose:Transpose:

jiij ac TTT ABAB )(

TTT BABA )(

If If AAT A is symmetricA is symmetric

Examples:Examples:

52

16

51

26T

852

316

83

51

26T


Matrices

Determinant:Determinant:

1315213

52det

Example:Example:

A must be squareA must be square

3231

222113

3331

232112

3332

232211

333231

232221

131211

detaa

aaa

aa

aaa

aa

aaa

aaa

aaa

aaa

122122112221

1211

2221

1211det aaaaaa

aa

aa

aa


Matrices

IAAAA nnnnnnnn

11

Inverse:Inverse: A must be squareA must be square

1121

1222

12212211

1

2221

1211 1

aa

aa

aaaaaa

aa

Example:Example:

61

25

28

1

51

261

10

01

280

028

28

1

51

26.

61

25

28

1

51

26.

51

261


2D Vector),( 21 xxv

PP

x1x1

x2x2

vv

Magnitude:Magnitude: 22

21|||| xx v

Orientation:Orientation:

1

21tanx

x

||||,

||||||||21

vvv

v xxIs a unit vectorIs a unit vector

If If 1|||| v , , v Is a UNIT vectorIs a UNIT vector


Vector Addition

),(),(),( 22112121 yxyxyyxx wv

vvww

V+wV+w


Vector Subtraction

),(),(),( 22112121 yxyxyyxx wv

vvww

V-wV-w


Scalar Product

),(),( 2121 axaxxxaa v

vv

avav


Inner (dot) Product

vv

ww

22112121 .),).(,(. yxyxyyxxwv

The inner product is a The inner product is a SCALAR!SCALAR!

cos||||||||),).(,(. 2121 wvyyxxwv

wvwv 0.


Orthonormal Basis

),( 21 xxv

1||||

1||||

j

i

jiv .. 21 xx

PP

x1x1

x2x2

vv

iijj )1,0(

)0,1(

j

i 0ji

12121 0.1.)...(. xxxxx ijiiv

22121 1.0.)...(. xxxxx jjijv


Vector (cross) Product

wvu

The cross product is a The cross product is a VECTOR!VECTOR!

ww

vv

uu

0)(

0)(

wwvwuwu

vwvvuvuOrientation:Orientation:

sin|||||||||.|| ||u|| wvwv Magnitude:Magnitude:


Vector Product Computation

),,(),,( 321321 yyyxxx wvu

kji

kji

u

)()()( 122131132332

321

321

yxyxyxyxyxyx

yyy

xxx

ww

vv

uu

i=(1,0,0)i=(1,0,0)j=(0,1,0)j=(0,1,0)k=(0,0,1)k=(0,0,1) 1k

1j

1i

i.j=0 , i.k=0 , i.j=0 , i.k=0 , j.k=0j.k=0

2D Geometrical Transformations


2D Translation

tt

PP

P’P’


2D Translation Equation

PP

xx

yy

ttxx

ttyy

P’P’tt

tPP ),(' yx tytx

),(

),(

yx tt

yx

t

P


2D Translation using Matrices

PP

xx

yy

ttxx

ttyy

P’P’tt

),(

),(

yx tt

yx

t

P

1

1

0

0

1' y

x

t

t

ty

tx

y

x

y

xP

tt PP


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!


Back to Cartesian Coordinates:

• Divide by the last coordinate and eliminate it. For example,

)/,/,/(0 ),,,(

)/,/(0 ),,(

wzwywxwwzyx

zyzxzzyx


2D Translation using Homogeneous Coordinates

PP

xx

yy

ttxx

ttyy

P’P’tt

1100

10

01

1

' y

x

t

t

ty

tx

y

x

y

x

P

)1,,(),(

)1,,(),(

yxyx tttt

yxyx

t

P tt

PP

PTP ' T


Scaling

PP

P’P’


Scaling Equation

PP

xx

yy

SSxx.x.x

P’P’SSyy.y.y

1100

00

00

1

' y

x

s

s

ys

xs

y

x

y

x

P

)1,,(),('

)1,,(),(

ysxsysxs

yxyx

yxyx

P

P

SPSP '


Scaling & Translating

PP

P’=S.PP’=S.P

P’’=T.P’P’’=T.P’

P’’=T.P’=T.(S.P)=(T.S).PP’’=T.P’=T.(S.P)=(T.S).P

SS

TT


Scaling & TranslatingP’’=T.P’=T.(S.P)=(T.S).PP’’=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


Translating & Scaling Scaling & Translating

P’’=S.P’=S.(T.P)=(S.T).PP’’=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


Rotation

PP

PP’’


Rotation Equations

Counter-clockwise rotation by an angle Counter-clockwise rotation by an angle

y

x

y

x

cossin

sincos

'

'

PP

xx

Y’Y’PP’’

X’X’

yy R.PP'


Degrees of Freedom

R is 2x2 R is 2x2 4 elements4 elements

BUT! There is only 1 degree of freedom: BUT! There is only 1 degree of freedom:

1)det(

R

IRRRR TT

The 4 elements must satisfy the following constraints:The 4 elements must satisfy the following constraints:

y

x

y

x

cossin

sincos

'

'


Scaling, Translating & Rotating

Order matters!Order matters!

P’ = S.PP’ = S.PP’’=T.P’=(T.S).PP’’=T.P’=(T.S).PP’’’=R.P”=R.(T.S).P=(R.T.S).P P’’’=R.P”=R.(T.S).P=(R.T.S).P

R.T.S R.T.S R.S.T R.S.T T.S.R … T.S.R …


3D Rotation of PointsRotation around the coordinate axes, Rotation around the coordinate axes, counter-clockwisecounter-clockwise::

100

0cossin

0sincos

)(

cos0sin

010

sin0cos

)(

cossin0

sincos0

001

)(

z

y

x

R

R

R

PP

xx

Y’Y’PP’’

X’X’

yy

zz


3D Rotation (axis & angle)

0

0

0

sin)cos1(cos

angle ,

12

13

23

233231

322

221

31212

1

321

nn

nn

nn

nnnnn

nnnnn

nnnnn

nnn T

IIR

n


3D Translation of Points

Translate by a vector t=(tTranslate by a vector t=(txx,t,tyy,t,txx))TT::

1000

100

010

001

z

y

x

t

t

t

T

PPxxY’Y’

PP’’

x’x’

yyzz

z’z’

tt

Linear Algebra Review. 6/26/2015Octavia I. Camps2 Why do we need Linear Algebra? We will associate coordinates to –3D points in the scene –2D points in.

Documents

t slide

d translation t p p

camps23 scaling p p

camps28 rotation p p

angle p x y p x y slide

p x y p x y z slide

x p s y

j slide