linear algebra review - university of maryland · 2015. 9. 10. · linear algebra review. 09-sep-15...

Linear Algebra Review

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


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


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


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


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


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


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


Vector Addition

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

vw

V+w


Vector Subtraction

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

vw

V-w


Scalar Product

),(),( 2121 axaxxxaa v

v

av


Inner (dot) Product

v

w

22112121 .),).(,(. yxyxyyxxwv

The inner product is a SCALAR!

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

wvwv 0.


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


Vector (cross) Product

wvu

The cross product is a VECTOR!

w

v

u

0)(

0)(

wwvwuwu

vwvvuvuOrientation:

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


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

2D Geometrical Transformations


2D Translation

t

P

P’


2D Translation Equation

P

x

y

tx

ty

P’t

tPP ),(' yx tytx

),(

),(

yx tt

yx

t

P


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


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

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


Scaling

P

P’


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 '


Scaling & Translating

P

P’=S.P

P’’=T.P’

P’’=T.P’=T.(S.P)=(T.S).P

S

T


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


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


Rotation

P

P’


Rotation Equations

Counter-clockwise rotation by an angle

y

x

y

x

cossin

sincos

'

'

P

x

Y’P’

X’

y R.PP'


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

'

'


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 …


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


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


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

linear algebra review - university of maryland · 2015. 9. 10. · linear algebra review. 09-sep-15...

Documents