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

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

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

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

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

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

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

09-Sep-15 Octavia I. Camps 9

Vector Addition

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

vw

V+w

09-Sep-15 Octavia I. Camps 10

Vector Subtraction

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

vw

V-w

09-Sep-15 Octavia I. Camps 11

Scalar Product

),(),( 2121 axaxxxaa v

v

av

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.

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

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:

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

2D Geometrical Transformations

09-Sep-15 Octavia I. Camps 17

2D Translation

t

P

P’

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

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

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!

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

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

09-Sep-15 Octavia I. Camps 23

Scaling

P

P’

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 '

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

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

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

09-Sep-15 Octavia I. Camps 28

Rotation

P

P’

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'

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

'

'

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 …

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

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

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