Multimedia Programming 07: Image Warping Keyframe Animation Departments of Digital Contents Sang Il...

Multimedia Programming 07:

Image WarpingKeyframe Animation

Departments of Digital Contents

Sang Il Park

Outline

• Review• Image Warping

– Translation

• Keyframe Animation

Review

• Image Warping– Scaling– Rotation– Together with a 2X2 matrix

Image Processing 2-2

Image Warping(Translation)

Lots of slides taken from Alexei Efros

Review: 2x2 Matrices

• What types of transformations can be represented with a 2x2 matrix?

2D Identity?

yyxx

''

yx

yx

1001

''

2D Scale around (0,0)?

ysy

xsx

y

x

*'

*'

y

x

s

s

y

x

y

x

0

0

'

'

Review: 2x2 Matrices

• What types of transformations can be represented with a 2x2 matrix?

2D Rotate around (0,0)?

yxyyxx

*cos*sin'*sin*cos'

y

x

y

x

cossin

sincos

'

'

2D Shear?

yxshy

yshxx

y

x

*'

*'

y

x

sh

sh

y

x

y

x

1

1

'

'

Review: 2x2 Matrices

• What types of transformations can be represented with a 2x2 matrix?

2D Mirror about Y axis?

yyxx

''

yx

yx

1001

''

2D Mirror over (0,0)?

yyxx

''

yx

yx

1001

''

Review: 2x2 Matrices

• What types of transformations can be represented with a 2x2 matrix?

2D Translation?

y

x

tyy

txx

'

'

Only linear 2D transformations can be represented with a 2x2 matrix

NO!

All 2D Linear Transformations

• Linear transformations are combinations of …– Scale,– Rotation,– Shear, and– Mirror

• Properties of linear transformations:– Origin maps to origin– Lines map to lines– Parallel lines remain parallel– Ratios are preserved– Closed under composition

y

x

dc

ba

y

x

'

'

yx

lkji

hgfe

dcba

yx''

Translation

• How can we represent translation?

• How can we represent it as a matrix?

y

x

tyy

txx

'

'

Homogeneous Coordinates

•Homogeneous coordinates– represent coordinates in 2 dimensions with a 3-

vector

1

coords shomogeneou y

x

y

x

( 동차좌표 )

Homogeneous Coordinates

• Q: How can we represent translation as a 3x3 matrix?

• A: Using the rightmost column:

100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx

'

'

Translation•Example of translation

tx = 2ty = 1

11100

10

01

1

'

'

y

x

y

x

ty

tx

y

x

t

t

y

x

Homogeneous Coordinates

Homogeneous Coordinates

• Add a 3rd coordinate to every 2D point– (x, y, w) represents a point at location (x/w,

y/w)– (x, y, 0) represents a point at infinity– (0, 0, 0) is not allowed

Convenient coordinate system to represent many useful transformations

1 2

1

2(2,1,1) or (4,2,2) or (6,3,3)

x

y

Basic 2D Transformations

• Basic 2D transformations as 3x3 matrices

1100

0cossin

0sincos

1

'

'

y

x

y

x

1100

10

01

1

'

'

y

x

t

t

y

x

y

x

Translate

Rotate

1100

00

00

1

'

'

y

x

s

s

y

x

y

x

Scale

Affine Transformations

• Affine transformations are combinations of …– Linear transformations, and– Translations

• Properties of affine transformations:– Origin does not necessarily map to origin– Lines map to lines– Parallel lines remain parallel– Ratios are preserved– Closed under composition– Models change of basis

wyx

fedcba

wyx

100''

( 유사변환 )

Projective Transformations

• Projective transformations …– Affine transformations, and– Projective warps

• Properties of projective transformations:– Origin does not necessarily map to origin– Lines map to lines– Parallel lines do not necessarily remain parallel– Ratios are not preserved– Closed under composition– Models change of basis

wyx

ihgfedcba

wyx

'''

( 사영변환 )

Matrix Composition

• Transformations can be combined by matrix multiplication

wyx

sysx

tytx

wyx

1000000

1000cossin0sincos

1001001

'''

p’ = T(tx,ty) R() S(sx,sy) p

Sequence of composition1. First, Scaling2. Next, Rotation3. Finally, Translation

Inverse Transformation

1

'

'

100

0cossin

0sincos

1

y

x

y

x

1

'

'

100

10

01

1

y

x

t

t

y

x

y

x

Translate

Rotate

1

'

'

100

0/10

00/1

1

y

x

s

s

y

x

y

x

Scale

Inverse Transformation

wyx

sysx

tytx

wyx

1000000

1000cossin0sincos

1001001

'''

p’ = T(tx,ty) R() S(sx,sy) p

'

'

'

100

10

01

100

0cossin

0sincos

100

0/10

00/1

w

y

x

ty

tx

sy

sx

w

y

x

P = S-1(sx,sy) R-1() T-1(tx,ty) p’

2D image transformations

These transformations are a nested set of groups• Closed under composition and inverse is a member

Keyframe Animation

Keyframe Animation

• Keyframes?– Define starting and ending

points of any smooth transitions

• The workflow of traditional hand-drawn animation– 1. Key frames

by senior key artists– 2. ‘clean-up’ and

‘inbetweens’ by inbetweeners keyframes inbetweens

Keyframe Animation

The starting keyframe The ending keyframe The completed animation

Keyframing: issues

• How do you know where to put the keyframes?

• Interface: How do you actually specify the key values?

• Inbetween: How do you make frames between keys

Making inbetweens

• Interpolation

StartTime = 0

EndTime = 1

InbetweenTime = α

(0 ≤ α ≤1)

? ?

Making inbetweens

• Linear Interpolation

StartTime = 0

EndTime = 1

InbetweenTime = α

(0 ≤ α ≤1)

1-α α

State vector

• How to represent a state?– By Numbers! (or Parameters!)

– Examples)• Color: R,G,B (H,S,V)• Scale: Sx, Sy• Rotation: θ• Translation: Tx, Ty• Position of vertices• …

State vector

• A state vector: a collection of parameters– Example with 5 parameters

• Scale: Sx, Sy• Rotation: θ• Translation: Tx, Ty

y

x

y

x

T

T

s

s

v

Interpolation of state vector

• Keyframes:

0

0

0

0

0

0

y

x

y

x

T

T

s

s

v

y

x

y

x

T

T

s

s

1

1

1

1

1

1 v

10 , vv

(at time=0) (at time=1)

Interpolation of state vector

• What is the state vector at time = α ?

)1( 10 vvv

1-α α

0v1v

α

Interpolation of state vector

• What is the state vector at time = α ?

)1(

)1(

)1(

)1(

)1(

)1(

10

10

10

10

10

10

yy

xx

yy

xx

TT

TT

ss

ss

vvv

1-α α

0v1v

α