Image warping/morphing

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Image warping/morphing

Digital Visual Effectsg Yung-Yu Chuang

with slides by Richard Szeliski, Steve Seitz, Tom Funkhouser and Alexei Efros

Image warping Image warping

Image formation

B

A

Sampling and quantization

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What is an image

• We can think of an image as a function, f: R²R:

f( ) i th i t it t iti ( ) – f(x, y) gives the intensity at position (x, y) – defined over a rectangle, with a finite range:

f [ b] [ d]  [0 1]

• f: [a,b]x[c,d]  [0,1]

f x y

• A color image ^{( ,} ⁾

( , ) ( , )

r x y

f x y g x y

 

 

  

( , ) ( , )

( , )

f y g y

b x y

 

 

A digital image

• We usually operate on digital (discrete) images:

S l th 2D l id – Sample the 2D space on a regular grid

– Quantize each sample (round to nearest integer)

f l h

• If our samples are D apart, we can write this as:

f[i ,j] = Quantize{ f(i D, j D) }

• The image can now be represented as a matrix of integer values

o tege values

Image warping

image filtering: change range of image ( ) h(f( ))

g(x) = h(f(x))

f

h

g

h(y)=0.5y+0.5

x

h

x

image warping: change domain of image

f

g(x) = f(h(x))

h(y)=2y

f

h

g

x x

Image warping

image filtering: change range of image ( ) h(f( ))

f g

g(x) = h(f(x))

h(y)=0.5y+0.5 h

f g

image warping: change domain of image

f

g(x) = f(h(x))

h([x,y])=[x,y/2]

h

f

g

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Parametric (global) warping

Examples of parametric warps:

translation rotation aspect

affine perspective

cylindrical

Parametric (global) warping

T T

p = (x,y) p’ = (x’,y’)

• Transformation T is a coordinate-changing machine: p’ = T(p)

p (x,y) p (x ,y )

machine: p T(p)

• What does it mean that T is global?

Is the same for any point p – Is the same for any point p

– can be described by just a few numbers (parameters)

R t T t i ’ M*

• Represent T as a matrix: p’ = M*p

 

 



 

 

 



 x x

' M

' 

 

 

 

 y' y

Scaling

• Scaling a coordinate means multiplying each of its components by a scalar

its components by a scalar

• Uniform scaling means this scalar is the same for all components:

for all components:

f g







 x ' 2 x



x 



 



 

 

 





y y ' 2

 

 



 y

 2

Scaling

• Non-uniform scaling: different scalars per

component:

      '  

component:

 

 



 



 



 

 

 



 



 





' ' y g x y

f x







 x ' 2 x



  



   y y

 

 



 

 

 





y y ' 0 . 5

x  2, y  0.5

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Scaling

• Scaling operation:

x '  ax by y ' 

• Or, in matrix form:











 



 



 



 



 

 

 





y x b a y

x

0 0 '

'











 y 0 b y

scaling matrix S scaling matrix S What’s inverse of S?

2-D Rotation

• This is easy to capture in matrix form:

   

__ _







   

^_^^_^ ^_^



 

 



 





y x y

x



cos sin

sin cos

'

'_ _

   

__ _

y sin  cos y

R

• Even though sin() and cos() are nonlinear to ,

’i li bi ti f d

R

– x’ is a linear combination of x and y – y’ is a linear combination of x and y

Wh i h i f i ?

• What is the inverse transformation?

– Rotation by –

T

– For rotation matrices, det(R) = 1 so R^11 R^T

2x2 Matrices

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

represented with a 2x2 matrix?

2D Identity?

' x' 1 0x y

y x x 

'

' 







 

 



y x y

x

1 0

0 1 '

'

2D Scale around (0,0)?

x s

x' *  ' s 0  y

s y

x s x

y

x

* '





 







 



 



 





y x s s y

x

y x

0 0 '

'

y    ^y 

2x2 Matrices

2D Rotate around (0,0)?

* i

*

'   _ _ _ _ i ___ _

y x

y

y x

x

* cos

* sin '

* sin

* cos '









 



 







 



 





 





y x y

x



 cos sin

sin cos

' '

2D Shear?

y x sh y

y sh x x

y x









* '

*

' 



 







 







 





y x sh

sh y

x x

1 1

' ' y

x sh

y _y  y shy 1 y

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2x2 Matrices

2D Mirror about Y axis?

' x'  1 0x

y y

x x 

'

' 







 

 



y x y

x

1 0

0 1 '

'

2D Mirror over (0,0)?

y y

x x 

'

' 







 

 

 



y x y

x

1 0

0 1 '

'

y

y y  y

All 2D Linear Transformations

• Linear transformations are combinations of …

Scale – Scale, – Rotation, – Shear andShear, and – Mirror

• Properties of linear transformations:

– Origin maps to origin – Lines map to linesLines map to lines

– Parallel lines remain parallel

– Ratios are preserved^p

 x'   a b   x 

– Closed under composition

 

 



 



 



 

 

 





y x d c

b a y

x ' '











 y c d y

2x2 Matrices

• What types of transformations can not be represented with a 2x2 matrix?

2D Translation?

t '

y x

t y y

t x x







 '

' NO!

Only linear 2D transformations

b t d ith 2 2 t i

can be represented with a 2x2 matrix

Translation

• Example of translation

Homogeneous Coordinates









 









































































1 1 1 0 0

1 0

0 1

1 ' '

y x y

x

t y

t x y x t t y

x













1 0 0 1 1 1

t_x= 2 t_y_y= 1

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Affine Transformations

• Affine transformations are combinations of …

Li f i d

– 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

 

 

 

 

 

  x

f d

c b a x

' '

– Models change of basis

 

 



 



 





 



 



 w

y f e d w

y

1 0 0 '













Projective Transformations

• Projective transformations …

Affine transformations and – Affine transformations, and – Projective warps

P ti f j ti t f ti

• Properties of projective transformations:

– Origin does not necessarily map to origin l

– Lines map to lines

– Parallel lines do not necessarily remain parallel – Ratios are not preserved

 

 

 

 

 

  x' a b c x

– Models change of basis

 

 



 



 





 



 



 w

y i h g

f e d w

y ' '











 w g h i w

Image warping

• Given a coordinate transform x’ = T(x) and a source image I(x) how do we compute a source image I(x), how do we compute a transformed image I’(x’) = I(T(x))?

T(x)

I(x) I’(x’)

x x’

(x)

I(x) I (x )

Forward warping

• Send each pixel I(x) to its corresponding location x’ T(x) in I’(x’)

location x’ = T(x) in I’(x’)

T(x)

I(x) I’(x’)

x x’

(x)

I(x) I (x )

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Forward warping

fwarp(I, I’, T) {

{

for (y=0; y<I.height; y++) for (x=0; x<I width; x++) { for (x=0; x<I.width; x++) {

(x’,y’)=T(x,y);

I’(x’ y’) I(x y);

I’(x’,y’)=I(x,y);

} }

} I I’

T

x

x’

Forward warping

Some destination may not be covered not be covered

Many source pixels could map Many source pixels could map

to the same destination

Forward warping

• Send each pixel I(x) to its corresponding location x’ T(x) in I’(x’)

location x’ = T(x) in I’(x’)

• What if pixel lands “between” two pixels?p p

• Will be there holes?

• Answer: add “contribution” to several pixels

• Answer: add contribution to several pixels, normalize later (splatting)

h(x)

f(x) g(x’)

x x’

(x)

f(x) g(x )

Forward warping

fwarp(I, I’, T) {

{

for (y=0; y<I.height; y++) for (x=0; x<I width; x++) { for (x=0; x<I.width; x++) {

(x’,y’)=T(x,y);

Splatting(I’ x’ y’ I(x y) kernel);

Splatting(I’,x’,y’,I(x,y),kernel);

} }

} I I’

T x

x’

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Inverse warping

• Get each pixel I’(x’) from its corresponding location x T^-1(x’) in I(x)

location x = T¹(x’) in I(x)

T^-1(x’)

I(x) I’(x’)

x x’

(x )

I(x) I (x )

Inverse warping

iwarp(I, I’, T) {

{

for (y=0; y<I’.height; y++) for (x=0; x<I’ width; x++) { for (x=0; x<I’.width; x++) {

(x,y)=T^-1(x’,y’);

I’(x’ y’) I(x y);

I’(x’,y’)=I(x,y);

} }

} I T^-1 I’

x

x’

Inverse warping

• Get each pixel I’(x’) from its corresponding location x T^-1(x’) in I(x)

location x = T¹(x’) in I(x)

• What if pixel comes from “between” two pixels?

• Answer: resample color value from

• Answer: resample color value from interpolated (prefiltered) source image

f(x) g(x’)

x x’

f(x) g(x )

Inverse warping

iwarp(I, I’, T) {

{

for (y=0; y<I’.height; y++) for (x=0; x<I’ width; x++) { for (x=0; x<I’.width; x++) {

(x,y)=T^-1(x’,y’);

I’(x’ y’) Reconstruct(I x y kernel);

I’(x’,y’)=Reconstruct(I,x,y,kernel);

} }

} I T^-1 I’

x

x’

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Inverse warping

• No hole, but must resample

Wh l h ld k f i

• What value should you take for non-integer coordinate? Closest one?

Inverse warping

• It could cause aliasing

Reconstruction

• Reconstruction generates an approximation to the original function Error is called aliasing the original function. Error is called aliasing.

l l

sampling reconstruction

sample value

l i i sample position

Reconstruction

• Computed weighted sum of pixel neighborhood;

output is weighted average of input where output is weighted average of input, where weights are normalized values of filter kernel k



^k⁽^q ⁾^q

color=0;

 



i i

i i i

q k

q q p k

) (

color=0;

weights=0;

for all q’s dist < width width

d

q

d = dist(p, q);

w = kernel(d);

p

d color += w*q.color;

weights += w;

p Color = color/weights;

q

p.Color = color/weights;

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Triangle filter Gaussian filter

Sampling

band limited

Reconstruction

The reconstructed function is obtained by interpolating The reconstructed function is obtained by interpolating among the samples in some manner

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Reconstruction (interpolation)

• Possible reconstruction filters (kernels):

t i hb – nearest neighbor – bilinear

bi bi – bicubic

– sinc (optimal reconstruction)

Bilinear interpolation (triangle filter)

• A simple method for resampling images

Non-parametric image warping

• Specify a more detailed warp function

S li h i l fl ( i l i )

• Splines, meshes, optical flow (per-pixel motion)

Non-parametric image warping

• Mappings implied by correspondences

I i

• Inverse warping

? P’

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Non-parametric image warping

' '

'

' w A w B w C P  _A  _B  _C C

w B w A w

P  

Barycentric coordinate

C w B w A w

P _A  _B  _C

P P’

Barycentric coordinates

3 1

2 1

3 3 2 2 1 1





 t t t

A t A t A t P

3 1

2 1t t t

Non-parametric image warping

' '

'

' w A w B w C P  _A  _B  _C C

w B w A w

P  

Barycentric coordinate

C w B w A w

P _A  _B  _C

Non-parametric image warping

) 2

(r e ^^r

  ^

Gaussian ^^P^ _K1



^k^Xi(^P')^^Xⁱ

radial basis function

) log(

)

(r r² r

thin plate 

spline

K i

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Image warping

• Warping is a useful operation for mosaics, video matching view interpolation and so on

matching, view interpolation and so on.

An application of image warping:

face beautification

Data-driven facial beautification Facial beautification

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Facial beautification Facial beautification

Training set

• Face images

92 C i f l

– 92 young Caucasian female – 33 young Caucasian male

Feature extraction

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Feature extraction

• Extract 84 feature points by BTSM D l i l i 234D di

• Delaunay triangulation -> 234D distance vector (normalized by the square root of face area)

tt l t f

BTSM

scatter plot for

all training faces

234D vector

Beautification engine

Support vector regression (SVR)

• Similar concept to SVM, but for regression RBF k l

• RBF kernels

• f_b(v)

Beautification process

• Given the normalized distance vector v, generate a nearby vector v’ so that generate a nearby vector v so that

f_b(v’) > f_b(v)

• Two optionsp

– KNN-based – SVR-basedSVR based

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KNN-based beautification

5.1 5.1 4.3

4.3

3 1 3 1

44 55

3.1 3.1 v

v' 44..55

4.6 55..33 4.6

SVR-based beautification

• Directly use f_b to seek v’

• Use standard no-derivative direction set method for minimization

• Features were reduced to 35D by PCA

SVR-based beautification

• Problems: it sometimes yields distance vectors corresponding to invalid human face

corresponding to invalid human face

• Solution: add log-likelihood term (LP)

• LP is approximated by modeling face space as a pp y g p multivariate Gaussian distribution ^{’s i-th}

component

u’s projection

in PCA space i-th eigenvalue

in PCA space

PCA

λ₂

λ₁

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Embedding and warping Distance embedding

• Convert modified distance vector v’ to a new face landmark

face landmark

1 if i d j b l t diff t f i l f t 1 if i and j belong to different facial features 10 otherwise

• A graph drawing problem referred to as a stress i i i ti bl l d b LM l ith minimization problem, solved by LM algorithm for non-linear minimization

Distance embedding

• Post processing to enforce similarity transform for features on eyes by minimizing

for features on eyes by minimizing

SVR K=5

K=3 Original

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Results (in training set) User study

Results (not in training set) By parts

eyes

mouth full

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Different degrees

50% 100%

Facial collage

Results

• video

Image morphing

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Image morphing

• The goal is to synthesize a fluid transformation from one image to another

from one image to another.

• Cross dissolving is a common transition between cuts, but it is not good for morphing because of the ghosting effects.

i g #1 di l i g image #2

image #1 dissolving image #2

Artifacts of cross-dissolving

http://www.salavon.com/

Image morphing

• Why ghosting?

M hi i di l i

• Morphing = warping + cross-dissolving

shape (geometric)

color (photometric) (geometric) (photometric)

Image morphing

cross-dissolving

image #1 image #2

morphing

warp warp

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Morphing sequence Face averaging by morphing

average faces

Image morphing

create a morphing sequence: for each time t

1 C t i t di t i fi ld (b 1. Create an intermediate warping field (by

interpolation)

2 Warp both images towards it 2. Warp both images towards it

3. Cross-dissolve the colors in the newly warped images

images

t=0 t=0.33 t=1

An ideal example (in 2004)

t=0 morphingt=0.25t=0.75t=0.5 t=1

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An ideal example

middle face (t=0.5)

t=0 t=1

Warp specification (mesh warping)

• How can we specify the warp?

1 S if di li t l i t

1. Specify corresponding spline control points

interpolate to a complete warping function

easy to implement but less expressive easy to implement, but less expressive

Warp specification

• How can we specify the warp

2 S if di i t 2. Specify corresponding points

• interpolate to a complete warping function

Solution: convert to mesh warping

1 D fi t i l h th i t 1. Define a triangular mesh over the points

– Same mesh in both images!

N h i l i l d

– Now we have triangle-to-triangle correspondences 2. Warp each triangle separately from source to destination

– How do we warp a triangle?

– 3 points = affine warp!

– Just like texture mapping

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Warp specification (field warping)

• How can we specify the warp?

3 S if di t

3. Specify corresponding vectors

• interpolate to a complete warping function

• The Beier & Neely Algorithm

Beier&Neely (SIGGRAPH 1992)

• Single line-pair PQ to P’Q’:

Algorithm (single line-pair)

• For each X in the destination image:

1 Fi d th di

1. Find the corresponding u,v

2. Find X’ in the source image for that u,v

3 d ti ti I (X) I (X’)

3. destinationImage(X) = sourceImage(X’)

• Examples:

Affine transformation Affine transformation

Multiple Lines

X X D_i X_i^' X_i D  

length = length of the line segment, dist = distance to line segment

The influence of a, p, b. The same as the average of X_i’

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Full Algorithm Resulting warp

Comparison to mesh morphing

• Pros: more expressive

C d d l

• Cons: speed and control

Warp interpolation

• How do we create an intermediate warp at time t?

time t?

– linear interpolation for line end-points

B t li t ti 180 d ill b 0 – But, a line rotating 180 degrees will become 0

length in the middle

One solution is to interpolate line mid point and – One solution is to interpolate line mid-point and

orientation angle

t=0

t=1 t=1

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Animation Animated sequences

• Specify keyframes and interpolate the lines for the inbetween frames

the inbetween frames

• Require a lot of tweaking

Results

Michael Jackson’s MTV “Black or White”

Multi-source morphing

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Multi-source morphing References

• Thaddeus Beier, Shawn Neely, Feature-Based Image Metamorphosis, SIGGRAPH 1992, pp35-42.

• Detlef Ruprecht, Heinrich Muller, Image Warping with Scattered Data Interpolation, IEEE Computer Graphics and Applications, March 1995 pp37-43

March 1995, pp37 43.

• Seung-Yong Lee, Kyung-Yong Chwa, Sung Yong Shin, Image Metamorphosis Using Snakes and Free-Form Deformations, SIGGRAPH 1995

SIGGRAPH 1995.

• Seungyong Lee, Wolberg, G., Sung Yong Shin, Polymorph: morphing among multiple images, IEEE Computer Graphics and Applications, g p g , p p pp , Vol. 18, No. 1, 1998, pp58-71.

• Peinsheng Gao, Thomas Sederberg, A work minimization approach to image morphing The Visual Computer 1998 pp390 400 to image morphing, The Visual Computer, 1998, pp390-400.

• George Wolberg, Image morphing: a survey, The Visual Computer, 1998, pp360-372.

• Data-Driven Enhancement of Facial Attractiveness, SIGGRAPH 2008

Project #1: image morphing Project #1: image morphing

Reference software

• Morphing software review

I d 30 d lF t M h i i

• I used 30-day evaluation version.

You can use any one you like.

FantaMorph

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