Optical flow

Motion estimation

Digital Visual Effectsg Yung-Yu Chuang

with slides by Michael Black and P. Anandan

Motion estimation

• Parametric motion (image alignment) T ki

• Tracking

• Optical flow

Parametric motion

direct method for image stitching

Tracking

Optical flow

Three assumptions

• Brightness consistency S i l h

• Spatial coherence

• Temporal persistence

Brightness consistency

Image measurement (e g brightness) in a small region Image measurement (e.g. brightness) in a small region remain the same although their location may change.

Spatial coherence

• Neighboring points in the scene typically belong to the

• Neighboring points in the scene typically belong to the same surface and hence typically have similar motions.

• Since they also project to nearby pixels in the image, Since they also project to nearby pixels in the image, we expect spatial coherence in image flow.

Temporal persistence

The image motion of a surface patch changes gradually over time.

Image registration

Goal: register a template image T(x) and an input image I(x) where x (x y)^T (warp I so that it

image I(x), where x=(x,y)^T. (warp I so that it matches T)

Image alignment: I(x) and T(x) are two images

Tracking: T(x) is a small patch around a point p in Tracking: T(x) is a small patch around a point p in

the image at t. I(x) is the image at time t+1.

O ti l fl T( ) d I( ) t h f i

Optical flow: T(x) and I(x) are patches of images at t and t+1.

warp

T I

warp fixed

Simple approach (for translation)

• Minimize brightness difference

 

²

 



^{  }



y x

y x T v

y u x

I v

u E

,

) 2

, ( )

, (

) , (

Simple SSD algorithm

For each offset (u, v) ( )

compute E(u,v);

Choose (u, v) which minimizes E(u,v);

Problems:

Problems:

• Not efficient N b i l

• No sub-pixel accuracy

Lucas-Kanade algorithm

Lucas Kanade algorithm

Newton’s method

• Root finding for f(x)=0 March x and test signs

• March x and test signs

• Determine Δx (small→slow; large→ miss)

Newton’s method

• Root finding for f(x)=0

Newton’s method

• Root finding for f(x)=0 T l ’ i

Taylor’s expansion:







 1 '' ( )

²

) (

' )

( )

( f f f

f

₀

 

₀



₀

 '' (

₀

)

²

 

) 2 (

' )

( )

( x  f x f x  f x 

f



 ) ( ) ' ( )

( x f x f x

f ( x

₀

  )  f ( x

₀

)  f ( x

₀

) 

f   

) (x f

) (

'

) (

n n

n

f x

x

 f

 

) (

'

) (

1

n n

n

f x

x x f

x

_

 

)

( x

_n

f

Newton’s method

• Root finding for f(x)=0

) (

f

) (

'

) (

n n

n

f x

x

 f

 

) (

_n

f

x₀ x₁

x₂

Newton’s method

pick up x=x₀ iiterate

) (x

compute

f

) ( '

) (

x f

x x   f



update x by x+Δx til

until converge

Finding root is useful for optimization because Minimize g(x) → find root for f(x)=g’(x)=0

Minimize g(x) find root for f(x) g (x) 0

Lucas-Kanade algorithm

 



^{  }

 I x u y v T x y v

u

E( , )

 

( , ) ( , )



²

y x

y y

,

) (

) (

) (

y

x vI

uI y

x I v

y u x

I((  ,, y  ))  (( ,, y))  _x  _y

 



^{  }



_ 

I(x, y) T (x, y) uI_x vI_y



²

y x

y

y x

y

,

) , ( )

, (

 



^{  }

 

 

y x

y x

x I x y T x y uI vI

u I E

,

) , ( )

, ( 2

0

 



^{  }

 

 

y x

y x

y I x y T x y uI vI

v I

E 2 ( , ) ( , )

0 v _x,y

Lucas-Kanade algorithm

 



^{  }

 

 E I_x I x y T x y uI_x vI_y )

, ( )

, ( 2

0

_  

 _{x y} ^x y y ^{x y}

u _, ( , ) ( , )

 



^{ }

E I I x y T x y uI vI

) (

) (

2

0 ^

_ 

^{  }



 

y x

y x

y I x y T x y uI vI

v _, 2I ( , ) ( , ) 0



 

 





















 



y

x x y

x y

x y

x

x

y x I y

x T I v

I u

I I

y x I y

x T I v

I I u

I

2

, ,

,

2

) (

) (

) , ( )

, (

 















y x

y y

x

y y

x

y

xI u I v I T x y I x y

I

, ,

,

) , ( )

, (

 

^

 





²

  ^{ }

 

^











 



 























y x

x y

x

y x y

x

x

y x I y

x T I

y x I y

x T I v

u I

I I

I I I

, 2

, ,

2

) (

) (

) , ( )

, (

 



 



 







 





  

y x

y y

x

y y

x

y

xI I v I T x y I x y

I

, , ,

) , ( )

, (

Lucas-Kanade algorithm

iterate

shift I(x y) with (u v) shift I(x,y) with (u,v)

compute gradient image I_x, I_y

compute error image T(x y) I(x y) compute error image T(x,y)-I(x,y) compute Hessian matrix

solve the linear system solve the linear system (u,v)=(u,v)+(∆u,∆v)

til

until converge

 



 





 





^Ix2



^IxIy



^Ix

^

^{T (x, y) I(x, y)}

^

 





 



 

 



 









 

















y y x

x

y y

x

y x

y x y

x

x

y x I y

x T I

y y

v u I

I I

, 2

, ,

) , ( )

, (

) , ( )

, (



 

 ^{x,y x,y x,y}

Parametric model

 



^{  }

 I x u y v T x y v

u

E( , )

 

( , ) ( , )



²

y x

y y

,

) (

) (

) (

 



^

 W(x;p) x

p) ( ) ( ) ²

( I T

E Our goal is to find

p to minimize E(p)

 



x

x p)

W(x;

p) ( ) ( )

( I T

E

d

x 

 

p to minimize E(p) for all x in T’s domain

T y x

y

x p d d

d y

d

x ,  ( , )



 







  p)

translation W(x;

x xy

xx y

d x d

d

1 







 





 Ax d p)

affine W(x;

T

y yy

yx

d y d

d ,

1 1

 







 

 d Ax

p)

affine W(x;

T y x

yy yx

xy

xx d d d d d

d

p  ( , , , , , )

Parametric model

 



^{ }

x

x Δp)

p

W(x; ) ( ) ²

( T

minimize I

x

with respect to ^Δp

W p Δp p) W

W(x;

Δp) p

W(x; 

 





) (

)

( Δp

p p) W

W(x;

Δp) p

W(x; 

 



 I

I

p Δp W p) x

W(x; 





 

 I

I( )

p x 









 



 



 

 W Δp x

p) W(x;

2

) ( )

( I T

minimize



I 

 

 

x

p p p)

( ; ) ( )

(

Parametric model

image gradient

warped image target image



 W ²

image gradient







 



 



 



x

x p Δp

p) W

W(x; ) ( )

( I T

I

Jacobian of the warp



































x x

x x

p W p

W p

W p

W

W 





 



 









  







 



 



 



n y y

y

n y

p W p

W p

W

p p

p p

W p p

W



2 1

2 1



 

 p p₁ p₂ pⁿ

Jacobian matrix

• The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function

partial derivatives of a vector-valued function.

m

F : R

n

 R )

, ,

( x

₁

x

₂

x

_n

F 

)) ,

, ( ),

, ,

( ),

, ,

(

( f

₁

x

₁

x

₂

 x

_n

f

₂

x

₁

x

₂

 x

_n

 f

_m

x

₁

x

₂

 x

_n





  f  f or

) ,

,

(

_{1 2 n}

F

x x x

J 

   

 











x

n

f x

f 

¹

1 1

) ,

, (

or

2

1

f f

m

f 





 



 







_{m m}

n

f f 

¹

  )

, ,

( x

₁

x

₂

 x

_n

  

 



  

_n

m m

x

x 

1

Δx x

x Δx

x ) ( ) ( )

( F J

_F

F   

Jacobian matrix

   ^{0 , 0 , 2} 

³

: R      R

F t  r sin  cos 







cos

sin sin

r v

r u

 ) 

, , ( )

, ,

( r t u v

F   



 t t t

v r cos 













 















u u

u

t t

r t













 









  ) , ,

(    

v v

v

u u

r r u

J_F





 



 















v v

r v







 

 sin sin cos sin sin cos sin sin

cos cos

cos sin

























r r

r r



 



 cos  sin 0

cos s

s cos s

s

















r

Parametric model

image gradient

warped image target image



 W ²

image gradient







 



 



 



x

x p Δp

p) W

W(x; ) ( )

( I T

I

Jacobian of the warp



































x x

x x

p W p

W p

W p

W

W 





 



 









  







 



 



 



n y y

y

n y

p W p

W p

W

p p

p p

W p p

W



2 1

2 1



 

 p p₁ p₂ pⁿ

Jacobian of the warp



   



 W_x W_x W_x W_x













  













 



y y

y

n y

x

W W

W

p p

p W

p p

W





2 1

For example for affine



   



 p p₁ p₂ pⁿ

For example, for affine



   











  _{xx xy x} d_xx x d_xy y d_x y

d x d

d (1 )

p) 1

W(x; 

 

   

 



 







 



y yy

yx y

yy

yx y d x d y d

d d

d (1 )

1 1 p)

W(x;



 



 





1 0

0 0

0 1

0 0

y x

y x

p W



 y

p

d_xx d_yx d_xy d_yy d_x d_y

Parametric model







 



 W Δp x

p) W(x;

2

) ( )

( I T

min

I

arg 



 

  

x

x p Δp

p)

W(x; ) ( )

( I T

I



^{ W  (W( ))  W Δ ( )}

0 I I I T

T

min

Δp

arg







 



 



 

 



 







 



x

x p Δp

p) W p W(x;

W ( ) ( )

0 I I I T

T

 



 ^



 







 

 ^

x

p) W(x;

p x H W

Δp ¹ I T( ) I( )

T







^{ WT  W }







 







 



 







 



x p

W p

H I W I

(Approximated) Hessian

Lucas-Kanade algorithm

iterate

1) warp I with W(x;p) 1) warp I with W(x;p)

2) compute error image T(x,y)-I(W(x,p)) 3) compute gradient image with W(x p)^I 3) compute gradient image with W(x,p) 4) evaluate Jacobian at (x;p)

5) compute ^p

W





W

I

I

5) compute

6) compute Hessian

7) t

p

I



 W ^T

7) compute 8) solve

9) d t b

 

  ^



 







 

x

p) W(x;

p x

W T( ) I( ) I

Δp

9) update p by p+Δ

until converge

Δp

 





 

 ^ W x W(x;p)

H

Δp ¹ I T( ) I( )

T

 



 ^

 

 



x

p) W(x;

p x H

Δp I T( ) I( )

 

 ^{ }



 







 

 ^

x

p) W(x;

p x H W

Δp ¹ I T( ) I( )

T

Coarse-to-fine strategy

J J^{w fi} I

ain

J I

J ^warp J^{w refine} I

a

+ 

J ^warp J^{w refine} I

a

pyramid construction

pyramid construction

a

+

J ^warp J^{w refine} I

 a

+

a

out

Application of image alignment

Direct vs feature-based

• Direct methods use all information and can be very accurate but they depend on the fragile very accurate, but they depend on the fragile

“brightness constancy” assumption.

• Iterative approaches require initialization

• Iterative approaches require initialization.

• Not robust to illumination change and noise images

images.

• In early days, direct method is better.

• Feature based methods are now more robust and potentially faster

and potentially faster.

• Even better, it can recognize panorama without initialization

initialization.

Tracking

Tracking

Tracking

(u, v)

I(x,y,t) (u, v) I(x+u,y+v,t+1)

Tracking

0 )

, , ( )

1 ,

,

(x  u y  v t   I x y t  brightness constancy I

0 )

, , ( )

, , ( )

, , ( )

, , ( )

, ,

(x y t  uI x y t  vI x y t  I x y t  I x y t 

I _{x y t}

0 )

, , ( )

, , ( )

, ,

(x y t  vI x y t  I x y t 

uI_{x yy t}

 0



 I v I u

I_xu  I_yv  I_t  0 optical flow constraint equation I optical flow constraint equation

Optical flow constraint equation

Multiple constraints

Area-based method

• Assume spatial smoothness

Area-based method

• Assume spatial smoothness

 



^{ }



y x

t y

xu I v I

I v

u

E( , ) ²

y x,

Area-based method

must be invertible must be invertible

Area-based method

• The eigenvalues tell us about the local image structure

structure.

• They also tell us how well we can estimate the fl i b th di ti

flow in both directions.

• Link to Harris corner detector.

Textured area

Edge

Homogenous area

KLT tracking

• Select features by

M i f b i di i il i





 , )  ( _{1 2} min

• Monitor features by measuring dissimilarity

Aperture problem

Aperture problem

Aperture problem

Demo for aperture problem

• http://www.sandlotscience.com/Distortions/Br eathing Square htm

eathing_Square.htm

• http://www.sandlotscience.com/Ambiguous/Ba b l Ill i ht

rberpole_Illusion.htm

Aperture problem

• Larger window reduces ambiguity, but easily violates spatial smoothness assumption

violates spatial smoothness assumption

KLT tracking

http://www ces clemson edu/~stb/klt/

http://www.ces.clemson.edu/ stb/klt/

KLT tracking

http://www ces clemson edu/~stb/klt/

http://www.ces.clemson.edu/ stb/klt/

SIFT tracking (matching actually)

Frame 0  Frame 10

SIFT tracking

Frame 0  Frame 100

SIFT tracking

Frame 0  Frame 200

KLT vs SIFT tracking

• KLT has larger accumulating error; partly

because our KLT implementation doesn’t have because our KLT implementation doesn t have affine transformation?

SIFT i i i l b t

• SIFT is surprisingly robust

• Combination of SIFT and KLT (example)

http://www.frc.ri.cmu.edu/projects/buzzard/smalls/

Rotoscoping (Max Fleischer 1914)

1937

Tracking for rotoscoping

Tracking for rotoscoping

Waking life (2001)

A Scanner Darkly (2006)

• Rotoshop, a proprietary software. Each minute of animation required 500 hours of work

of animation required 500 hours of work.

Optical flow

Optical flow

Single-motion assumption

Violated by

M i di i i

• Motion discontinuity

• Shadows

• Transparency

• Specular reflection

• Specular reflection

• …

Multiple motion

Multiple motion

Simple problem: fit a line

Least-square fit

Least-square fit

Robust statistics

• Recover the best fit for the majority of the data

data

• Detect and reject outliers

Approach

Robust weighting

T t d d ti

Truncated quadratic

Robust weighting

G & M Cl

Geman & McClure

Robust estimation

Fragmented occlusion

Regularization and dense optical flow

• Neighboring points in the scene typically belong to the

• Neighboring points in the scene typically belong to the same surface and hence typically have similar motions.

• Since they also project to nearby pixels in the image, Since they also project to nearby pixels in the image, we expect spatial coherence in image flow.

Input image Horizontal ti

Vertical motion

motion motion

Application of optical flow

video video

matching

Input for the NPR algorithm

Brushes

Edge clipping

Gradient

Smooth gradient

Textured brush

Edge clipping

Temporal artifacts

Frame-by-frame application of the NPR algorithm

Temporal coherence