# Motion estimation

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## Motion estimation

Digital Visual Effects Yung-Yu Chuang

with slides by Michael Black and P. Anandan

(2)

### Motion estimation

• Parametric motion (image alignment)

• Tracking

• Optical flow

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### Parametric motion

direct method for image stitching

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### Three assumptions

• Brightness consistency

• Spatial coherence

• Temporal persistence

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### Brightness consistency

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

(8)

### Spatial coherence

• 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, we expect spatial coherence in image flow.

(9)

### Temporal persistence

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

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### 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

matches T)

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

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.

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

T fixed

I warp

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### Simple approach (for translation)

• Minimize brightness difference

y x

y x T v

y u x

I v

u E

,

) 2

, ( )

, (

) , (

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### Simple SSD algorithm

For each offset (u, v) compute E(u,v);

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

Problems:

• Not efficient

• No sub-pixel accuracy

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### Newton’s method

• Root finding for f(x)=0

• March x and test signs

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

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### Newton’s method

• Root finding for f(x)=0

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### Newton’s method

• Root finding for f(x)=0 Taylor’s expansion:

0 0 0 2

0

0

0

0

n n

n

1

n n n

n

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### Newton’s method

• Root finding for f(x)=0

x0 x1

x2

n n n

(18)

### Newton’s method

pick up x=x0 iterate

compute

update x by x+Δx until converge

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

(19)

y x

y x T v

y u x

I v

u E

,

) 2

, ( )

, (

) , (

y

x vI

uI y

x I v

y u x

I( , ) ( , )

y x

y

x vI

uI y

x T y

x I

,

) 2

, ( )

, (

### 

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

(20)

### 

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

y x

y y

x

y y

x

y x

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

x

y x

y y

x

y 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

) , ( )

, (

) , ( )

, (

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### Lucas-Kanade algorithm

iterate

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

compute gradient image Ix, Iy

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

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

until converge

y x

y y x

x

y x

y y

x

y 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

) , ( )

, (

) , ( )

, (

(22)

y x

y x T v

y u x

I v

u E

,

) 2

, ( )

, (

) , (

### 

x

x p)

W(x;

p) ( ) ( ) 2

( I T

E

T y x

y

x p d d

d y

d

x , ( , )



p)

translation W(x;

T y x

yy yx

xy xx

y yy

yx

x xy

xx

d d

d d

d d

p

y x d

d d

d d

d

) ,

, ,

, ,

(

, 1 1

1





Ax d p)

affine W(x;

Our goal is to find p to minimize E(p) for all x in T’s domain

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### 

x

x Δp)

p

W(x; ) ( ) 2

( T

minimize I

with respect to Δp

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( )





x

x p Δp

p) W W(x;

2

) ( )

( I T

minimize I

(24)

### 





x

x p Δp

p) W W(x;

2

) ( )

( I T

I

image gradient

Jacobian of the warp warped image

n y y

y

n x x

x

y x

p W p

W p

W

p W p

W p

W

p W p W p

W

2 1

2 1

target image

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### Jacobian matrix

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

m

n

2 1

2 1

2 2

1 1

2 1

n m

n n

n

2 1

2 1

2 1

n m

n F

n m m

n

1

1 1

1

F

(26)

3

















 

 

0 sin

cos

cos sin

sin cos

sin sin

sin sin

cos cos

cos sin

) , , (

r

r r

r r

v v

r v

u u

r u

t t

r t r

JF

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### 





x

x p Δp

p) W W(x;

2

) ( )

( I T

I

image gradient

Jacobian of the warp warped image

n y y

y

n x x

x

y x

p W p

W p

W

p W p

W p

W

p W p W p

W

2 1

2 1

target image

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### Jacobian of the warp

For example, for affine









y yy

yx

x xy

xx y

yy yx

x xy

xx

d y

d x

d

d y

d x

y d x d

d d

d d

d

) 1

( ) 1

( 1 1

p) 1 W(x;

n y y

y

n x x

x

y x

p W p

W p

W

p W p

W p

W

p W p W p

W

2 1

2 1





1 0

0 0

0 1

0 0

y x

y x

p W

dxx dyx dxy dyy dx dy

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



x

x p Δp

p) W W(x;

2

) ( )

( I T

I

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

### 

x p

W p

H I W I

T

(Approximated) Hessian

Δp

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### Lucas-Kanade algorithm

iterate

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) 4) evaluate Jacobian at (x;p)

5) compute

6) compute Hessian 7) compute

8) solve

9) update p by p+

until converge

p W

p

W

I

x

p) W(x;

p x

W T( ) I( ) I

T

Δp

Δp

x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

I

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x W(x;p)

p H W

Δp 1 I T( ) I( )

T

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x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

(33)

x W(x;p)

p H W

Δp 1 I T( ) I( )

T

(34)

x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

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x W(x;p)

p H W

Δp 1 I T( ) I( )

T

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x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

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x W(x;p)

p H W

Δp 1 I T( ) I( )

T

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x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

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x W(x;p)

p H W

Δp 1 I T( ) I( )

T

(40)

x

p) W(x;

p x H W

Δp 1 I T( ) I( )

T

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x W(x;p)

p H W

Δp 1 I T( ) I( )

T

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### Coarse-to-fine strategy

J warp Jw refine I

ain

a

+

J warp Jw refine I

a

a

+ J

pyramid construction

J warp Jw refine I

a

+

I

pyramid construction

out

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### Direct vs feature-based

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

“brightness constancy” assumption.

• Iterative approaches require initialization.

• Not robust to illumination change and noise images.

• In early days, direct method is better.

• Feature based methods are now more robust and potentially faster.

• Even better, it can recognize panorama without initialization.

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### Tracking

0 )

, , ( )

1 ,

,

(x u y v t I x y t 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

uIx y t

0

y t

xu I v I

I optical flow constraint equation brightness constancy

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### Area-based method

• Assume spatial smoothness

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### Area-based method

• Assume spatial smoothness

y x

t y

xu I v I

I v

u E

,

) 2

, (

(52)

### Area-based method

must be invertible

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### Area-based method

• The eigenvalues tell us about the local image structure.

• They also tell us how well we can estimate the flow in both directions.

• Link to Harris corner detector.

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### KLT tracking

• Select features by

• Monitor features by measuring dissimilarity

, ) ( 1 2 min

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### Demo for aperture problem

• http://www.sandlotscience.com/Distortions/Br eathing_Square.htm

• http://www.sandlotscience.com/Ambiguous/Ba rberpole_Illusion.htm

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### Aperture problem

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

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### KLT tracking

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

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### KLT tracking

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

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### SIFT tracking (matching actually)

Frame 0  Frame 10

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### SIFT tracking

Frame 0  Frame 100

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### SIFT tracking

Frame 0  Frame 200

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### KLT vs SIFT tracking

• KLT has larger accumulating error; partly

because our KLT implementation doesn’t have affine transformation?

• SIFT is surprisingly robust

• Combination of SIFT and KLT (example)

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

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### A Scanner Darkly (2006)

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

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### Single-motion assumption

Violated by

• Motion discontinuity

• Shadows

• Transparency

• Specular reflection

• …

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### Robust statistics

• Recover the best fit for the majority of the data

• Detect and reject outliers

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### Regularization and dense optical flow

• 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, we expect spatial coherence in image flow.

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Input image Horizontal motion

Vertical motion

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video

matching

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### Temporal artifacts

Frame-by-frame application of the NPR algorithm

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### References

• B.D. Lucas and T. Kanade, An Iterative Image Registration Technique with an Application to Stereo Vision, Proceedings of the 1981 DARPA Image

Understanding Workshop, 1981, pp121-130.

• Bergen, J. R. and Anandan, P. and Hanna, K. J. and Hingorani, R., Hierarchical Model-Based Motion Estimation, ECCV 1992, pp237-252.

• J. Shi and C. Tomasi, Good Features to Track, CVPR 1994, pp593-600.

• Michael Black and P. Anandan, The Robust Estimation of Multiple Motions:

Parametric and Piecewise-Smooth Flow Fields, Computer Vision and Image Understanding 1996, pp75-104.

• S. Baker and I. Matthews, Lucas-Kanade 20 Years On: A Unifying

Framework, International Journal of Computer Vision, 56(3), 2004, pp221 - 255.

• Peter Litwinowicz, Processing Images and Video for An Impressionist Effects, SIGGRAPH 1997.

• Aseem Agarwala, Aaron Hertzman, David Salesin and Steven Seitz,

Keyframe-Based Tracking for Rotoscoping and Animation, SIGGRAPH 2004, pp584-591.

Updating...

## References

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