CSCE 643 Computer Vision:
Lucas-Kanade Registration
Jinxiang Chai
Appearance-based Tracking
Slide from Robert Collins
Image Registration
• This requires solving image registration
problems
• Lucas-Kanade is one of the most popular
frameworks for image registration
- gradient based optimization
- iterative linear system solvers
- applicable to a variety of scenarios, including optical flow
estimation, parametric motion tracking, AAMs, etc.
Pixel-based Registration:
Optical flow
Will start by estimating motion of each pixel separately
Then will consider motion of entire image
Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?
Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?
– Solve pixel correspondence problem
• given a pixel in H, look for nearby pixels of the same
color in I
Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?
– Solve pixel correspondence problem
• given a pixel in H, look for nearby pixels of the same
color in I
Key assumptions
– color constancy: a point in H looks the same in I
• For grayscale images, this is brightness constancy
– small motion: points do not move very far
This is called the optical flow problem
Optical Flow Constraints
Let’s look at these constraints more closely
– brightness constancy: Q: what’s the equation?
Optical Flow Constraints
Let’s look at these constraints more closely
– brightness constancy: Q: what’s the equation?
H(x,y) - I(x+u,v+y) = 0
Optical Flow Constraints
Let’s look at these constraints more closely
– brightness constancy: Q: what’s the equation?
H(x,y) - I(x+u,v+y) = 0
– small motion: (u and v are less than 1 pixel)
• suppose we take the Taylor series expansion of I:
Optical Flow Constraints
Let’s look at these constraints more closely
– brightness constancy: Q: what’s the equation?
H(x,y) - I(x+u,v+y) = 0
– small motion: (u and v are less than 1 pixel)
• suppose we take the Taylor series expansion of I:
Optical Flow Equation
Combining these two equations
Optical Flow Equation
Combining these two equations
Optical Flow Equation
Combining these two equations
Optical Flow Equation
Combining these two equations
Optical Flow Equation
Combining these two equations
In the limit as u and v go to zero, this becomes
exact
Optical Flow Equation
How many unknowns and equations per pixel?
Optical Flow Equation
How many unknowns and equations per pixel?
Intuitively, what does this constraint mean?
Optical Flow Equation
How many unknowns and equations per pixel?
Intuitively, what does this constraint mean?
– The component of the flow in the gradient direction is
determined
– The component of the flow parallel to an edge is
unknown
Optical Flow Equation
How many unknowns and equations per pixel?
Intuitively, what does this constraint mean?
– The component of the flow in the gradient direction is
determined
– The component of the flow parallel to an edge is
unknown
I
Ambiguity
Ambiguity
Stripes moved
upwards 6 pixels
Stripes moved
left 5 pixels
Ambiguity
Stripes moved
upwards 6 pixels
Stripes moved
left 5 pixels
How to address this problem?
Solving the Aperture Problem
How to get more equations for a pixel?
– Basic idea: impose additional constraints
• most common is to assume that the flow field is smooth locally
• one method: pretend the pixel’s neighbors have the same (u,v)
– If we use a 5x5 window, that gives us 25 equations per pixel!
RGB Version
How to get more equations for a pixel?
– Basic idea: impose additional constraints
• most common is to assume that the flow field is smooth locally
• one method: pretend the pixel’s neighbors have the same (u,v)
– If we use a 5x5 window, that gives us 25 equations per pixel!
Lukas-Kanade Flow
Prob: we have more equations than unknowns
Lukas-Kanade Flow
Prob: we have more equations than unknowns
Solution: solve least squares problem
Lukas-Kanade Flow
Prob: we have more equations than unknowns
Solution: solve least squares problem
– minimum least squares solution given by solution
(in d) of:
Lukas-Kanade Flow
– The summations are over all pixels in the K
x K window
– This technique was first proposed by Lukas
& Kanade (1981)
Lukas-Kanade Flow
When is this Solvable?
• ATA should be invertible
• ATA should not be too small due to noise
– eigenvalues l1 and l2 of ATA should not be too small
• ATA should be well-conditioned
– l1/ l2 should not be too large (l1 = larger eigenvalue)
Lukas-Kanade Flow
When is this Solvable?
• ATA should be invertible
• ATA should not be too small due to noise
– eigenvalues l1 and l2 of ATA should not be too small
• ATA should be well-conditioned
– l1/ l2 should not be too large (l1 = larger eigenvalue)
Look familiar?
Lukas-Kanade Flow
When is this Solvable?
• ATA should be invertible
• ATA should not be too small due to noise
– eigenvalues l1 and l2 of ATA should not be too small
• ATA should be well-conditioned
– l1/ l2 should not be too large (l1 = larger eigenvalue)
Look familiar? Harris Corner detection criterion!
Edge
Bad for motion estimation
- large l1, small l2
Low Texture Region
Bad for motion estimation:
- gradients have small magnitude
- small l1, small l2
High Textured Region
Good for motion estimation:
- gradients are different, large magnitudes
- large l1, large l2
Good Features to Track
This is a two image problem BUT
– Can measure sensitivity by just looking at one of the
images!
– This tells us which pixels are easy to track, which are
hard
• very useful later on when we do feature tracking...
For more detail, check
“Good feature to Track”, Shi and Tomasi, CVPR 1994
Errors in Lucas-Kanade
What are the potential causes of errors in this
procedure?
– Suppose ATA is easily invertible
– Suppose there is not much noise in the image
Errors in Lucas-Kanade
What are the potential causes of errors in this
procedure?
– Suppose ATA is easily invertible
– Suppose there is not much noise in the image
When our assumptions are violated
– Brightness constancy is not satisfied
– The motion is not small
– A point does not move like its neighbors
• Optical flow in local window is not constant.
Errors in Lucas-Kanade
What are the potential causes of errors in this
procedure?
– Suppose ATA is easily invertible
– Suppose there is not much noise in the image
When our assumptions are violated
– Brightness constancy is not satisfied
– The motion is not small
– A point does not move like its neighbors
• Optical flow in local window is not constant.
Revisiting the Small Motion
Assumption
Is this motion small enough?
– Probably not—it’s much larger than one pixel (2nd order terms
dominate)
– How might we solve this problem?
Iterative Refinement
Iterative Lukas-Kanade Algorithm
1. Estimate velocity at each pixel by solving LucasKanade equations
2. Warp H towards I using the estimated flow field
- use image warping techniques
3. Repeat until convergence
Idea I: Iterative Refinement
Iterative Lukas-Kanade Algorithm
1. Estimate velocity at each pixel by solving LucasKanade equations
2. Warp H towards I using the estimated flow field
- use image warping techniques
3. Repeat until convergence
Idea II: Reduce the Resolution!
Coarse-to-fine Motion
Estimation
u=1.25 pixels
u=2.5 pixels
u=5 pixels
image H
Gaussian pyramid of image H
u=10 pixels
image II
image
Gaussian pyramid of image I
Coarse-to-fine Optical Flow
Estimation
run iterative L-K
Upsample & warp
run iterative L-K
.
.
.
image JH
Gaussian pyramid of image H
image II
image
Gaussian pyramid of image I
Errors in Lucas-Kanade
What are the potential causes of errors in this
procedure?
– Suppose ATA is easily invertible
– Suppose there is not much noise in the image
When our assumptions are violated
– Brightness constancy is not satisfied
– The motion is not small
– A point does not move like its neighbors
• Optical flow in local window is not constant.
Lucas Kanade Tracking
• Assumption of constant flow (pure translation)
for all pixels in a larger window might be
unreasonable
Lucas Kanade Tracking
• Assumption of constant flow (pure translation)
for all pixels in a larger window might be
unreasonable
• However, we can easily generalize LucasKanade approach to other 2D parametric motion
models (like affine or projective)
Beyond Translation
So far, our patch can only translate in (u,v)
What about other motion models?
– rotation, affine, perspective
Warping Review
Figure from Szeliski book
Geometric Image Warping
w(x;p) describes the geometric relationship
between two images:
x
Input Image
x’
Transformed Image
 w x ( x; p ) 

x '  w ( x; p )  
 w ( x; p ) 
 y

Geometric Image Warping
w(x;p) describes the geometric relationship
between two images:
(x)
(x’)
Input Image
Transformed Image
 w x ( x; p ) 

x '  w ( x; p )  
 w ( x; p ) 
 y

Warping parameters
Warping Functions
Translation:
Affine:
Perspective:
 x  p1 

w ( x ; p )  
 y  p2 
 p1 x  p 2 y  p 3 

w ( x ; p )  
 p4 x  p5 y  p6 


w ( x; p )  



p1 x  p 2 y  p 3 

p7 x  p8 y  1 
p4 x  p5 y  p6 

p7 x  p8 y  1 
Image Registration
Find the warping parameter p that minimizes the intensity
difference between template image and the warped
input image
Image Registration
Find the warping parameter p that minimizes the intensity
difference between template image and the warped
input image
Again, we can formulate this as an optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Image Registration
Find the warping parameter p that minimizes the intensity
difference between template image and the warped
input image
Again, we can formulate this as an optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
The above problem can be solved by many gradient-based
optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc.
Image Registration
Find the warping parameter p that minimizes the intensity
difference between template image and the warped
input image
Again, we can formulate this as an optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
The above problem can be solved by many gradient-based
optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x
Taylor series expansion
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x
Taylor series expansion


W
p  H ( x)
 I ( w ( x ; p ))   I
p


x
2
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p


x
 I
I  
 x
Taylor series expansion
I 

y 
Image gradient
2
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p


x
 I
I  
 x
Taylor series expansion
I 

y 
Image gradient
 w x
 w   p1
 
p
 w y
 p
 1
...
...
w x 
p n 

w y 
 p 2 
Jacobian matrix
2
w
1

p
0
0

1
x

p
0
y
1
0
0
0
0
x
y
w
……
translation
0

1
affine
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x
- Intuition?
Taylor series expansion


W
I
(
w
(
x
;
p
))


I

p

H
(
x
)


p


2
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x
Taylor series expansion


W
I
(
w
(
x
;
p
))


I

p

H
(
x
)


p


2
- Intuition: a delta change of p results in how much change
I
x
of pixel values at pixel w(x;p)!  p
Image Registration
Mathematically, we can formulate this as an
optimization problem:
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Similar to optical flow:
arg min
p
 arg min
p
2


I
(
w
(
x
;
p


p
))

H
(
x
)

x

x
Taylor series expansion


W
I
(
w
(
x
;
p
))


I

p

H
(
x
)


p


2
- Intuition: a delta change of p results in how much change
I
x
of pixel values at pixel w(x;p)!  p
- An optimal  p that minimizes color inconsistency
between the images.
Gauss-newton Optimization
arg min
p

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p


2
Rearrange
Gauss-newton Optimization
arg min
p
 arg min
p

x

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p

2


W

I

p

(
H
(
x
)

I
(
w
(
x
;
p
)))



p


2
Rearrange
Gauss-newton Optimization
arg min
p
 arg min
p

x

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p

2


W

I

p

(
H
(
x
)

I
(
w
(
x
;
p
)))



p


A
b
2
Rearrange
Gauss-newton Optimization
arg min
p
 arg min
p

x

x


W
p  H ( x)
 I ( w ( x ; p ))   I
p

2
T
x
Rearrange


W

I

p

(
H
(
x
)

I
(
w
(
x
;
p
)))



p


A
p  (
2

W  
 W  1

I

I

 
 ) (

p

p
x

 

(ATA)-1
b
T

W 

I

 ( H ( x )  I ( w ( x ; p )))

p


ATb
Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image
T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image
3. Warp the gradient  I with w(x;p)
T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image
3. Warp the gradient  I with w(x;p)
w
4. Evaluate the Jacobian  p at (x;p)
T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image
3. Warp the gradient  I with w(x;p)
w
4. Evaluate the Jacobian  p at (x;p)
5. Compute the  p using linear system solvers
Lucas-Kanade Registration
T
p  (
x

W  
 W  1

I

I

 
) 

p

p
x

 

T

W 

I

 ( H ( x )  I ( w ( x ; p ))

p


Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image
3. Warp the gradient  I with w(x;p)
w
4. Evaluate the Jacobian  p at (x;p)
5. Compute the  p using linear system solvers
6. Update the parameters p  p   p
Lucas-Kanade Algorithm
Iteration 1:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 2:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 3:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 4:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 5:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 6:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 7:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 8:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Iteration 9:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
Lucas-Kanade Algorithm
Final result:
H(x)
I(w(x;p))
H(x)-I(w(x;p))
How to Break Assumptions
• Small motion
• Constant optical flow in the window
• Color constancy
Break the Color Constancy
How to deal with illumination change?
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Break the Color Constancy
How to deal with illumination change?
arg min
p
2


I
(
w
(
x
;
p
))

H
(
x
)

x
Issue: corresponding pixels do not have consistent values
due to illumination changes?
Break the Color Constancy
How to deal with illumination change?
- linear models
B
H ( x)  H 0 ( x) 
lH
i
i
( x)
i 1
- can model gain and bias (H1=H0, H2= const., other
zeros)
H ( x)  H ( x)  c
0
Linear Model
Can also model the appearance of a face under
different illumination using a linear combination of
base images (PCA):
- recording images under different illumination
- applying PCA to recorded images to model illumination in recorded
images
- Images under unknown illuminations can be represented as a
weighted combination of precomputed illumination image templates
B
H ( x)  H 0 ( x) 
lH
i
i 1
i
( x)
Linear Model
Can also model the appearance of a face under
different illumination using a linear combination of
base images (PCA):
- recording images under different illumination
- applying PCA to recorded images to model illumination in recorded
images
- Images under unknown illuminations can be represented as a
weighted combination of precomputed illumination image templates
B
H ( x)  H 0 ( x) 
lH
i
i
( x)
i 1
Mean
Eigen vectors
Linear Model
Can also model the appearance of a face under
different illumination using a linear combination of
base images (PCA):
- recording images under different illumination
- applying PCA to recorded images to model illumination in recorded
images
- Images under unknown illuminations can be represented as a
weighted combination of precomputed illumination image template
Unknown
weights/parameters
B
H ( x)  H 0 ( x) 
lH
i
i
( x)
i 1
Mean
Eigen vectors
Linear Model
Can also model the appearance of a face under
different illumination using a linear combination of
base images (PCA):
mean
face
lighting
variation
Image Registration
Similarly, we can formulate this as an optimization
problem:
arg min
p ,l

x

 A0 ( w ( x ; p )) 

B

i 1

l i Ai ( w ( x ; p ))  H ( x ) 

2
Image Registration
Similarly, we can formulate this as an optimization
problem:
arg min
p ,l

x

 A0 ( w ( x ; p )) 

Geometric warping
B

i 1

l i Ai ( w ( x ; p ))  H ( x ) 

2
Illumination variations
Image Registration
Similarly, we can formulate this as an optimization
problem:
arg min
p ,l

 A0 ( w ( x ; p )) 


x
B

i 1

l i Ai ( w ( x ; p ))  H ( x ) 

For iterative registration, we have
arg min
p ,l

x

 A0 ( w ( x ; p   p )) 

B

i 1

( l i   l i ) Ai ( w ( x ; p   p ))  H ( x ) 

2
2
Image Registration
Similarly, we can formulate this as an optimization
problem:
arg min
p ,l

 A0 ( w ( x ; p )) 


x

l i Ai ( w ( x ; p ))  H ( x ) 

B

i 1
2
For iterative registration, we have
arg min
p ,l
 arg min
p ,l

x

x

 A0 ( w ( x ; p   p )) 

B

i 1

W
A
(
w
(
x
;
p
))


A
p 
0
0

p


( l i   l i ) Ai ( w ( x ; p   p ))  H ( x ) 

B
 l
i 1
B
i
Ai ( w ( x ; p )) 

i 1
2
Taylor series expansion
2

l i ( Ai ( x )   Ai
 p )  H ( x )   high order
p

W
Gauss-newton optimization
arg min
p ,l

x

W
p 
 A 0 ( w ( x ; p ))   A 0

p

B
 l
i 1
B
i
Ai ( w ( x ; p )) 

i 1

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

W
2
Gauss-newton optimization

arg min
p ,l
 arg min
p ,l
x

x

W
p 
 A 0 ( w ( x ; p ))   A 0

p


W
p 
  A0
p

B
 l
i 1
i
Ai ( w ( x ; p )) 
i 1
B

B
i 1
W
B
Ai ( w ( x ; p ))  l i 
 l A
i
i 1

i
p

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

2

l i Ai ( w ( x ; p ))  H ( x ) 

2
W
B
 p  A 0 ( w ( x ; p )) 

i 1
Gauss-newton optimization

arg min
p ,l
 arg min
p ,l
let
x

x
 p
q   
l 

W
p 
 A 0 ( w ( x ; p ))   A 0

p


W
p 
  A0
p

similarly
B
 l
i
Ai ( w ( x ; p )) 
i 1
B

B
i 1
W
B
Ai ( w ( x ; p ))  l i 
i 1
 p 

 q  
 l 
 l A
i
i 1

i
p

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

2

l i Ai ( w ( x ; p ))  H ( x ) 

2
W
B
 p  A 0 ( w ( x ; p )) 

i 1
Gauss-newton optimization

arg min
p ,l
 arg min
p ,l
let
x

x
 p
q   
l 

W
p 
 A 0 ( w ( x ; p ))   A 0

p


W
p 
  A0
p

similarly
B
 l
i
Ai ( w ( x ; p )) 
i 1
B

B
i 1
W
B
Ai ( w ( x ; p ))  l i 
i 1

 l A
i
i 1
i
p

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

2

l i Ai ( w ( x ; p ))  H ( x ) 

2
B
 p  A 0 ( w ( x ; p )) 
 p 

 q  
 l 
arg min
p ,l
2


J

q

b

x
W

i 1
Gauss-newton optimization
arg min
p ,l
 arg min
p ,l
let

x

x
 p
q   
l 

W
p 
 A 0 ( w ( x ; p ))   I

p


W
p 
 I
p

 l
B
i
Ai ( w ( x ; p )) 
i 1
B

similarly
B
i 1
W
B
Ai ( w ( x ; p ))  l i 
i 1

 l A
i
i 1
i
p

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

2

l i Ai ( w ( x ; p ))  H ( x ) 

2
W
B
 p  A 0 ( w ( x ; p )) 

i 1
 p 

 q  
 l 
arg min
p ,l
2


J

q

b

x
B


W
J    A 0   l i  Ai )
, A1 ( w ( x ; p )),..., A B ( w ( x ; p ) 
p
i 1


Jacobian matrix
B
b  H ( x )  A0 ( w ( x ; p ))   l i Ai ( w ( x ; p ))
i 1
Error image
Gauss-newton optimization
arg min
p ,l
 arg min
p ,l
let

x

x
 p
q   
l 

W
p 
 A 0 ( w ( x ; p ))   I

p


W
p 
 I
p

 l
B
i
Ai ( w ( x ; p )) 
i 1
B

similarly
B
i 1
W
B
Ai ( w ( x ; p ))  l i 
i 1

 l A
i
i 1
i
p

l i ( Ai ( w ( x ; p ))   Ai
p )  H ( x)
p

2

l i Ai ( w ( x ; p ))  H ( x ) 

2
W
B
 p  A 0 ( w ( x ; p )) 

i 1
 p 

 q  
 l 
arg min
p ,l
2


J

q

b

x
B


W
J    A 0   l i  Ai )
, A1 ( w ( x ; p )),..., A B ( w ( x ; p ) 
p
i 1


B
b  H ( x )  A0 ( w ( x ; p ))   l i Ai ( w ( x ; p ))
i 1
Jacobian matrix
Update equation:
Error image
1
q  ( J J ) J b
T
T
Results with Illumination Changes
[Hagar and Belhumeur 98]
Applications: 2D Face Registration
Face registration using active appearance models
AAM for Image registration
• Goal: automatic detection of facial features
from a single image
AAM for Image registration
• Goal: automatic detection of facial features
from a single image
• Solution: register input image against a
template constructed from a prerecorded
facial image database
AAM: database construction
• Construct a database of images (e.g.,
faces) with variations
http://www2.imm.dtu.dk/~aam/datasets/datasets.html
AAM: Feature Labeling
• Label all database images by identifying key
facial features
AAM: Feature Labeling
• Label all database images by identifying key
facial features
• So how to build a template based on labeled
database images?
Key idea
• Decouple image variation into shape
variation and appearance variation
• Model each of them using PCA
• A combined model consists of a linear
shape model and a linear appearance
model
Shape Variation
Shape Variation modeling
• A linear shape model consists of a triangle base face
mesh s0 plus a linear combination of shape vectors,
s1,…,sn
Shape Variation modeling
• A linear shape model consists of a triangle base face
mesh s0 plus a linear combination of shape vectors,
s1,…,sn
A long vector stacking
positions of vertices
Shape Variation modeling
• A linear shape model consists of a triangle base face
mesh s0 plus a linear combination of shape vectors,
s1,…,sn
Mean shape
Shape Variation modeling
• A linear shape model consists of a triangle base face
mesh s0 plus a linear combination of shape vectors,
s1,…,sn
Eigen vectors
Shape Variation modeling
• A linear shape model consists of a triangle base face
mesh s0 plus a linear combination of shape vectors,
s1,…,sn
Shape parameter
Appearance Variation modeling
• A linear appearance model consists of a base
appearance image A0 defined on the pixels inside the
base mesh s0 plus a linear combination of m appearance
images Ai also defined on the same set of pixels.
Appearance Variation modeling
• A linear appearance model consists of a base
appearance image A0 defined on the pixels inside the
base mesh s0 plus a linear combination of m appearance
images Ai also defined on the same set of pixels.
Defined in base mesh
(mean shape)!
Model Instantiations
• A new image can be generated via AAM
Image Registration with AAM
• Analysis by synthesis: estimate the optimal parameters
by minimizing the difference between input image and
synthesized image
2
-
min
p
Input image
Synthesized image
Image Registration with AAM
• Analysis by synthesis: estimate the optimal parameters
by minimizing the difference between input image and
synthesized image
2
-
min
p
Input image
Synthesized image
• Solve the problem with iterative linear system solvers
- for details, check “active appearance model revisited”, Iain Matthews and Simon
Baker , IJCV 2004
Iterative Approach
Pros and Cons of AAM Registration
• It can register facial images from different peoples, different facial
expressions and different illuminations
• The quality of results heavily depends on training datasets
• Gradient-based optimization is prone to local minima
• It often fails when face is under extreme deformation, pose, or
illumination
• Needs to figure out a better way to measure the distance between
input image and template image (e.g., gradients and edges)
Lucas-Kanade for Image
Alignment
Pros:
– All pixels get used in matching
– Can get sub-pixel accuracy (important for good
mosaicing!)
– Relatively fast and simple
– Applicable to optical flow estimation, parametric
motion tracking, and AAM registration
Cons:
– Prone to local minima
– Relative small movement
Beyond 2D Tracking/Registration
So far, we focus on registration between 2D
images
The same idea can be used in registration
between 3D and 2D (model-based tracking)
We will go back to this when we talk about modelbased 3D tracking (e.g., head, human body and
hand gesture)