1. CSCE 643 Computer Vision: Lucas-Kanade Registration
Jinxiang Chai

2. Appearance-based Tracking
Slide from Robert Collins

3. 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.

4. Pixel-based Registration: Optical flow
Will start by estimating motion of each pixel separately
Then will consider motion of entire image

5. Problem Definition: Optical Flow
How to estimate pixel motion from image H to image I?

6. 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

7. 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

8. Optical Flow Constraints
Let’s look at these constraints more closely
brightness constancy: Q: what’s the equation?
A: H(x,y) = I(x+u, y+v)

9. Optical Flow Constraints
Let’s look at these constraints more closely
brightness constancy: Q: what’s the equation?
A: H(x,y) = I(x+u, y+v)
H(x,y) - I(x+u,v+y) = 0

10. Optical Flow Constraints
Let’s look at these constraints more closely
brightness constancy: Q: what’s the equation?
A: H(x,y) = I(x+u, y+v)
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:

11. Optical Flow Constraints
Let’s look at these constraints more closely
brightness constancy: Q: what’s the equation?
A: H(x,y) = I(x+u, y+v)
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:

12. Optical Flow Equation
Combining these two equations

13. Optical Flow Equation
Combining these two equations

14. Optical Flow Equation
Combining these two equations

15. Optical Flow Equation
Combining these two equations

16. Optical Flow Equation Combining these two equations
In the limit as u and v go to zero, this becomes exact

17. Optical Flow Equation How many unknowns and equations per pixel?
A: u and v are unknown, 1 equation

18. Optical Flow Equation How many unknowns and equations per pixel?
Intuitively, what does this constraint mean?
A: u and v are unknown, 1 equation

19. 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
A: u and v are unknown, 1 equation

20. 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
A: u and v are unknown, 1 equation

21. Ambiguity

22. Ambiguity
Stripes moved upwards 6 pixels
Stripes moved left 5 pixels

23. Ambiguity How to address this problem? Stripes moved upwards 6 pixels
Stripes moved left 5 pixels
How to address this problem?

24. 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!

25. 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!

26. Lukas-Kanade Flow
Prob: we have more equations than unknowns

27. Lukas-Kanade Flow Prob: we have more equations than unknowns
Solution: solve least squares problem

28. 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:

29. Lukas-Kanade Flow
The summations are over all pixels in the K x K window
This technique was first proposed by Lukas & Kanade (1981)

30. 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)

31. 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?

32. 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!

33. Edge
Bad for motion estimation
- large l1, small l2

34. Low Texture Region Bad for motion estimation:
- gradients have small magnitude
- small l1, small l2

35. High Textured Region Good for motion estimation:
- gradients are different, large magnitudes
- large l1, large l2

36. 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

37. 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

38. 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.

39. 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.

40. 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?

41. Iterative Refinement Iterative Lukas-Kanade Algorithm
Estimate velocity at each pixel by solving Lucas-Kanade equations
Warp H towards I using the estimated flow field
- use image warping techniques
Repeat until convergence

42. Idea I: Iterative Refinement
Iterative Lukas-Kanade Algorithm
Estimate velocity at each pixel by solving Lucas-Kanade equations
Warp H towards I using the estimated flow field
- use image warping techniques
Repeat until convergence

43. Idea II: Reduce the Resolution!

44. Coarse-to-fine Motion Estimation
Gaussian pyramid of image H
Gaussian pyramid of image I
image I
image H
u=10 pixels
u=5 pixels
u=2.5 pixels
u=1.25 pixels
image H
image I

45. Coarse-to-fine Optical Flow Estimation
Gaussian pyramid of image H
Gaussian pyramid of image I
image I
image H
run iterative L-K
Upsample & warp
run iterative L-K .
image J
image I

46. 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.

47. Lucas Kanade Tracking
Assumption of constant flow (pure translation) for all pixels in a larger window might be unreasonable

48. Lucas Kanade Tracking
Assumption of constant flow (pure translation) for all pixels in a larger window might be unreasonable
However, we can easily generalize Lucas-Kanade approach to other 2D parametric motion models (like affine or projective)

49. Beyond Translation So far, our patch can only translate in (u,v)
What about other motion models?
rotation, affine, perspective

50. Warping Review
Figure from Szeliski book

51. Geometric Image Warping
w(x;p) describes the geometric relationship between two images: x x’
Input Image
Transformed Image

52. Geometric Image Warping
w(x;p) describes the geometric relationship between two images:
(x)
(x’)
Input Image
Transformed Image
Warping parameters

53. Warping Functions
Translation:
Affine:
Perspective:

54. Image Registration
Find the warping parameter p that minimizes the intensity difference between template image and the warped input image

55. 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:

56. 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:
The above problem can be solved by many gradient-based optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc. 

57. 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:
The above problem can be solved by many gradient-based optimization algorithms:
- Steepest descent
- Gauss-newton
- Levenberg-marquardt, etc

58. Image Registration
Mathematically, we can formulate this as an optimization problem:

59. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion

60. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion

61. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion
Image gradient

62. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion
translation
affine
Image gradient
Jacobian matrix ……

63. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion
- Intuition?

64. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion
- Intuition: a delta change of p results in how much change of pixel values at pixel w(x;p)!

65. Image Registration
Mathematically, we can formulate this as an optimization problem:
Similar to optical flow:
Taylor series expansion
- Intuition: a delta change of p results in how much change of pixel values at pixel w(x;p)!
- An optimal that minimizes color inconsistency between the images.

66. Gauss-newton Optimization
Rearrange

67. Gauss-newton Optimization
Rearrange

68. Gauss-newton Optimization
Rearrange A b

69. Gauss-newton Optimization
Rearrange A b
(ATA)-1
ATb

70. Lucas-Kanade Registration
Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))

71. Lucas-Kanade Registration
Initialize p=p0:
Iterate:
1. Warp I with w(x;p) to compute I(w(x;p))
2. Compute the error image

72. Lucas-Kanade Registration
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 with w(x;p)

73. Lucas-Kanade Registration
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 with w(x;p)
4. Evaluate the Jacobian at (x;p)

74. Lucas-Kanade Registration
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 with w(x;p)
4. Evaluate the Jacobian at (x;p)
5. Compute the using linear system solvers

75. Lucas-Kanade Registration
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 with w(x;p)
4. Evaluate the Jacobian at (x;p)
5. Compute the using linear system solvers
6. Update the parameters

76. Lucas-Kanade Algorithm
Iteration 1:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

77. Lucas-Kanade Algorithm
Iteration 2:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

78. Lucas-Kanade Algorithm
Iteration 3:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

79. Lucas-Kanade Algorithm
Iteration 4:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

80. Lucas-Kanade Algorithm
Iteration 5:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

81. Lucas-Kanade Algorithm
Iteration 6:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

82. Lucas-Kanade Algorithm
Iteration 7:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

83. Lucas-Kanade Algorithm
Iteration 8:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

84. Lucas-Kanade Algorithm
Iteration 9:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

85. Lucas-Kanade Algorithm
Final result:
H(x)
I(w(x;p))
H(x)-I(w(x;p))

86. How to Break Assumptions
Small motion
Constant optical flow in the window
Color constancy

87. Break the Color Constancy
How to deal with illumination change?

88. Break the Color Constancy
How to deal with illumination change?
Issue: corresponding pixels do not have consistent values due to illumination changes?

89. Break the Color Constancy
How to deal with illumination change?
- linear models
- can model gain and bias (H1=H0, H2= const., other zeros)

90. 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

91. 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
Mean
Eigen vectors

92. 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
Mean
Eigen vectors

93. 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

94. Image Registration
Similarly, we can formulate this as an optimization problem:

95. Image Registration
Similarly, we can formulate this as an optimization problem:
Geometric warping
Illumination variations

96. Image Registration
Similarly, we can formulate this as an optimization problem:
For iterative registration, we have

97. Image Registration
Similarly, we can formulate this as an optimization problem:
For iterative registration, we have
Taylor series expansion

98. Gauss-newton optimization

99. Gauss-newton optimization

100. Gauss-newton optimization
let
similarly

101. Gauss-newton optimization
let
similarly

102. Gauss-newton optimization
let
similarly
Jacobian matrix
Error image

103. Gauss-newton optimization
let
similarly
Jacobian matrix
Error image
Update equation:

104. Results with Illumination Changes
[Hagar and Belhumeur 98]

105. Applications: 2D Face Registration
Face registration using active appearance models

106. AAM for Image registration
Goal: automatic detection of facial features from a single image

107. 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

108. AAM: database construction
Construct a database of images (e.g., faces) with variations

109. AAM: Feature Labeling
Label all database images by identifying key facial features

110. AAM: Feature Labeling
Label all database images by identifying key facial features
So how to build a template based on labeled database images?

111. 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

112. Shape Variation

113. Shape Variation modeling
A linear shape model consists of a triangle base face mesh s0 plus a linear combination of shape vectors, s1,…,sn

114. 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

115. 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

116. 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

117. 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

118. 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.

119. 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)!

120. Model Instantiations
A new image can be generated via AAM

121. 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

122. Image Registration with AAM
Analysis by synthesis: estimate the optimal parameters by minimizing the difference between input image and synthesized image
Solve the problem with iterative linear system solvers
- for details, check “active appearance model revisited”, Iain Matthews and Simon Baker , IJCV 2004 2 -
min p
Input image
Synthesized image

123. Iterative Approach

124. 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)

125. 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

126. 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 model-based 3D tracking (e.g., head, human body and hand gesture)